optimization—theory and applications || duality and upper semicontinuity of set valued functions

CHAPTER 17

Duality and Upper Semicontinuity of Set Valued Functions

17.1 Convex Functions on a Set

We shall consider here a real valued function F(u) defined and everywhere finite on a set U of R". If U is convex (Section 8.4), then F is said to be convex in U if Ulo U2 E U, ° ~ ~ ~ 1, implies F(~Ul + (1 - ~)U2) ~ ~F(Ul) + (1 - ~)F(U2). The function F is said to be extended (Section 8.5) if we take F = + 00 in Rn - U. With obvious conventions the convexity of F in R" is equivalent to the statement that U is convex and F is convex in U. As mentioned in Section 8.5, the set Q = [(ZO, u) 1+ 00 > ZO > F(u), u E U] is said to be the epigraph of F, or epi F.

17.l.i. The extended function F is convex in R" if and only if epi F is convex.

17.l.ii. The extended function F is lower semicontinuous in R" if and only if epi F is closed (cf. (8.S.v)).

17.1.iii. If U is a convex set in R" and F(u), u E U, a given real valued function, then F(u) is convex if and only if Uj E U, Aj ~ 0, j = 1, ... , v, v finite, Al + ... + Av = 1, Uo = Lj= 1 AjUj implies F(uo) ~ Lj= 1 AjF(Uj).

This is a corollary of (8.4.i). Note that F is said to be concave in U if U is convex and Ul' U2 E U, ° ~ ~ ~ 1, implies F(~Ul + (1 - ~)U2) ~ ~F(Ul) + (1 - ~)F(U2)' that is, -F is convex. From (17.1.iii) we derive that a function F(u), u E U, on a convex set U, is "affine", that is, of the form F(u) = r + Li biui, if and only if it is both convex and concave in U.

Note that if Pi ~ 0, i = 1, ... ,N, N ~ 2, are arbitrary numbers with PI + ... + PN > 0, then the relation above for convex functions can be

474 L. Cesari, Optimization—Theory and Applications© Springer-Verlag New York Inc. 1983

17.1 Convex Functions on a Set 475

written in the equivalent form

F ~ , (PIUI + ... + PNUN) PIF(UI) + ... + PNF(UN)

PI + ... + PN PI + ... + PN

which is sometimes referred to as lensen's inequality. A linear (affine) scalar function z(u) = r + bIu l + ... + bnun, u ERn, is

said to be a (nonvertical) supporting hyperplane ( or plane) of F(u), u E U, at a point U E U, provided F(u) = z(u) and F(u) ~ z(u) for all u E U.

As usual in this book, n-vectors b = (bb ... ,bn), u = (u l , ..• ,un) are thought of as column vectors, and the inner product therefore is written in any of the forms Lj bjuj = b*u = b . u.

17.l.iv. If U is a convex subset of Rn, and F(u), u E U, a given real valued convex function, then F(u) has a supporting plane at every interior point u ofU.

Proof. We know already that the set Q = [(z,u)lz ~ F(u), u E U] c Rn+1 is convex, and by (8.4.iv) there exists some supporting hyperplane to the convex set Q at the point (z, U), z = F(U), say pOz + p' u - c = ° with pO, p = (pI, ... ,pn) real, pOz + p . u - c = 0, and pOz + p . u - c ~ ° for all u E U and z ~ F(u). Let us prove that pO #- 0. Indeed, if pO = 0, then we have p . u - c = 0, p . u - c ~ ° for all u E U. If U I #- u is any point of Rn and e real, then for u(e) = WI + (1 - e)u we have u(e) ~ u as e ~ 0, and p . u(e) -c = ep . (UI - u). Since u E int U, then both u(e), u( - e) belong to U for lei sufficiently small, and yet one of the two numbers p . u(±e) - c is negative, a contradiction. We have proved that pO #- 0. Actually, we must have pO> 0, since pOz+ p' u- c~o for all z~ F(u). Finally, if we take z(u) =( - p' u+ c)/po, then z(u) = F(u) and F(u) ~ z(u) for all u E U. 0

Given a set U, we denote as usual by int U the subset of its interior points.

If U has no interior points, that is, int U = 0, statement (17.l.iv) has the following implication. First, let us denote by R the linear variety of Rn of minimum dimension r containing U. Then, U eRe Rn, ° ~ r ~ n. If U is reduced to a single point, then R = U and r = O. Otherwise, 1 ~ r ~ n, and we denote by Rint U the certainly nonempty set of points of U which are interior to U with respect to R. Thus, int U c Rint U cUe R eRn. Statement (17.l.iv) has the following corollary.

17.1.v. Under the same hypotheses as in (17.1.iv), F(u) has a supporting plane at every point u E Rint U.

17.1.vi. Under the same hypotheses as in (l7.l.iv), F(u) is continuous at every point U E

Rint U. In particular, if U = Rn, then F is continuous in Rn.

Proof· We may well assume that U is not a single point; that is, 1:0; r:O; n, and Rint U #- 0. Let u be any point u E Rint U, and let z = c + P . u be some supporting

476 Chapter 17 Duality and Upper Semicontinuity of Set Valued Functions

plane at U, so that F(u) = c + P . U. Assume, if possible, that for some (J > 0 and some sequence of points Uk e Rint U with Uk ~ U as k ~ 00, we have F(Uk) - F(u) ~ - (J for all k. Then, F(uJ ~ c + P . Uk' and hence - (J ~ F(Uk) - F(Ii) ~ P . (Uk - Ii). As k - 00,

we have - (J ~ 0, a contradiction. Assume now, if possible, that for some (J > 0 and sequence of points Uk e Rint U with Uk - Ii as k - 00, we have F(uJ - F(Ii) ~ (J for all k. Then we can choose r points Vj e U, j = 1, ... ,r, independent in Rn, such that !vi - iii = (j > 0, j = 1, ... ,r, and Ii = Lj r-lvj. Since Uk - Ii, we have Uk = Lj A.jkVj with A.jk - r- l as k - oo,j = 1, ... , r. If A.k = min[A.jk,j = 1, ... ,r] then 0 ~ A.k ~ r-1, Uk = LiA.jk - A.JVj + (A.kr)u, and hence

F(Uk) ~ L(A.jk - A.k)F(vJ + (A.kr)F(U), j

where A.jk - A.k ~ 0, A.kr - 1. For all k sufficiently large, we have then F(Uk) ~ F(Ii) + (J/2, a contradiction. This proves that F is continuous at every point of Rint U. 0

Statements (17.1.iv) and (17.1.v) cannot be made stronger so as to include points of U - Rint U. Indeed, the function F(u), -1 ~ U ~ 1, defined by F(u) = 0 for -1 < U < 1 and F( -1) = F(I) = 1 is convex but not continuous at the end points U = 0 and U = 1. The function F(u) = -(1 - U2)1/2, -1 ~ U ~ 1, is convex and continuous on [ -1,1] but has no "supporting plane" (ofthe form z = p . U + c) at the end points, U = ± 1.

17.1.vii. Under the same hypotheses as in (17.1.iv), F(u) is bounded below on every bounded part K of U.

Proof. Indeed, if K contains more than one point, then K contains some point Ii e Rint U, and if z(u) = p . U + c is a supporting plane at Ii, then F(u) ~ p . U + c for all U eKe U, and p . U + c has a finite lower bound on K. 0

17.1.viii. Under the same hypotheses as in (17.1.iv), F(u) is upper semicontinuous at every Ii e U - Rint U along any segment s issuing from Ii and contained in U.

Proof. Let s be the segment s = liuo, s c U. Assume, if possible, that there is a sequence of points Uk esc U, Uk - Ii as k - 00, with F(Uk) ~ F(U) + (J for all k for some (J > O. Then all points interior to the segment s are certainly points of Rint U, say u = (1 - ex)1i + exuo, 0 < ex < 1, and since F(u) ~ (1 - ex)F(U) + exF(uo), we see that F is bounded above on s. Since hk = Uk - Ii ~ 0 as k - 00, there is a sequence of numbers Pk > 1 with Pk - 00, Pkhk - 0 as k - 00. Hence, the points Uk = Ii + Pk(Uk - Ii), k = 1, 2, ... , are on the half straight line from Ii containing s, and Uk ~ Ii as k - 00. Thus, Uk e s, Uk e Rint U for all k sufficiently large, and the following relations hold:

Uk = P; luI. + P; l(Pk - 1)u,

F(Uk) ~ P; 1 F(u;') + P; l(Pk - I)F(Ii),

F(u;') ~ PkF(Uk) - (Pk - I)F(Ii) ~ F(Ii) + Pk(J.

Hence F(u;') ~ + 00 as k ~ 00, a contradiction since F is bounded above on s. We have proved that F is upper semicontinuous at Ii along s. 0

17.1.ix.1f U is a convex subset of R n, if F(u), u e U, is a given real valued convex function on U, and if its epigraph Q = [(ZO, u)lzO ~ F(u), u e U] = epi F c R n+ 1, is closed and convex, then the function F(u) is lower semicontinuous at every point Ii e U - Rint U, and therefore continuous on every segment s issuing from Ii and contained in U.

17.1 Convex Functions on a Set 477

Proof. Assume, if possible, that there is a number (J > 0 and points u, Uk, k = 1, 2, ... , with u E V - Rint V, Uk E V, F(uJ < F(u) - (J for all k. Take ZO = F(u), and note that all points (ZO - (J, uJ are in Q = epi F. Then, as k -+ 00, we see that (ZO - (J, u) is in the closed set Q, a contradiction, since (z, u) E Q if and only if Z ~ ZO = F(u). The last part of the statement is now a consequence of (17.l.viii) 0

17.l.x. If V is a convex subset of R", if F(u), u E V, is a given real valued function on V, and if epi F is closed and convex, then F is convex and lower semicontinuous.

Proof. Because of (17.l.i), F is convex if and only if epi F is convex. If epi F is convex, then F is continuous at every U E Rint V by (17.l.v), and if epi F is also closed, then F is lower semicontinuous at every U E V - Rint V by (17.l.ix), and thus F is lower semicontinuous in all of V. 0

A function F(u), u E V, convex on a convex set V, may not be continuous at the points of V - Rint V, even if the set Q is closed, as the following example shows. Take V = [(u,v)IO::;; u::;; 1, v ~ 0, (u _1)2 + v2 ::;; 1]; F(u,v) = v if 0::;; u::;; 1,0::;; v::;; u; F(u, v) = (2U)-I(U2 + v2) if 0 < u < 1, u ::;; v ::;; (1 - (1 - U)2)1/2. Obviously, V is convex, F is convex in (u, v), but F is not continuous at (0,0), since F(O, 0) = 0,

F(u, (1 - (1 - U)2)1/2) = 1 for all 0 < u < 1.

Given a convex set VcR" and a scalar function F(u), u E V, we say that F is convex at the point. u E V provided F(u)::;; L.I= 1 AjF(u) for any convex combination u =

LJ=1 AjUj of points Uj E V,j = 1, ... , v (Aj ~ 0, Al + ... + Av = 1, v ~ 2 any integer).

