an unconstrained minimization method for solving low rank

An unconstrained minimization method for solvinglow rank SDP relaxations of the max cut problem

L. Grippo∗, L. Palagi∗, V Piccialli∗

∗Universita degli Studi di Roma “La Sapienza”Dipartimento di Informatica e Sistemistica “A. Ruberti”

Via Ariosto 25 - 00185 Roma - Italy

e-mail (Grippo): [email protected] (Palagi): [email protected]

e-mail (Piccialli): [email protected]

Abstract

In this paper we consider low-rank semidefinite programming (LRSDP)relaxations of the max cut problem. Using the Gramian represen-tation of a positive semidefinite matrix, the LRSDP problem istransformed into the nonconvex nonlinear programming problemof minimizing a quadratic function with quadratic equality con-straints. First, we establish some new relationships among thesetwo formulations and we give necessary and sufficient conditions ofglobal optimality. Then we propose a continuously differentiableexact merit function that exploits the special structure of the con-straints and we use this function to define an efficient and globallyconvergent algorithm for the solution of the LRSDP problem. Fi-nally, we test our code on an extended set of instances of the maxcut problem and we report comparisons with other existing codes.

Keywords: semidefinite programming - low-rank factorization - maxcut problem - nonlinear programming - exact penalty functions

2 L. Grippo, L. Palagi, V. Piccialli

1 Introduction

This paper concerns the solution of large scale Semidefinite Programming(SDP) problems arising as relaxations of the max cut problem in a graph.Given a simple undirected graph G = (V, E) weighted on the edges, themax cut problem consists in finding a partition of vertices such that thesum of the weights on the edges between the two parts of the partitionis maximum. The max cut problem is a well known NP-hard problem,and good bounds can be obtained by using convex SDP relaxations [17].The simplest SDP relaxation of the max cut problem is of the form:

min trace (QX)diag(X) = e (1)X º 0, X ∈ Sn,

where the data matrix Q is an n× n real symmetric matrix, trace (QX)denotes the trace-inner product of matrices, diag(X) is the vector ofdimension n containing all the diagonal elements of X and the n × nmatrix variable X is required to be symmetric and positive semidefinite,as indicated with the notation X º 0, X ∈ Sn.Several algorithms have been proposed in the literature for solving SDPproblems, many of them belonging to the interior point class (see for ex-ample the survey [27] and references therein). In alternative to interiorpoint methods, a recent trend has been developing algorithms based onnonlinear programming reformulations of the SDP problem. The firstidea goes back to Homer and Peinado [22] who use the change of vari-ables Xij = vT

i vj/‖vi‖‖vj‖ for the elements of X, to transform problem(1) into an unconstrained optimization problem in the new variablesvi ∈ Rn for i = 1, . . . , n. In particular, they define a parallel compu-tational scheme, in order to cope with the large dimensionality of thenew problem. Burer and Monteiro in [6] propose a variant of Homer andPeinado’s approach where they use the change of variables X = LLT

where L is a lower triangular matrix.More recently, Burer and Monteiro in [7, 8] recast a general linear SDPproblem as a low rank semidefinite programming problem (LRSDP) byapplying the change of variables X = V V T , where V is a n × r, r < n,rectangular matrix. The value of r is chosen by exploiting the resultproved in Barvinok [1] and Pataki [31], that states that there exist anoptimal solution of a linearly constrained SDP problem with rank rsatisfying r(r + 1) ≤ m, where m is the number of linear constraints.For the solution of the LRSDP problem, Burer and Monteiro propose anaugmented Lagrangian method, which requires the solution of a sequence

Method for solving LRSDP relaxations of the max cut 3

of unconstrained problems for different values of a penalty parameter andof the Lagrange multipliers estimates.

In this paper we focus on the max cut SDP relaxation (1), and on thecorresponding LRSDP problem. In fact, we also consider the reducedproblem where we replace the variable X with a rectangular matrix V ofdimension n× r. First, we study optimality conditions and we establishnecessary and sufficient conditions expressed in terms of the Lagrangemultipliers for guaranteeing that a stationary point of the Lagrangianfunction of LRSDP problem yields a global minimizer that solves theoriginal SDP problem. In particular, we show that known sufficientoptimality conditions [7] can be proven to be also necessary.

Then we define a new unconstrained differentiable exact merit func-tion for the computation of stationary points of the LRSDP problem.The augmented Lagrangian approach introduced in [7], although quiteeffective in practice, has two intrinsic drawbacks, since a sequence ofunconstrained minimizations need to be performed, and some a poste-riori assumptions on the behavior of the sequence are needed in orderto prove global convergence. The exact penalty method defined in thispaper overcomes both these drawbacks. Indeed, we need only a singleunconstrained minimization of the merit function for a fixed sufficientlysmall value of the penalty parameter, and we can prove the global conver-gence of the algorithm without imposing any assumption on the behaviorof the generated sequence. Therefore, we feel that, at least in the par-ticular case of the problem arising from the relaxation of max cut, ourapproach fills the gap left by the seminal work of Burer and Monteiro[7, 8].

The paper is organized as follows. In Section 2, we report some knownresults on the max cut problem and we state the non linear relaxationthat will be addressed in the paper. In particular, we use a Kroneckerproduct notation to reformulate the problem in a standard NLP form.In Section 3, we review the main optimality conditions for the non linearprogramming problem and we state necessary and sufficient conditionsfor global optimality. In Section 4 we recast the original equality con-strained problem as an unconstrained one, using a penalty function ap-proach that exploits the special structure of the problem. In Section 5 wedefine a globally convergent algorithm for the computation of stationarypoints of the LRSDP problem and finally in Section 6 we report extensivenumerical results on standard instances of the max cut problem.


1.1 Notation and terminology

We denote by IRn the space of real n−dimensional columns vector andby IRn×m the space of real n×m matrices. By Sn we indicate the spaceof real n× n symmetric matrices.Given two n×n square matrices Q and A we define the usual trace-innerproduct by letting

trace (QA) =n∑

i=1

n∑

j=1

qijaij ,

and we indicate by ‖A‖ the induced Frobenius norm:

‖A‖2 = trace (AT A).

If v ∈ IRn, ‖v‖ is intended as the Euclidean norm of v. Given a squarematrix A ∈ IRn×n, we denote by diag(A) the vector of dimension ncontaining all the diagonal elements of A. Given a vector a ∈ IRn, wedenote by Diag(a) the diagonal square matrix of dimension n, with theelements of a on the diagonal. Moreover, we indicate by ei the vector ofzeroes elements except for the i-th equal to one, by e the vector of allones, and we set Eii = Diag(ei), Ip = Diag(e) with e ∈ IRp.In the paper, we make use of the Kronecker product ⊗ (see, for instance,[24]). We recall that given two matrices A m × n and B p × q, theKronecker product A⊗B is the mp× nq matrix given by

A⊗B =

a11B a12B . . . a1nB...

......

...am1B an2B . . . amnB

.

The basic properties of the Kronecker product are the following identities

A⊗B ⊗ C = (A⊗B)⊗ C = A⊗ (B ⊗