17.l.xi. If V is a convex subset of R", and F(u), u E V, a given real valued function, then F(u) is convex at an interior point u of V if and only if F(u) has a supporting plane at U.

Proof Suppose F is convex at the point u E int V. Then, the smallest convex set co Q containing Q = [(z,u)lz ~ F(u), u E V] = epi Fe R"+ 1 is the set of all points (z, u) =

L.I= 1 AiZj' u) with (Zj' u) E Q, Aj ~ 0, Al + ... + Av = 1, v finite. Now, (z, u) ¢ co Q if Z < F(u), since for every convex combination (z, u) = LJ= 1 AiZj' Uj) with u = Ii, u = Ll= 1 AjUj, we have Z = :L AjZj ~ Lj AjF(uj) ~ F(U), so Z ~ F(u). Hence, (F(u), u) is a boundary point of co Q. Then there is a hyperplane V = [(z,u)lpoz + p' u - c = 0] C

R"+ 1 such that PoF(u) + P . u - c = 0 and PoZ + P . u - c ~ 0 for all (z, u) E co Q. For every convex combination Ii = Lj AjUj and numbers Zj ~ F(uj), we have (Zj' u) E

co Q, and POZj + P . Uj - C ~ O. Therefore, Po[Lj AjZj] + P . u - c ~ 0, PoF(u) + P . u -c = 0, and Po[LAjZj - F(u)] ~ O. Since the expression in brackets is nonnegative because ofthe convexity of Fat u, we conclude that Po ~ O. But Po = 0 implies P . u - c ~ 0 for all u E V with P . Ii - c = 0; hence P . (u - u) ~ 0 for all u E V, which is impossible. Thus, Po > 0, and the hyperplane V can be written in the form Z = b . u + r, with b =

- p/Po, r = c/Po, and Z ~ b . u + r for all (z, u) E co Q, F(u) = b . u + r. Thus, z(u) =

b . u + r is a supporting plane for F(u) at u = Ii. Conversely, if F(u) has a supporting plane z(u) = b . u + r at Ii E V, then for every convex combination Ii = Lj AjUj of points Uj E V we have Lj AjF(u) ~ Lj AjZ(U) = L [b' Uj + r ]Aj = b . Ii + r = F(u), and F is convex at U. Statement (17.l.xi) is thereby proved. 0

Remark. The sufficiency part of (17.l.xi) can be stated in a stronger form as follows: If F(u)::;; L.I = 1 A~(U) for any convex combination u = LJ= 1 AjUj of points Uj E V, j = 1, ... , v, Aj ~ 0, Al + ... + Av = 1, and all possible v, 2 ::;; v ::;; n + 2, then F(u) has a supporting plane at Ii. Indeed, in the proof above, and by force of Caratheodory's


theorem (S.4.iii) in Rn+ 1, we can restrict ourselves to the convex combinations with v~n+2.

17.2 The Function T(x; z)

For the sake of simplicity we denote by x the vector variable which in other sections we denote by (t,x). Let A be a given subset of the x-space RV, for every x E A let Q(x) be a given subset ofthe z-space Rn, and let M denote the set M = [(x,z)lx E A, ZE Q(x)] C Rv +n. For every x eA let Q(x) denote a subset of Rn + 1 whose projection on the z-space is Q(x), and assume that (a) for every (ZO, z) E Q(x) and Zlo ~ ZO we also have (ZlO, z) E Q(x). Let M denote the set if = [(x,zO,z)l(zO,z) e Q(x), x E A] C Rv +n+1•

For every x E A and z ERn, let T(x, z) denote

(17.2.1) T(x, z) = inf[ ZO I (ZO, z) E Q(x)], - 00 :::; T(x, z) :::; + 00

Then T(x, z) = + 00 for every x E A, z ERn - Q(x); and - 00 :::; T(x, z) < + 00

for every x E A, z e Q(x). Thus, T(x, z) is defined in M, and T(x, z) < + 00

everywhere in M. Note that the graph of Q(x) is M and the graph of Q(x) is if. Obviously if c epi T. If all sets Q(x) are closed and bounded below, then for every x E A, T(x, z) is finite, T(x, z) e Q(x), min holds instead of inf in (17.2.1), and if = epi T. We shall also consider the extended function T defined in the whole of W+ n by taking T(x, z) = + 00 everywhere in Rv+n - M.

Note that the convexity of Q(x) implies the convexity of Q(x), but Q(x) may not be closed even if Q is closed. For instance, Q = [(ZO, z) I ZO ~ (tan Z)2, -n/2 < z < n/2] c R2 is convex and closed, while Q = [z l-n/2 < z < n/2] c R 1 is convex but not closed.

As before, we denote by R = R(x) a linear variety in Rn of minimum dimension r containing Q(x); thus, Q(x) eRe R", 0:::; r:::; n. As usual we denote by int Q(x) the set of all z E Rn which are interior to Q(x) with respect to Rn, and by Rint Q(x) the set of all points z which are interior to Q(x) with respect to R; thus

int Q(x) c Rint Q(x) c Q(x) eRe Rn.

The results of Sections 8.5B and 8.5e apply here, in particular the final statement of Remark 4 of Section 8.5B: If we take for Q(x) to be the empty sets for x E W - A, then the set epi T is closed if and only if T(x, u) is lower semicontinuous in Rv+n, and ifand only if the sets Q(x), x E RV have property (K) in RV.

Having in view the properties of the sets A and M, the following more detailed statement holds.

17.2.i. (a) If the sets Q(x) have property (K) in A, then the real valued function T(x, z), (x, z) E M, is lower semicontinuous in M. The converse is also true if M is closed. (b) If the extended function T is lower semicontinuous in RV x Rn, and the sets Q(x) are closed, then the sets Q(x) have property (K) in A. The

17.2 The Function T(x;z) 479

converse is also true if A is closed. (e) Finally, if A and M are closed, then the extended function T is lower semicontinuous in Rdn if and only if the set epi T is closed, and if and only if the sets Q(x) have property (K) in A.

Proof. The first part could be derived from the second one. However, we prove the two parts independently. Let us prove the first part.

Let us assume that the sets Q(x) have property (K) in A, and let us prove that T is lower semicontinuous everywhere in M. Let (x, z) be a point of M. If T(x, z) = - 00

there is nothing to prove. Let ZO = T(x, z) be finite, and let us assume, if possible, that T is not lower semicontinuous at (x, z). Then, there is a (1 > 0 and a sequence of points (Xk' zJ E M with (Xk, zJ --+ (x, z) as k --+ 00, and T(Xb Zk) < ZO - (1 for all k. By property (a) we derive that (zo - (1, zJ E Q(Xk) for all k, and, given e > 0, also (zo - (1, Zk) E Q(x, 26) for all k sufficiently large, where Q(x, 26) is the union of all Q(x), x E A, with Ix - xl ~ 26 (cf. Section 8.5). Then (ZO - (1, z) E cl Q(x,26). By property (K) we derive {,zo - (1, z) E

Q(x), a contradiction, since ZO = T(x, z). An analogous argument holds if T(x, z) = + 00.

We have proved that T is lower semicontinuous in M. Conversely, assume that T is lower semicontinuous in M. Let Xo be a point of A,

and let us prove that the sets Q(x) have property (K) at Xo. Let (zg, zo) be a point of nd cl Q(xo,b). Then there is a sequence of points (z~, Zk,Xk) with zg --+ zg, Zk --+ Zo, Xk --+ Xo, Xk E A, (z~, Zk) E Q(Xk), (Xb Zk) E M, T(Xb Zk) ~ z~, and by the lower semicontinuity of T at (xo, zo) E cl M = M, we have T(xo, zo) ~ zg, or (zg, zo) E Q(xo). We have proved that the sets Q(x) have property (K) in A. Thus we have proved (a).

Let us prove the second part of(17.2.i). Here the graph ofQ is epi T, and by Remark 2 after (8.5.iii) we know that x --+ Q(x) certainly has property (K) in A if the graph of Q(x) (that is, epi T) is closed, and that the converse is also true if A is closed. On the other hand, from (17.1.ii), we know that epi T is closed if and only if the extended function T is lower semicontinuous in W+ n• This proves (b). The last part of (17.2.i) is only a corollary of (a), (b), and (17.l.ii). With Q(x) the empty set and Q(x) = R" for x E RV - A, then parts (a), (b), (c) hold for A = RV as necessary and sufficient conditions with no restrictions. 0

17.2.ii. If Q(x) is convex, then either T(x, z) = - 00 for all z E Rint Q(x); or T(x, z) is finite everywhere in Q(x) and a convex function of z in Q(x), T(x, z) is bounded below on every bounded subset of Q(x), and T(x, z) is continuous on the convex set Rint Q(x), open with respect to R. Finally, if Q(x) is convex and closed and T(x, z) > - 00 for all Z E Q(x), then T(x,z) is lower semicontinuous at every point Z E Q(x) - Rint Q(x), hence everywhere in Q(x).

Proof. If Q(x) is a single point, then r = 0, Rint Q(x) = 0, and nothing has to be proved. Assume that Q(x) is not a single point. Then, 1 ~ r ~ n, and Rint Q(x) #- 0. Let z be any point z E Rint Q(x). Assume that, at some point z I E Q(x), z I #- z, we have T(x, z I) =

- 00, and let us prove that T(x, z) = - 00. For any integer k, there are points (z~, Z dE Q(x) with z~ < -k, k = 1, 2, .... Take .l. = ZI - z, and choose b > 0 so small that Z2 = z - .l.b E Rint Q(x). Take any point (z~, Z2) E Q(x), and note that all points

(IXZ~ + (1 - IX)Z~, IXZ2 + (1 - IX)ZI),

belong to Q(x). In particular, for IX = (1 + b)-I, we have

IXZ2 + (1 - IX)ZI = IX(Z - .l.b) + (1 - IX)ZI

o ~ IX ~ 1,

= Z - (1 - IX)(Z - Zl) - lX.l.b = z + .l.(1 - IX - IXb) = Z,

T(x,z) ~ IXZ~ + (1 - IX)Z~ ~ (1 + b)-IZ~ - (1 - (1 + b)-I)k,


where the last term approaches -00 as k- 00; hence T(x,z) = -00. Since x is any point of Rint Q(x), we have proved the first part of (17.2.ii).

The remaining parts of (17.2.ii) are now a consequence of the definitions and state-ments (17.1.vi, vii, ix). 0

In the next few lines we show by examples that the cases considered in (17.2.ii) can actually occur, and in particular they can occur in the situation which interests control theory, where fo(t, x, u),f(t,x, u) = (fl, ... ,J.) are continuous functions of(t, x) in A and of a control variable u. Precisely, let A be a given subset of the tx-space Rl +n, for every (t,x) E A let V(t, x) be a given subset ofthe u-space Rm, let M denote the set of all (t, x, u) E

R2+n+m with (t,x) E A, u E V(t,x), let fo and f be defined on M, and take

Q(t, x) = [(ZOz) I ZO ~ fo(t, x, u), z = f(t, x, u), u E V(t, x)] c Rn+ 1,

Q(t, X) = [ZIZ = f(t,x, U), u E V(t, X)] eRn.

Let A be closed, M closed, fo and f continuous on M. The first of the two cases mentioned in (17.2.ii) may actually occur even in situations where the sets Q have property (Q) at x. Indeed, take m = n = 1, fo = u, f = 0, V = R. Then Q = [zlz = 0], Q = [(zo, z) I ZO E R, z = 0], and T = - 00. As another example, take n = 1, m = 2, u, v control variables, fo = u, f = sin v, V = [(u, v) E R2]. Then Q = [z 1-1 :::;; z:::;; 1], Q = [(zO,z)lzO E R, -1:::;; z:::;; 1], and T(z) = -00 for all -1:::;; z:::;; 1. In both cases, Q and Q are fixed, closed, convex sets, and certainly have property (Q). As a third example, take n = 1, m = 2, u, v control variables, fo = (l - sin2 v)u, f = sin v, V = [(u, v) E R2]. Then Q = [zl-1 :::;; z:::;; 1] and Q = [(zO,z)lzO E R if -1 < z < 1; ZO ~ 0 if z = ± 1]. Finally, T(z) = - 00 for -1 < z < 1, T(z) = 0 for z = ± 1.

The following example shows that T(x, z) may not be lower semicontinuous on Q(X) ifthe set Q(x) is not closed. As usual, we shall denote by [g(P)]h the function of P which has the value g(P) if g(P) < h, and the value h if g(P) ~ h. Now, take n = 1, m = 2, U, v control variables, fo = [(1 - sin2 v)u] -h f = sin v, V = [(u, v) E R2]. Then, Q = [zl-1 :::;; z:::;; 1], Q = [(zO,z)lzO ~ -1 if -1 < z < 1; ZO ~ Oifz = ± 1]. Finally, T(z) = -1 for -1 < z < 1, T(z) = 0 for z = ± 1, and the set Q is not closed.

The following example shows that, even if the sets Q(x) is closed and convex, the function T(x, z) may not be continuous at the points z E Q(x) - Rint Q(x). Let Q be the convex set [(e,'1)IO:::;; e :::;; 1, '1 ~ 0, (e - 1)2 + '12:::;; 1], and let T(e,'1) be defined by taking T = '1 for 0:::;; e:::;; 1, 0:::;; '1 :::;; e, T = (2e)-1(e 2 + '12) for 0 < e < 1, e:::;; '1 :::;; (1 - (1 - e)2)1/2. As we have seen in Section 17.1, T(e, '1) is convex and bounded in Q, and continuous in Q except at the point (e = 0, '1 = 0). Now let us define fo, f, V. First, let V be the union ofthe two disjoint sets

V 1 = [(u, v, w)IO:::;; u:::;; 1, -1 :::;; v:::;; u - 1, w ~ 0], V 2 = [(u,v,w)IO:::;; u:::;; 1, u:::;; v:::;; (1- (1- U)2)1/2, w ~ 0].

Let a(w) = (w + 1)-1, w ~ O. Finally, let us define the functions fo(u,v, w), ft(u,v, w), f2(u, v, w), continuous on V = V 1 U V 2, by taking fl = U, f2 = V + 1, fo = v + 1 on V1,and

a(w) + (1 - a(w»(u2 + v2) fo = a(w) + 2(1 - a(w»u

on V 2. Then, if Q denotes the corresponding set

Q = [(zO,e,'1)lzO ~ fo, e = fl' '1 = f2' (U,V, w) E V = V 1 U V 2]

17.3 Seminormality 481

and T(~,rll = inf[zOI(zO, ~,'1) E Q],

then T is exactly the convex function defined above on Q, and Q is convex and closed, but T is discontinuous.

The following example shows that, at a point Z E Q(x) - Rint Q(x) the supporting plane of Q(x) may be vertical even if Q(x) is convex and closed, Q(x) is convex and compact, and T(x, z) continuous on Q(x). Indeed, take

Q = [(u,v)lu2 + v2 :;; 1], T = -(1- u2 _ V2)112,

U = Q, II = U, j~ = v, 10 = T,

- ° I ° Q = [(z , u, v) z ~ T, (u, v) E u].

17.3 Seminormality

Let A c R', Q(x) c R", Q(x) C R"+ I be the sets introduced in the previous sections, and T(x, z), x E A, z E R", the corresponding (extended) real valued function defined by (17.2.1 ).

For every X E A and b > 0 let Q(x; b) denote the set

Q(x;b) = U Q(x), XENO(X)

where Nix) is the set of all x E A with Ix - xl :;; b. We say that condition (IJ() is satisfied at a point (x, z), x E A, z E R", provided

(IJ() if (ZO, z) E n cl co Q(x; b), then Z E Q(x).

Thus, condition (IJ() is a necessary condition for property (Q). Note that whenever Q(x) = Rn for every x E A, this condition (IJ() is trivially satisfied.

This case, Q(x) = R" for all x E A, is the usual case for classical problems of the calculus of variations, with x replaced by (t, x), T replaced by j~(t, x, x'), and fo defined in A x R".

We shall now introduce the following condition (X) at a point (x, z), x E A, z E Q(x):

X. For every e > 0 there are numbers b > 0 and r real, and a real vector P = (PI' ... , Pn), such that

(Xd T(x,z) ~ r + P' z for all z E Q(x) and all x E No(x) n A; (X 2 ) T(x, z) < r + P . z + B.

For short, we shall say that Tis seminormal at (x, z) if properties (IJ() and (X) hold at (x, z). We say that Tis seminormal at x E A if properties (IJ() and (X) hold at the points (x, z) for all z E Q(x). Semi normality in A then means that properties (IJ(), (X) hold at all (x, z), x E A, z E Q(x).

Finally, we say that property (X') holds at (x, z) provided

X'. For every e > 0 there are numbers b > 0, v> 0, r real, and a real vector P =

(PI' ... ,p") such that

(X'd T(x,z) ~ r + p' z + viz - zl for all z E Q(x), and all x E No(x) in A; (Xz) T(x, z) < r + P . Z + B.


Again, we shall say that T is normal at (x, z) if properties (0() and (X') hold at (x, z). We say that T is normal at x E A if properties (0() and (X') hold at the points (x, z) for all z E Q(x). We say that T is normal in A if properties (0() and (X') hold at all points (x, z), x E A, z E Q(x).

17.4 Criteria for Property (Q)

As above, A is a closed subset of the x-space R V , and for each x E A a subset Q(x) of R" is given. Let M denote the set M = [(x, z) Ix E A, z E Q(x)] C RV+". For every x E A let Q(x) denote a subset of R"+ 1 whose projection on the z-space R" is Q(x), and such that, if (ZO, z) E Q(x), z'o > zO, then (z'o, z) E Q(x). For every x E A let T(x, z) = inf[ ZO I (ZO, z) E Q(x)], - 00 :$ T(x, z) :$ + 00, z E R". Then T(x, z) < + 00 for x E A, z E Q(x); T(x,z) = + 00 for x E A, z E R" - Q(x).

Criterion 1. Let A be closed, T(x,z) lower semicontinuous on M, and Q(x) = [(zO,z)1 ZO ~ T(x, z)J. If there is a real valued junction ¢((), 0:$ ,< + 00, bounded below, such that ¢(W( --> + 00 as ( --> + 00, (zo, z) E Q(x) implies ZO ~ ¢(Izi), and the set Q(x) is convex, then the sets Q(x) have property (Q) at X.

This is a corolIary of (1O.5.i).

Criterion 2. Let A be closed, and for any x E A let Q(x), Q(x) be given sets in R" and R"+ 1 such that Q(x) is the projection of Q(x) on the z-space. Let Q(x) = [(ZO, z)lzO ~ T(x, z), z E Q(x)], x E A. If T satisfies properties (0() and (X) at a point x E A, then the sets Q(x) have property (Q) at X.

Proof. We assume that, for a given x E A, T satisfies conditions (0() and (X) at every (x, z), z E Q(x), and we prove that the sets Q(x) satisfy condition (Q) at x (and hence Q(x) is closed and convex). We have only to prove that, if¥ = (zo, z) E no clco Q(x; £5), then ¥ = (zo, z) E Q(x). From condition (0() we know already that z E Q(x).

For ¥ = (zo, z) E no cl co Q(x;£5) and any £5 > ° we certainly have ¥ E cl co Q(x; £5) and thus there are points Z = (ZO, z) E co Q(x, £5) at a distance as small as we want from ¥ = (ZO, z). Thus, there is a sequence of numbers £5k > ° and of points Zk = (z2, Zk) E

co Q(x; £5d such that £5k --> 0, Zk --> ¥ as k --> 00. In other words, for every integer k, there is a system of points Xk E N Ok(X), zt = (z2r, zk) E Q(xk), and numbers Ak ~ 0, y = 1, ... /1, such that

(17.4.1) zk E Q(xk),

where Lv ranges over y = 1, ... ,/1, and xl --> x, Zk --> ¥, z2 --> zO, Zk --> Z as k --> 00,

y = 1, ... ,fJ. By Caratheodory's theorem we may take fJ = n + 2. Given 8 > 0, by conditions (XI) and (X z) there is a neighborhood No(x) of x in A, and numbers r,

b = (b l , .•. ,b"), such that

(17.4.2)

(17.4.3)

f(x,z) = T(x,z) - r - b· z ~ ° for all x E Nix) and z E Q(x);

f(x, z) = T(x, z) - r - b . z :$ 8.

17.4 Criteria for Property (Q) 483

For k sufficiently large, so that IXk - xl :s; 0, y = 1, ... ,/J., we have now from (17.4.1), (17.4.2)

As k -> 00, we obtain Zo ~ r + b . z; hence, by (17.4.3),

Zo ~ r + b . z ~ T(x, z) - e.

Here e > 0 is arbitrary; hence Zo ~ T(x, z). This shows that ~ = (ZO, z) E Q(x). We have proved that the sets Q(x) satisfy property (Q) at x. 0

The following criteria are better expressed in terms of control theory. Here A is a subset of the x-space W, for every x E A a subset U(x) is given in the u-space Rm, u =

(ul , •.• ,um), and M denotes the set [(x, u) I x E A, u E U(x)] C Rv +rn• Let fo(x, u), f(x, u) =

(fl' ... ,f,,) be given functions on M.

Criterion 3. Let A be closed, M closed, fo(x, u), f(x, u) = (fl' ... ,f,,) continuous on M, and assume that 1 and f are of slower growth than fo as lul-> + 00 uniformly in a closed neighborhood N 6o(X) of x in A. If the set Q(x) is convex, then the sets Q(x) have property (Q) at x.

The proof is analogous to the one for (1O.5.i) and is left as an exercise for the reader. Here we say that 1 and f are of slower growth than fo as lul-> + 00 uniformly in N ~o(x) provided, given e > 0, there is N such that for all lui ~ N and x E N ~(x) we have 1 :s; efo(x, u),lf(x, u)1 < efo(x, u).

In the following Criterion 4 we shall assume U = Rrn, M = A x Rrn, and, as in Section 17.3, we shall say that the real valued function fo(x, u), u E Rrn, x E A, is is seminormal in u at a point x E A provided for every U E Rrn the following condition (X) holds: Given e > 0 there are 0> 0, and r, b = (b l , • .• ,brn) real (which may all depend on U and e), such that foC-x, it) < r + b . u + 8, and fo(x, u) ~ r + b . u for all u E Rift and x E N(x) n A. Note that if fo(x, u) is seminormal in u at a point x E A, and ro, bo = (bOl ' ••• ,born) are real numbers, then also fo(x, u) - ro - bo . u is seminormal in u at the point x.

Criterion 4. Let A be closed, U = Rift, M = A x Rm, fo(x, u) continuous on M, and f = B(x)u + C(x), where the entries of the matrices Band C are continuous on A. If fo is seminormal in u at a point x E A, and there are numbers ro, bo = (bOl , ••• ,born) real and 00 > 0, (J? 0, such that fo(x, u) ~ ro + bo . u + (Jlul for all x E N ~o(x) and u E Rift, then the sets Q(x) = [(ZO, z) I ZO ~ fo(x, u), z = f(x, u), u E Rm] c Rn+ 1 have property (Q) at x.

Proof. We know that fo(x, u) - ro - bo . u ~ (Jlul for all x E N 6o(X) and u E Rrn. By replacing fo with fo - r ° - bo . u if necessary, we see that it is not restrictive to assume fo ~ (Jlul for all x E N 6ix) and u E Rift. We have to prove thati = (ZO, z) E n~ cl co Q(x; 0) implies Z E Q(x). Let z be a given point ~ = (zO, z) E n 6 cl co Q(x; 0), and let us prove that ~ E Q(x). For every 0 > 0 we have ~ E cl co Q(x; 0), and thus, for every 0 > 0, there are points Z = (ZO, z) E co Q(x; 0) at a distance as small as we want from ¥ = (ZO, z). Thus, there is a sequence of numbers Ok > 0 and points Zk = (z2, Zk) E co Q(x; Ok) such that Ok -> 0, Zk -> ¥ as k -> 00. In other words, for every integer k, there are a system of


points x); EN J.(:X), y = 1, ... ,v, say v = n + 2, corresponding points Zk = (z~Y, zk) E Q(xk), points Uk E Rm, and numbers A.I, 0 :.::;; Ak :.::;; 1, y = 1, ... , v, such that

1 = I Ak, Zk = I Am, z~ = I AM, Zk = I A);Zk,

(17.4.4) zl: = !(X);, un = B(xDuk + C(xD,

where y = 1, ... , v, k = 1,2, ... , where Iy ranges over y = 1, ... , v, xl: E NJ.('x), and where x); --+ X, Zk --+ ~, z~ --+ zo, Zk --+ Z as k --+ 00, y = 1, ... , v.

By hypothesis !o(x, u) ~ ulul for all x E N Jo(x), If k is sufficiently large so that Dk :.::;; 'Do, and hence Ix); - xl :.::;; Dk < Do, then because' = ulul is a convex function in u, we have

(17.4.5) z~ = I AkZ~Y ~ I Ak!o(Xk, un ~ I AkUlukl ~ u II AMI· y y y y

Thus, IIy Aku);1 :.::;; u- 1 z~, where z~ --+ Zo as k --+ 00. This proves that Iy AkUk, k = 1,2, ... , is a bounded sequence of points of Rm. By a suitable extraction, there is a subsequence, say still [k], such that Uk = Iy AkUk --+ /1 E Rm as k --+ 00.

From the third relation (17.4.4) where z~ --+ zo, z~y ~ 0, 0 :.::;; A); :.::;; 1, we derive that each of the v sequences [AkZ~Y, k = 1, 2, ... ], y = 1, ... , v, is bounded. From the fifth relation (17.4.4) we then derive that

(17.4.6) AkZ~y ~ Ak!O(Xk, un ~ AkUlukl,

and hence Aklukl:.::;; u-lAI:Z~y. Thus, each of the v sequences [AM, k = 1,2, ... ], y =

1, ... , v, is bounded. If we denote by LI); the expression

Llk = Ak[B(xk)uk + C(xnJ - Ak[B(x)Uk + C(x)]

= [B(xD - B(X)]AkU); + Ak[c(xD - C(x)],

and because of the continuity of Band C, since Xk --+ X, 0 :.::;; Ak :.::;; 1, we conclude that Llk --+ 0 as k --+ 00, y = 1, ... , v.

Given e > 0, by the seminormality of j~(x, u) in U at X, and for the point z E Rm determined above, there are numbers D' > 0 and r, b = (bl> ... , bm ) real such that

!o(x,u) ~ r + b· U for all x E NJ,(x), U E Rm,

!o(x, /1) :.::;; r + b ' /1 + e.

Now we have, for k sufficiently large,

z~ = I A.kZ~Y ~ I Akfo(Xk, un ~ I Ak[r + b . uk] = r + b . Uk y

= r + b ' /1 + b . (Uk - /1) ~ fo(x, /1) + b . (Uk - U) - e,

Zk = I A);Zk = I Ak[B(Xk)Uk + C(Xk)]

= I Ak[B(x)Uk + C(x)] + ILl); = B(X)Uk + C(x) + I Lit Y y

At the limit as k --+ 00, we obtain

ZO ~ !o(x, /1) - e, z = B(x)/1 + C(x),

and because e > 0 is arbitrary, also ZO ~ fo(x,l1), z = !(x, u); hence z = (ZO, z) E Q(x). Criterion 4 is thereby proved. 0

17.4 Criteria for Property (Q) 485

Criterion 5. Let A be closed, U = Rm, M = A x Rm, fo(x, u), f(x, u) continuous on M. Let x E A, and Na(x) be a neighborhood of x in A such that: (1) for every e > 0 there is a constant J1.. > 0 such that \f(x, u)\ ::;; J1.. + efo(x, u) for all x E N a(x), and (2) there is an increasing function A(C), 0 ::;; C < + 00, with A(C) --+ + 00 as C --+ 00, such that fo(x, u) ;;::: A(\u\) for all x E Nix) and u E Rm. If Q(x) is convex, then the sets Q(x) have property (Q) at x.

Proof. The proof proceeds as for Criteri<m 4 up to relation (17.4.5), which becomes here

(17.4.7)

Let us divide the v sequences [uk, k = 1,2, ... J or indices y = 1, ... , v into two categories. The first category is the one for which [un is bounded; then by an extraction we can assume that Uk --+ uY E Rm as k --+ 00. Let us put in the second category the remaining ones, for which [un is unbounded, and then by a further extraction we can assume that \Uk\--+ 00 as k --+ 00. Let sums I~, I~ range over the two categories of indices y. We may well assume by a further extraction that the sequences [An have a limit AY as k --+ 00,0::;; AY ::;; 1, y = 1, ... , v.

For the terms ofthe second category in the last member of(17.4.7) we have A(\Uk\)--+ + 00, while the first member in (17.4.7) is bounded, and the remaining terms remain bounded. Hence, Ak --+ AY = 0 as k --+ 00 for y of the second category. Furthermore AkA(\Uk\) is nonnegative and bounded; hence Alfo(Xk, un;;::: 0 for k sufficiently large, and (17.4.7) yields z~ ;;::: I~AUo(Xk, un. At the limit as k --+ 00 we have

(17.4.8) zo;;::: I' AYjO(X,UY).

We also have for all k

Zk = I AU(Xk, un = (I' + IN) (AU(Xk, un), Y Y Y

(17.4.9)

and for the indices of the second category and k sufficiently large we have

(17.4.10)

where the terms in parentheses are nonnegative and bounded, say ::;; M for all k, and AkJ1.. --+ 0 as k --+ 00. Thus, for k sufficiently large we have Ak\f(xk, un\ ::;; 2Me. Here e > 0 is arbitrary; thus, Ak\f(xk, un\--+ 0 as k --+ 00 for every index y of the second category. Relation (17.4.9) yields now, as k --+ 00,

(17.4.11)

From (17.4.10) and (17.4.11) we conclude that (ZO, z) E co Q(x) and hence (zo, z) E Q(x), since Q(x) is convex. Property (Q) at x is thereby proved. 0

Criterion 6. Let A be closed, U = Rm, M = A x Rm,fo(x, u) continuous in M, and f(x, u) =

Bu + C(x), where B is a constant matrix with rank B = m, and the entries of the matrix C are continuous in A. Let fo be seminormai in u at a point x E A. Then the sets Q(x) have property (Q) at x.

Criterion 7. Let A be closed, U = Rm, M = A x Rm,!o(x, u) continuous in M, and f(x, u) =

B(x)u + C(x), where the entries of the matrices Band C are continuous in A. Let fo be


seminormal in u at a point x E A. Let us further assume that there are numbers ro, bo =

(bOb ... ,bo,J and 150 > 0, (J > 0 such that fo(x,u) ~ ro + bo · f(x,u) + (Jlul for all x E

N 6o(x) and all u E Rm. Then the sets Q(x) have property (Q) at x.

Criterion 8. Let A be closed, U = Rm, M = A x Rm,fo(x, u) continuous in M, and f(x, u) = B(x)u + C(x), where the entries of the matrices B and C are continuous in A. Let fo be seminormal in u at a point x E A. Let us further assume that fo(x, u) - + 00 as lui- + 00

uniformly in some compact neighborhood N6o(X) ofx in A. Then the sets Q(x) have property (Q) at x.

Criterion 9. Let A be closed, U = Rm, M = A x Rm,fo(x, u) continuous in M, and f(x, u) =

B(x)u + C(x), where the entries of the matrices Band C are continuous in A. Let fo be normal in u at a point x E A. Let us further assume that rank B(x) = m. Then the sets Q(x) have property (Q) at x.

Criteria 1,2,3,4 were proved by Cesari [8], and Criterion 5 by Rupp [1]. Criteria 6, 7, 8, 9 were proved by Kaiser [3], and we refer to this paper for their proofs and for critical examples.

17.5 A Characterization of Property (Q) for the Sets Q(t, x) in Terms of Seminormality

As usual, let A be a given closed subset of the x-space W, for every x E A let Q(x) be a given subset of the z-space R", and let Q(x) be a subset of the zOz-space R"+ 1 whose projection on the z-space is Q(x). For every (x, z) E

A x R" let T(x,z) = inf[zOI(zO,z) E Q(x)]. We have now the following characterization of property (Q):

17.5.i(CESARI [14J). If T(x,z) > -00 in Q(x), and Q(x) = [(zO,z)lzO ~ T(x, z), Z E Q(x)J, then the sets Q(x) have property (Q) at x if and only if properties (oc) and (X) hold at the point x. Proof. First we note that if the set Q(x) is closed, then Q(x) = [(ZO, z) I ZO ~ T(x, z), z E Q(x)J; in other words, T(x, z) is a minimum and not a mere inf as in its definition.

For fixed x E A and () > 0 let us consider the sets

Q(x;{)= U Q(x)cR"+l,

(17.5.1) xeN6(X)

and projection on the z-space R",

Q*(x; () = co Q(x; () = co [ U Q(X)] c R". xe N6(X)

17.5 A Characterization of Property (Q) for the Sets Q(t, x)

Both sets Q*(x; c5) and Q*(x; c5) are convex, and

Q(x) c Q*(x; c5), Q(x) c Q*(x; c5).

As before, we consider the function T* analogous to T, or

(17.5.2) T*(x, c5; z) = inf[ ZO 1 (ZO, z) E Q*(x, c5)],

487

so that T*(x, c5, z) = + CfJ whenever z E Rn - Q*(x; c5), and - CfJ ~ T*(x, c5; z) < + CfJ for Z E Q*(x; c5). Moreover, we have T*(x, c5; z) ~ T(x; z).

We have already proved in Section 17.4 (Criterion 2) that conditions (a) and (X) are enough to guarantee property (Q). Let us assume that for a given x E A, T(x, z) > - CfJ for all z E Q(x), and that the sets Q(x) have property (Q) at X. We have to prove that T satisfies conditions (a) and (X) at all Z E Q(x). We have already noticed that condition (a) is a necessary condition for property (Q). Also, we know that the set Q(x) is closed and convex. Thus, Q(x) = [(ZO, z) 1 ZO ~ T(x, z), Z E Q(x)J.

Since T(x, z) > - CfJ for all Z E Q(x) by hypothesis, we know from (17.2.ii) that T(x, z) is a lower semicontinuous convex function of z in the convex set Q(x). We have already noticed that - CfJ ~ T*(x, z; c5) ~ T(x, z) < + CfJ

for all z E Q(x) and c5 > O. Now take any point Z E Q(x), and let ZO = T(x, z} Then, as noticed, the

point (ZO, z) belongs to Q(x). Given c > 0, the point P = (zo - c, z) is not on the closed set Q(x), and hence has a minimum distance I] from this set, with 0<1] ~ c. Since T(x, z) is lower semicontinuous at Z, there is some 1]', o < 1]' ~ 1]/2, such that T(x, z) > T(x, z) - 1]/3 for all z E Q(x) with Iz - zl ~ 1]'.

Let (J be the closed ball in W+ 1 with center P = (ZO - c, z) and radius 1'1'/3. Let (J 0 denote the projection of (J on the z-space; thus, (J 0 is the closed ball in W with center z and radius 1]'/3. We shall denote also by (J 1 the closed ball in W with center z and radius 21]'/3. Now let us consider the convex sets Q*(x; c5) = co Q(x; c5) defined in (17.5.1) and their relative function T*(x,c5;z) defined in (17.5.2).

Let us prove that there is some c5 0 > 0 such that

(17.5.3) o ~ T(x, z) - T*(x, c5; z) ~ 1]/3

for all 0 < c5 ~ c5 0 and z E (J 1 (\ Q*(x; c5). Indeed, in the contrary case there would be numbers c5k > 0 and points Zk E (J 1 C W, k = 1, 2, ... , with c5k -+ 0 as k -+ CfJ and T*(x, c5b zd < T(x, z) - 1]/3, and hence points (zg, Zk) E co Q*(x; c5k) with zg ~ T(x, z) - 1]/3 = ZO - 1]/3. Hence, for every c5 > 0 we have (zg, :k) E co Q(x, c5) for all k sufficiently large, and then also (ZO - 1]/3, zd E

co Q(x, c5). If z' is any point of accumulation of [Zk], we have z' E (J 10 (ZO -1]/3, z') E ~ co Q(x; c5), and by property (Q) also (ZO - 1]/3, z') E Q(x) = nb clco Q(x;c5). This implies T(x,z')~zo-I]/3 with Z'E(Jblz'-zl~ 21]'/3 < 1]', a contradiction. We have proved that, for some c5 0 > 0, relation (17.5.3) holds for all 0 < c5 ~ c5 0 and z E (J 1 (\ Q*(x; c5).


Let us prove that any two points

P = (ZO, z) E (J and P' = (z'O, z') E Q*(x; (jo)

have a distance {P, P'} 2:: ri' /3. Indeed, either P' is outside the cylinder [ZO E R, z E (J1], and then

{P',P} 2:: Iz' - zl2:: Iz' - zl-Iz - zl2:: 21'1'/3 - rl'/3 = r1'/3,

or P' is inside the cylinder above, and then by (17.5.3), for 0 < c5 < (jo,

z'o 2:: T*(x, (j; z') > T(x, z) - 21]/3 = ZO - 21]/3,

{P',P} 2:: z'o - ZO = [ZO - (ZO - 8)] + [z'O - ZO] + [ZO - 8 - ZO]

2:: 8 - 21]/3 -1]'/3 2:: 1]/3 -1]'/3 2:: 1]'/3.

Thus, the convex sets (J and Q*(x, (j) have a distance 2:: 1]'/3, and the same occurs for the closed convex sets (J and cl Q*(x, (j), (J compact. We conclude that there is some hyperplane 'It in Rn + 1 separating the two convex sets (J

and cl Q*(x, (j). This hyperplane 'It must intersect the vertical segment [ZO - 8 + 1]/3 ::;;

ZO ::;; zO, z = z] at some point (ZO = r, z = z), and 'It cannot be parallel to the ZO -axis, otherwise all points of the straight line z = z would be on 'It; in particular the center P of the ball (J is on 'It, and not all of (J can be separated from Q*(x; (j). Thus, 'It is of the form

'It:Z = r + b· (z - z) = (r - b . z) + b . z;

Q(x) as well as cl Q*(x; (j) is above 'It, and thus (ZO, z) E cl Q*(x; (j) implies ZO 2:: (r - b· z) + b· z. In other words, for 0 < (j::;; (jo, X E No(x), x E A, we have T(x, z) 2:: (r - b . z) + b . z. On the other hand, T(x, z) = ZO = (ZO - 8) + 8 < r + 8 = (r - b· z) + b . z + 8. We have proved that property (X) holds at the point x EA. Statement (17.5.i) is thereby proved. 0

17.6 Duality and Another Characterization of Property (Q) in Terms of Duality

A. The Dual Operation

We consider here extended real valued functions Tu, u ERn, that is, we allow T to have the values + 00 and - 00. In applying the usual definition of convexity, T((1 - OC)U1 + ocuz) ::;; (1 - OC)TU1 + ocTuz for all 0::;; oc ::;; 1, U b Uz E W, we may encounter some difficulties, since forms such as O( ± (0) and + 00 - 00 may occur. For such functions it would be more convenient to say that T is convex provided its epigraph is a convex subset of Rn+ 1,

17.6 Duality and Another Characterization of Property (Q) in Terms of Duality 489

where epigraph is defined as usual by epi T = [(y, u)1 Tu ~ y < + 00,

y i= - 00, U E R"]. However, we shall soon limit ourselves to functions T which never take the value - 00, which may take the value + 00, though Tu is not identically equal to + 00 (that is, Tu i= - 00, Tu t + 00, u E R"). For these functions the usual definition of convexity applies with the natural conventions (r + 00 = + 00, r( + (0) = + 00 for all r ~ 0, and + 00 + 00 = + 00 ).

For instance, for n = 1, T1u = + 00 for u < 0, T1u = - 00 for u ~ 0, then epi Tl = [(y, u) 1 ° ~ u < + 00, - 00 < y < + 00] is convex and closed; for n = 1, T 2u = +00 for u ~ 0, T 2u = -00 for u > 0, and then epi T2 = [(y,u)IO < u < + 00, - 00 < y < + 00] is convex and open; for n = 1, T3U = + 00 for u < 0, T3U = ° for u ~ 0, and then epi T3 = [(y, u)IO ~ u < 00, ° ~ y < + 00] is convex and closed.

The statements we have proved in Section 17.1 concerning convex functions in a set apply here, with obvious changes. Statements (17.1.i,ii) will be most relevant here, namely,

17.6.i. If Tu, u E R", is an extended real valued function, then epi T is convex if and only if T is convex, and epi T is closed if and only if T is lower semicontinuous in R".

In the following we will need the operation of closure of a function Tu, u E R", We denote by cl T the function defined by the relation epi( cl T) = cl(epi T). Thus, for n = 1, Tu = + 00 for u ~ 0, Tu = ° for u> 0, we have (cl T)(u) = + 00 for u < ° and (cl T)(u) = ° for u ~ 0.

Given T as before, we consider all pairs r, p, r E R, pER", such that - r + p . u ~ Tu for all u. In other words, we consider all half spaces st = [zo 2: - r + p . U, U E R"J with epi T c Sri.

17.6.ii. If Tu, u E R", is an extended real valued function, Tu t + 00, Tu i= - 00 for all u, and T is convex and lower semicontinuous in R", then for every UE R",

Tu = sup[ - r + p . ul - r + p . u ~ Tu for all u E R"]

or equivalently epi T = nst. Proof. Let {S+} be the family of all half spaces S+ = [(ZO, u)lpozO + p' u + c ~ 0] containing cl(epi T), and let {st} be the family of all half spaces considered above. Thus {S+} => {st}, and hence, by (8.4.vi), cl(epi T) = nS+ c nst. On the other hand, the sole half spaces in {S+} - {st} are those of the form [(ZO, u) 1 pu + c ~ 0], that is, Po = 0, or R x sto where sto = [ulpu + c ~ 0] c R" are the half spaces in R" whose intersection is the convex set cl U, and these do not affect epi T. Thus, since epi T is closed by (17.6.i), we have epi T = cl(epi T) = nS+ = nst. This proves (17.6.ii).

o


If Tu, u ERn, is any extended real valued function in Rn, then for every y ERn, we consider all r, if any, such that - r + y . u ~ Tu for all u ERn, and we take T*y = inf r. In other words, we take

(17.6.1) T*y = sup[y· u - TuluE Rn], yE Rn.

Indeed, if r = T*y, then r ~ y. u - Tu for all u, that is, Tu ~ -r + y. u for all u, and r = T*y is the inf of all numbers r for which this holds. Note that if - r + y . u ~ Tu for all u holds for no r E R, that is, the class of such r is empty, then T*y = + 00. We say that the extended function T* is the dual of T, and that the passage from T to T* is the dual operation.

The following examples may clarify: (a) Let n = 1, Tu = 0 if -1 ~ u ~ 1, Tu = + 00 if lui> 1, and then T*y = IYI for all - 00 < y < + 00. (b) Let n = 1, Tu = u, - 00 < u < + 00, and then T*y = 0 if y = 1, T*y = + 00 if y #= 1.

17.6.iii.lf Tu, u ERn, is an extended real valued function, Tu ¢. + 00, Tu #= - 00 for all u, and Tu is convex and lower semicontinuous in R", then

(17.6.2) epi T = n [(zO,u)lzO ~ y. u - T*y]. yeRn

Moreover T*y, Y ERn, is also an extended real valued function, T*y ¢. + 00,

T*y #= - 00 for all y, and T*y is convex and lower semicontinuous in Rn.

Proof. The first part is a corollary of (17.6.ii). Let us prove that T*y ¢. + 00.

Indeed, Tu #= - 00 for all u; hence epi T = nSri for a nonempty class {Sri}. If Sri = [(zo, u) 1-r + y . u, U ERn] is one of these half spaces, then T*y ~ 1', or T*y #= + 00, and thus T*y ¢. + 00. Let us prove that T*y #= - 00 for all y. Indeed, if T*y = - 00, it means that epi T is above any half space - r + y . u, or Tu == + 00, a contradiction. Let us prove that T*y is lower semicontinuous in R". Let r = T*y, and assume r finite. This means that for any e > 0 we have Tu ~ -r - e + y. u for all u ERn, while it is not true that Tu ~ - r + e + y . u for all u. Thus, there is some u such that

(17.6.3) -r - e + y. u ~ Tu ~ -r + e + y. u. Now take J > 0 such that Dlul < e, and any y ERn, Iy - yl ~ D. Then, for r = y . u - Tu we certainly have + r - y . u = - lu, and by addition with (17.6.3), also - r + r - e + (y - y) . u ~ 0 ~ - r + r + e + (y - y) . u. Here I(y - y) . ul ~ Iy - Yllul ~ Jlul < e, and therefore Ir - rl ~ 2e. Since - r + y . u = lu, we must have

T*y ~ r = r + (r - r) ~ T*y - 2e,

and this holds for all y ERn, Iy - yl ~ D. We have proved that T is lower semicontinuous at any y with Ty finite. An analogous argument holds at any y with ly = + 00.

Let us prove that T* is convex in Rn. Let Y1' Y2 be two points of Rn, let 0 ~ a ~ 1, and y = (1 - a)Y1 + OCY2. Let r1 = T*Yl' r2 = T*Y2, and as-


sume both r1, r2 finite. Then given e > 0 we have

Tu 2': - r 1 - e + y 1 . u,

for all u, and hence

Tu 2': - rz - e + Yz . u

Tu 2': -(1 - lX)r1 - IXr2 - e + [(1 - 1X)Y1 + IXY2] . u,

again for all u. This implies that

T*y = T*[(1 - IX)Y1 + IXY2] ~ (1 - lX)r1 + IXr z = (1 - IX)T*Y1 + IXT*Y2.

This proves the convexity of T* between any two points Y1' Yz where T* is finite. If one or both of rb r z are + 00, the argument is analogous. This proves (17.6.iii). 0

By (17.6.ii) applied to T*, we have now

T*y = sup[ - r + p . y 1- r + p . Y ~ T* Y for all Y ERn].

Moreover, we can repeat the process and define

T**z = SUp[z' Y - T*yly E R"], Z ERn,

and again T**z;j; + 00, T**z #- - 00 for all z, and T**is convex and lower semicontinuous in Rn. Moreover

17.6.iv. T** = T.

Proof. For every u E R", we know that Tu = sup[y . u - T*y 1 y ERn]; thus Tu 2': y' u - T*y for all y, and given e > 0 there is some y such that Tu < y' u - T*y + e. Now T**z = sup[z' y - T*yly E R"]. Hence, T**z 2': z . Y - T*y for all y E R", in particular T**u 2': u . Y - T*y for all y, and T**u 2': u . Y - T*Y. Finally,

Tu < y' u - T*y + e ~ y' u + (-u' y + T**u) + e,

or Tu < T**u + e, where e is arbitrary. Thus Tu ~ T**u. Analogously, we have T**u ~ U' Y - T*y + e for some y; hence T**u ~ U' Y + e + ( ~ y . u + Tu), or T**u ~ Tu + e and finally T**u ~ Tu. By comparison, we have Tu = T**u for all u ERn, where r = Tu, p = T**u are finite. An analogous proof holds in the cases where one of these numbers is + 00. D

If we denote by r 0 the family of all extended real valued functions Tu, u ERn, with Tu;j; + 00, Tu #- - 00 for all u ERn, T convex and lower semicontinuous in Rn, then we see that the operation T --+ T* maps r 0 into r 0,

and because of (17.6.iv) we conclude that this operation is onto and 1-1. It is to be noted that the dual operation T --+ T* defined by (17.6.1) is the operation by means of which we pass from the Lagrangian f to the Hamiltonian M in the calculus of variations and optimal control theory (see Section 17.7). This operation can be traced back to Legendre.


Remark. We shall encounter extended integrand functions T(t,x,z), which are finite everywhere on a measurable set M of the txz-space R1+2n, which are + 00 in R 1 + 2n - M, which are measurable in t for all (x,z), and such that, for almost all T, T(T, x, z) is a lower semicontinuous function in (x, z) and convex in z. Let A o denote the projection of M on the t-axis, which we assume for the sake of simplicity to be an interval, finite or infinite. Then, for almost all T E Ao, T(T, x, z) is not identically + 00, and never = - 00.

Let T*(t, x, y) be the dual of T(t, x, z) with respect to z, or T*(t, x, y) = supz[y . z - T(t, x, z)]. It is easy to see, in the frame of(17.6.iii), that T*(t,x, y) is measurable in t for all (x, y), and that, for almost all T E Ao, T*(I, x, y) is not identically + 00, never = - 00, and T*(t, x, y) is lower semicontinuous in (x, y) and convex in y.

B. The Operations /\ and V Given a family Tiu, U ERn, of functions Ti E r 0 depending on an index i ranging on an arbitrary index set I, we define the following basic "lattice" operations Vi and /\i:

V Ti(U) = sup Ti(u), i • i

/\ Ti(U) = sup[ - r + p . U [- r + p . U ~ Ti(U) for all U E R n and i E I]. i

On the other hand, if Qi' i E I, denotes a family of closed convex sets in Rn+ 1, then we

define the analogous operations Vi and Iv V Qi = cl co U Qi'

i i

With this notation, and for functions Ti E r 0, we have

(17.6.4) I:- (epi T;) = epi (y T). We may well have (Vi Ti)(U) = +00, as it happens for 1= {1,2}, T1u = +00 foru < 1, T1u = 0 for U ~ 1, T 2u = + 00 for U > -1, T2u = 0 for U ~ -1. Analogously, we may well have (/\i Ti)u = - 00 for some u. Indeed, for Tiu = - i, U ERn, i = 1, 2, ... , we have (/\i Ti)(U) = - 00 for all U ERn. However

17.6.v. If all Ti E r 0, i E I, and (Vi T;)u;:f= + 00, then Vi Ti E r o· If (/\i Ti)U # - 00

for all u, then /\i Ti E r o.

Proof. If (Vi Ti)U;:f= + 00, then the intersection of the convex closed sets epi Ti, i E I, is nonempty and thus necessarily closed and convex. Since Tiu # - 00 for all i and u, then (Vi Ti)u has the same property, and Vi Ti E r o. Analogously, if (/\i Ti)U # - 00

for all u, then consider the intersection of all half spaces Sri = [ZO ~ - r + p . U, U E RnJ with the property that Tiu ~ - r + p . U for all U and i. This intersection is not empty and necessarily closed and convex. As before, /\i Ti E r o. 0


17.6.vi. For Ti E rowe have

(17.6.5)

Proof. Here we have

(I: T) U = sup[ - r + p . U 1- r + p . U ~ Tiu for all u and i],

(I: Tir (p) = inf[rl- r + pu ~ Tiu for all u and iJ.

On the other hand

T1p = inf[rl-r + p . u ~ Tiu for all u],

(y n }P) = s~p inf[rl-r + p' u ~ Tiu for all u]

= inf[rl-r + p' u~· Tiu for all u and iJ. This proves the first part of (17.6.vi). The proof of the second part is analogous. 0

C. The Case of an Ordered Index Set I

We consider now the case in which the index set I is ordered by an ordered relation -< possessing a "least" element w. Concerning the relation we assume the following: (1) i -<j, j -< k implies i -< k; (2) given i, j E I, i, j 'I w, there is k E I, k 'I w, such that k -< i, k-<j.

Then, instead of the operation "sup" we have used above, we may use the operation

"lim sup" in terms of the operations V and I\, or V and A. Thus, instead of, say, Vi Qi = cl co Ui Qi we shall take

A V Qi = n cl co U Qi '*Wi~A '*ro i~A

and instead of Ai Ti we shall take

V ATi=SUpsup[-r+p'ul-r+pu~Ti(U),UER",i-<A] ;'4=00 i-<A 1*(0

In this connection we note that

UER",

is a convex lower semicontinuous function of u which depends monotonically on A, that is, A' -< A implies i".,(u) ~ i\(u), u E R". Moreover, by the use of the operation cl we have defined in Subsection A above, we also have:

17.6.vii. If the index set I is ordered by an order relation as stated, then

(17.6.6) cl inf i\u = /\ i\u. A A


Proof· First, let us prove that ,(u) = inf,l T,lU is a convex function of U in Rn. Indeed, if U1, Ul ERn, and z? = inf,l j\ui, i = 1,2, then given Il > 0 there are Ai E I such that z? ~ TA,ui < z? + Il, and since TAu is monotone in A, then we also have z? ~ TAui < z? + Il for all A -< Ai' i = 1,2. Finally, by property (2) of the order relation -<, there is some A' -< Ai> i = 1, 2, A' i= w, such that

o - 0 Zi ~ TA,ui < Zi + Il, i = 1,2.

For any a, 0 ~ a ~ 1, then

(i~f T.)((l - a)u1 + aUl) ~ T;..(1 - a)u1 + au 2 )

~ (1 - cx)T,l,u 1 + CXT,l,Ul

~ (1 - cx)z? + cxz~ + Il,

or ,((1 - a)u1 + auz) ~ (1 - cx),(Uj) + a,(uz) + e, and since Il is arbitrary we have , (1 - cx)u j + CXUl) ~ (1 - cx),(ud + cx,(uz).

Now note that, for all A, we have epi TA c epi(inf,l T,l)' Hence

and since the last set is convex, we also have

By applying the operation cl defined in Subsection A, we have

This is the same as saying that

(17.6.7)

To prove (17.6.6) we have only to prove that

(17.6.8) epi inf TA C cl co U epi TA, A A

since, by taking the closure of the set at the left, we have the relation opposite to (17.6.7), and thus equality must hold in (17.6.7). Suppose, if possible, that (17.6.8) is not true. Then we could strictly separate some point in the set on the left from the closed convex set on the right by means of a nonvertical closed hyperplane. Hence, for some constant c and some point U we would have

inf TAu < C < TAu A

for all A. Taking the infimum over A on the right then leads to a contradiction. This proves (17.6.vii). 0


D. Characterization of Property (Q) in Terms of Duality

Let us consider the situation where we have a family of sets Q(x) = epi T(x, u) = [(ZO, u) I z ~ T(x, u), u ERn], indexed by x E A. If Xo E A is a given point and x E A are preordered by their distance d(x, xo) from xo-that is, we say that x -< x' if x, x' E A, d(x,xo) < d(x',xo)-then Xo is the "least" element, and properties (1) and (2) hold. With this understanding, the "lim sup", in the sense mentioned in Subsection C, for the sets Q(x) at Xo is given by

(17.6.9) A V[Q(x)1 d(x, xo) ~ 15], ~>O

or equivalently n cl co U[Q(x)ld(x,xo) ~ 15]. ~<O

Thus, property (Q) at Xo reduces to

Q(xo) = A V[Q(x)ld(x,xo) ~ 15], 6>0

or equivalently epi T(xo,u) = A V[epi T(x,u)ld(x,xo) ~ 15],

6>0

or

(17.6.10) T(xo,u) = V A[T(x,u)ld(x,xo) ~ 15], 6>0

that is, property (Q) is the "upper semicontinuity" of the same sets with respect to the

operations V and A (or the analogous property for the functions T(x, u), u ERn, in terms of the operators A and V).

17.6.viii (THEOREM; GooDMAN [1])./f T(x, U) > - 00 for all x and u, then the sets Q(x) have property (Q) at x if and only if the dual function T*(x, y) satisfies the relation

(17.6.11) T*(xo,Y) = A V[T*(x,y)ld(x,xo) ~ 15]. ~

Indeed, by taking the dual of both members of (17.6.10) and using relations (17.6.5) we have

T*(xo, u) = (V A[T(x, u), d(x, Xo) ~ 15] )* 6>0

= A (A[T(x,u), d(x,xo) ~ 15])* ~>o

= A V[T*(x, y), d(x, Xo) ~ 15]. ~>o

17.6.ix (THEOREM; GOODMAN [I])./f T(x, u) > - 00 for all x and u, then the sets Q(x) have property (Q) at x if and only if

(17.6.12) T*(xo, y) = c{!~~o sup T*(x, y) J


Indeed, by applying (17.6.6) to (17.6.11), we have

T*(xo,Y) = 1\ {V[T*(x,y)ld(x,xo) ~ b]} d

= cl inf{V[T*(x,y)ld(x,xo) ~ b]} d

= cl [lim sup T*(x, y)]. x-t-xo

It is to be noted that if the term in brackets in (17.6.12) is finite for all y, then it is necessarily closed (that is, the epigraph is closed), and in that case relation (17.6.12) just means upper semicontinuity of T*(x, y) at Xo.

17.6.x (THEOREM; GooDMAN [1]). If T(x, u) > - 00 for all x and u, then the sets Q(x) have property (Q) at Xo if and only if

T(xo, y) = [lim sup T*(xo, y)]*. X~Xo

Indeed, by taking the dual of both members of (17.6.12), we have

T(xo, y) = T**(xo, y) = (C{li~~~P T*(x, y)]r

= [lim sup T*(x, y)]* X-'Xo

with the stated conventions.

17.7 Characterization of Optimal Solutions in Terms of Duality

Let E be a linear space, let E* be its dual, so that (u, u*) is a bilinear symmetric operator, or E x E* ..... R. Let Tu be any real (extended) function defined on E, not identically equal to + 00, never equal to - 00, and convex in u. As in Section 17.6, we define the dual T* of T by taking, for every u* E E*,

(17.7.1) T*u* = sup(u,u*) - Tu). . Then T*u* is also a real extended function in E*. For every u E E let aTu = {u*} denote the collection of all elements u* E E* such that Tz;;:: Tu + (z - u, u*) for all u E E. In other words aTu is the set of all u* such that h(z) = Tu + (z - u, u*) is a supporting hyperplane to Tat u E E.

17.7.i. Given u* E E*, then T*u*, the supremum defined by (17.7.1), is actually attained if and only if there is some U E E such that

T*u* = (u, u*) - Tu, or equivalently, u* E aTu.

17.7 Characterization of Optimal Solutions in Terms of Duality 497

Proof. Note that Tu + T*u* ;::: <u, u*) for all u E E*, u E E,

and that Tu + T*u* = <u, u*)

whenever T*u* is the maximum of <u, u*) - Tu for all u E E (and also, whenever Tu is the maximum of <u, u*) - T*u* for all u* E E*). Note, in particular, that for u* = 0, T*(O) = suP. [ - Tu J = - inf Tu, and that T*(O) is attained, that is, T attains its minimum at some u E E if and only if T*(O) = - Tu and 0 E 8Tu. 0

Let us consider now the Lagrange problem of optimal control concerning the minimum of the integral I[ x, u] below with the constraints as indicated:

f,12 I[x, u] = fo(t, x(t), u(t)) dt,

I,

(17.7.2) dx/dt = f(t,x(t),u(t)),

(t, x(t)) E A, u(t) E U(t, x(t)), x(ttl = x 1 , x(tz) = xz,

x = (xt, ... ,x"), u = (ut, ... ,urn),

for which we have defined

(17.7.3) F oft, x, z) = inf[ Zo I (ZO, z) E Q(t, x)J

= inf[ ZO I ZO ;::: fort, x, u), z = f(t, x, u), u E U(t, x)],

and then (17.7.2) is transformed into the problem

(17.7.4) f,12

I[ x] = F oft, x(t), x'(t)) dt, I,

x'(t) E Q(t, x(t)),

x = (x 1 , .•• ,x"),

where Q(t, x) = f(t, x, U(t, x)), the projection of Q(t, x) on the z-space R", though we do not yet claim equivalence between problems (17.7.2) and (17.7.3). (In Chapters 9, 11 we have proved this equivalence under a number of very general assumptions.) To simplify our discussion here, we assume that A is a closed subset of the tx-space R"+ 1,

and that for almost all T of some interval A o (finite or infinite), with [t1' tzJ C A o, we have F o(T, x, z) > - 00 for all x, z. This certainly occurs if, for instance, F oft, x, z) ;::: -1/J(t) - clzl for some constant c and a locally integrable function 1/J(t) ;::: 0, a hypothesis we have often assumed in our existence theorems. We shall also assume that for every AC function x(t), t 1 :::;; t:::;; tz, tII t2 finite, [t 1,tZJ c Ao, the integral J:; Fo(t,x(t),x'(t))dt exists, finite or + 00. This certainly occurs under the hypothesis F ° ;::: -1/J(t) - clzl above, and also it occurs necessarily if F ° is convex in z and continuous in (t, x, z), as we have seen in (2.l7.i).

Note that in (17.7.4), and when searching for the minimum of the functional, the requirements (t, x(t)) E A, x'(t) E Q(t, x(t)) need not be actually stated explicitly, since whenever the same relations are not satisfied a.e. in [tII tz], then F ° = + 00 in a set of positive measure, and I[ x, u J = + 00.

Note that the requirement that for any (7, x) E A the set Q(T, x) be closed is equivalent to the hypothesis that min holds in (17.7.3) instead of inf. Analogously, the assumption that, for a given I, the sets Q(t, x) have property (K) with respect to x on A(I) is equivalent


to the asswnption that the (extended) function F 0«, x, z) is lower semicontinuous in (x, z) in R2" (cf. Section 17.2).

The Hamiltonian H and corresponding function M in Chapter 5 for problem (17.7.2) are (for Ao = 1)

H(t, x, u, A) = fo(t, x, u) + A . f(t, x, u),

M(t, x, z) = infUo(t, x, u) + A . f(t, x, u), U E U(t, x)],

and these relations, in terms of problem (17.7.4), become

H(t, x, z, A) = Fo(t,x,z) + A' z, z = (zl, ... ,z"),

M(t, x, A) = inf[ F o(t, x, z) + A . z, Z E R"]. (17.7.5)

Again, we have written z E R" instead of Z E Q(t,x), since the extended function F 0 is + 00 for Z ¢: Q(t, x), and so is M.

Now let Wo denote the class of all integrands F o(t, x, z), F 0 an extended function, measurable in t for all (x, z), such that for almost all 7, F 0(7, x, z) is lower semicontinuous in (x, z) and convex in z. Then, by (17.6.iii), F~(t, x, p) is also an integrand of the same class Wo, and since F~* = F 0, we have

(17.7.6) F~(t, x, p) = sup [p . z - F o(t, x, z)], ZER"

(17.7.7) F o(t, x, z) = sup [p . z - F~(t, x, p)]. pER"

Since F o(t, x, z) + A . z = H(t, x, z, A), we have also

or

(17.7.8)

F~(t, x, - p) = sup[ - p . z - F o(t, x, z), Z E R"]

= -inf[Fo(t,x,z)+p'z,zER"]

= -inf[H(t, x, z,p), z E R"] = -M(t,x,p),

FW,x, -p) = -M(t,x,p).

Thus, we have established the relation between the notation in Chapter 5 and that of the present chapter. In Chapter 5, as usual in the theory of optimal control, H is called the Hamiltonian; some authors would prefer to call F6(t,x,P) or M(t, x, A) the Hamiltonian. Note that whenever the sup in (17.7.6) is attained for almost all 7 by measurable functions p(t), x'(t), t1 ::; t::; t2 , then we have the relation

F~(t, x(t), p(t» = p(t) . x'(t) - F o(t, x(t), x'(t» (a.e.),

or briefly, using the notation xy for x' y, also Fo = px' - Ft or F~ + Fo = px'. As before, we consider now the analogous class W of all integrands F o(t, x, z) with

F 0 an extended function which is measurable in t for all (x, z), and such that for almost all 7 of some interval 10 c R, F 0(7, x, z) is lower semicontinuous and convex in (x, z) in R2". Then we can take duals in terms of the variable (x, z), that is,

(17.7.9) Go(t, y, w) = sup [w . x + y . z - F o(t, x, z)]. (x.z)

Then Go is in the same class W, and by duality, also

(17.7.10) F o(t, x, z) = sup [w . x + y . z - Go(t, y, w)]. (y,w)

17.7 Characterization of Optimal Solutions in Terms of Duality

By only changing notation, relations (17.7.9-10) become

Go(t,p,p') = sup [p" x + p' x' - Fo(t,x, x')]. (x,x')

Hence Go(t, p, p') + F o(t, x, x') 2': p' . x + P . x'

499

for all p, p', x, x', while equality holds if and only if max holds instead of sup in (17.7.9) or in (17.7.10).

Let [t1' tzJ be fixed numbers, and for any two points a, bERn let w = Q(a, b) be the class of all AC functions x(t) = (Xl, ... ,xn), t 1 ~ t ~ t2, with x(t d = a, x(t2) = b. Also, let J[x] denote the integral J[x] = S:; Go(t, x(t), x'(t)) dt. For any two AC functions p(t), x(t) in [t1' t2], say x E Q(a, b), p E Q(c, d), we have then

Go(t,p(t),p'(t)) + Fo(t, x(t), x'(t)) 2': x(t)· p'(t) + x'(t)· p(t),

and by integration J[p] + l[ x] 2': b . d - a . c.

Let w(a, b), w1(c,d) denote

w(a, b) = inf lex], w1(c,d) = inf J[p], fl(a,b) fl(b,d)

which we assume to be finite or + 00. Then

W1(C, d) + w(a, b) 2': b . d - a . c

or w1(c,d) 2': sup [a' (-c) + b· d - w(a,b)].

(a,b)

This is the dual operation on w(a, b). Indeed, the following theorem holds.

17.7.ii. The extended real functions in R 2n, w(a, b), and W1(C, d) are lower semicontinuous and convex in R2n, never equal to - 00, and not identically + 00, and moreover

w 1(c,d) = sup [a . (-c) + b· d - w(a,b)] = w*(-c,d), (a,b)

w(a, b) = sup [( - a) . c + b . d - W1(C, d)] = w*( - a, b). (c,d)

The following statement is now a corollary of the above considerations.

17.7.iii (THEOREM). For any given system of points tb t2, a, b, c, d, then lex] attains its minimum if and only if J[p] attains its minimum, and if and only if for some AC functions x(t), pet), t1 ~ t < t2, we have

J[p] = (b . d - a . c) - l[ x],

Go(t, p(t), p'(t)) = p'(t)x(t) + p(t)x'(t) - F oCt, x(t), x'(t)),

(p'(t), p(t)) E 8F o(t, x(t), x'(t)),

Note that if F 0 is differentiable, then 8F o(t, x(t), x'(t)) is the linear operator defined by

FoAt,x(t),x'(t))· h(t) + Fox-{t, x(t), x'(t)) . h'(t),

500 Chapter 17 Duality and Upper Semi continuity of Set Valued Functions

where Fox = (aFo/axi, i = 1, ... , n), Fox' = (aFo/ax'i, i = 1, ... , n), that is,

or

(ol[ x J)h = (<, [Fox' h + Fox' ' h'J dt, J<l

p'(t) = FoAt,x(t),x'(t», p(t) = FoAt,x(t),x'(t»,

and hence (d/dt)F ox' = Fox, that is,

t E [tb t2J (a,e.),

d dt Fox" = Fox" i = 1, .. , , n, t E [t b t zJ (a.e.).

On the other hand, if fo, f satisfy the conditions of one of the implicit function theorems, say (S.2,iii), or (S.3,vi) and subsequent remarks, then there is a measurable function u(t) such that, for A = - p(t), we have

F o(t, x(t), x'(t» = io(t, x(t), u(t», u(t) E U(t, x(t»,

X'i(t) =!;(t,x(t),u(t», tE [t 1,t2J (a,e.), i = 1",., n,

A;(t) = -p;(t) = -FoAt,x(t),x'(t»

= -[(a/oxi)Fo(t,x,X'(t))]FX(t)

= -[(o/ax;){j~(t,x'U(t» + I A./t)jj(t,x,U(t»}] j x=x«)

= -[j~At'X(t)'U(t» + ~ A)t)jjAt,X(t),U(t»]

= -Hxtft,x(t),u(tV(t».

Thus, we have also the canonical equations

dXi aH dAi

dt aJ.;' dt aH

i = 1, ... , n, t E [t l , t2J (a.e,),

17.8 Property (Q) as an Extension of Maximal Monotonicity

Let X be a real Hilbert space with inner product (x, y) and norm Ilxll = (x, X)I/2, A set valued function Q(x), x E A c X, Q(x) c X, is said to be monotone in A provided Xb X2 E A, Z1 E Q(xd, Z2 E Q(X2) implies (ZI - Z2, XI - x 2) ~ O. Here A is said to be the domain of the set valued function x -+ Q(x), Thus, for X = R, Q(x) = f(x) a single valued, real valued function, we have the usual definition of monotone nondecreasing functions.

A set valued function x -+ Q(x), X E A c X, Q(x) c X, is said to be maximal monotone provided it is monotone and for any other monotone function x -+ R(x), x E B c X, R(x) c: X, with A c: B, Q(x) c: R(x), we have A = B, Q(x) = R(x) for all x E A = B. In other words, neither the domain nor any of the sets Q(x) can be enlarged preserving monotonicity.

Here are a few examples of monotone maps with X = R: (1) The map x -+ f(x) with o ~ x ~ 1, f(x) = x is monotone but not maximal monotone, (2) The map x -+ f(x)

17.8 Property (Q) as an Extension of Maximal Monotonicity 501

with - 00 < x < + 00, f(x) = x is maximal monotone. (3) The map x -+ g(x) with - 00 < x < + 00 with g(x) = x for 0 :5; x :5; 1, g(x) = 1 for x ;::: 1, g(x) = 0 for x :5; 0 is maximal monotone. (4) The map x -+ Q(x), 0:5; x :5; 1, Q(x) = {O} for 0 < x < 1, Q(O) = [ - 00 < z :5; 0], Q(I) = [0 :5; z < + 00], is maximal monotone. (5) The map x -+ f(x), - 00 < x < + 00, with f(x) = 1 + x for x;::: 0, f(x) = -1 + x for x < 0 is monotone but not maximal monotone. (6) The map x -+ Q(x), with Q(x) = {I + x} for x> 0, Q(x) = { -1 + x} for x < 0, Q(O) = [ -1 :5; z:5; 1] is maximal monotone. (7) The map x -+ f(x), -1 < x < + 1, f(x) = x(1 - X2)-1 is maximal monotone.

We just mention here the following statement which summarizes a number of important properties of maximal monotone functions.

17.8.i. For a maximal monotone function x -+ Q(x), X E A c X, Q(x) c X, we have (a) A is convex; (b) For x E A the set Q(x) is convex and closed; (c) For Xo E int A, there are 150 > 0 and M 0 ;::: 0, which may depend on Xo such that x E A, Ilx - xoll :5; 15 0 , z E Q(x) implies Ilzll :5; M 0; (d) for Xl> X2 E A, Xl =f X2, then int Q(x l) (\ int Q(X2) = 0.

For proofs of the various parts ofthis statement we refer to Brezis [I] and Fitzpatrick, Hess, and Kato [1]. The following theorem, which we state and prove only for X = Rn

for the sake of simplicity, shows that property (Q) is essentially an extension of maximal monotonicity.

17.8.ii (SURYANARAYANA [8, 10]).Let x -+ Q(x), X E A c R", Q(x) eRn, be any maximal monotone set valued function. Then, for every Xo E Rint A, the map x -+ Q(x) has property (Q) at Xo in A. In particular, for A = Rn, the map has property (Q) in Rn.

Proof. Let 15 0 and Mo be the constants relative to property (c) of (17.8.i). Let Zo E nd clco Q(xo;b), where Q(xo;b) = U[Q(x)!xEA, Ix- xol:5; 15]. Then zoEclco Q(xo;b) for all 0 < 15 :5; 150 , and thus there are sequences 15k> Zk, k = 1,2, ... , with 15k > 0, 15k -+ 0, Zk -+ Zo as k -+ 00, Zk E co Q(xo; 15k), By Caratheodory's theorem, for each Zk there are convex combinations of points zl with zl E Q(xD, xl E A, Ixl- xol :5; 15k :5; 150 , and

where v is a fixed integer that we can take v = n + 1. Here 0:5; A~:5; 1, so each sequence [Al, k = 1,2, ... ] is bounded; hence there is a

subsequence, say still [k], such that Ai: -+ AY as k -+ 00, i' = 1, ... , v, and thus 0 :5; AY :5; 1, I;Y = 1. Each sequence [Zk, k = 1,2, ... ] also is bounded in Rn; hence we may take the subsequence in such a way that zk -+ zY E Rn as k -+ 00, i' = 1, ... , v, and finally, Zo = Iy AYZY,AY;::: O,Iy AY = 1. Now for any i' = 1, ... , v and k = 1, 2, ... , we have zl E Q(xD, and for any x E A and Z E Q(x) we also have (z); - z, xl - x) ;::: O. By the continuity of the inner product, we have, as k -+ 00, (zY - Z, Xo - x) ;::: 0 for all x E A and Z E Q(x). Since x -+ Q(x) is maximal monotone, we must have zY E Q(xo), i' = 1, ... , v. Finally, Q(xo) is convex, and hence Zo = Iy AYZY belongs to Q(xo), or Xo E A, Zo E Q(xo). This proves property (Q) at Xo in A. 0

Remark. The proof of(17.8.ii) we have given here is similar to the proofs in Section 10.5. A different proof of this theorem has been given by M. B. Suryanarayana [8, 10] for a maximal monotone set valued map x -+ Q(x), X E X, Q(x) c X, in any real Hilbert space X. Moreover, Suryanarayana has shown that property (Q) is also the extension of a large class of monotonicity type properties of set valued functions.


Bibliographical Notes

The material in Section 17.4 and in particular Criteria 1-4 for property (Q) are taken from L. Cesari [8,9]. Criterion 5 is due to R. D. Rupp [1]. The remaining criteria are taken for from P. J. Kaiser [3].

The concepts of normality and seminormality of scalar functions T(x, z), continuous in (x,z) and convex in z, were introduced by L. Tonelli in his early work in 1914 [I], and used systematically by him and later by E. 1. McShane [6-10]' We mention here that for A compact, Q(x) = R" for all x E A, and a scalar function T(x, z), x E A, z E Q(x) = R", continuous in A x R" and convex in z, the following holds: For x E A, then T(x, z) is normal in z at the point x if and only if the graph G of ZO = T(x, z), z E R", contains no straight line (see L. Tonelli [I] for T of class C1, and L. H. Turner [1] for T merely continuous). A proof of this statement is also in L. Cesari [8, p. 126] (see also L. Cesari [9, 10, 11]).

The necessary and sufficient condition (17.5.i) for property (Q) in terms of seminormality is due to L. Cesari [8, p. 134]'

The necessary and sufficient conditions for property (Q) in terms of duality in Section 17.6 are due to G. S. Goodman [1]. Because of these characterizations, results of C. Olech [8, 9], R. T. Rockafellar [6], and A. D. Ioffe [1] in terms of duality can be equivalently expressed in terms of property (Q) by the use of Goodman's results.

We have presented the results we needed on convexity in Sections 2.16-17, 8.4-5, and 17.1-3, and on duality in Section 17.6. For convex theory in general and duality in particular we must refer here to T. Bonnesen and W. Fenchel [I], 1. Ekeland and R. Temam [I], W. Fenche1 [1], C. Olech [3], C. Olech and V. Klee [1], R. T. Rockafellar [I; 1-10], and F. A. Valentine [I].

The concept of duality can be traced back to Legendre, and we have shown in Section 17.7 that duality is the same operation with which we pass from the Lagrangian to the Hamiltonian in the calculus of variations and in optimal control theory-an operation which had its origin and motivation in theoretical mechanics (see e.g. P. Appell [I]). Concerning Theorem (17.7.iii) we refer to R. T. Rockafellar [6].

In Section 17.8 we show that property (Q) is a generalization of the concept of maximal monotonicity of G. Minty and H. Brezis. The result is due to M. B. Suryanarayana [8], who proved it for maps in a real Hilbert space. The simple proof in Section 17.8 for finite dimensional spaces is similar to the one for (10.5.i). For maximal monotone maps see, e.g., H. Brezis [I]. Actually, M. B. Suryanarayana [10] has considered a large class of concepts of maximal monotone multi functions, including the one above and defined by means of analytic properties or in terms oflattice theory. All these properties imply property (Q). For lattice theory we refer to G. Birkhoff [I].

optimization—theory and applications || duality and upper semicontinuity of set valued functions

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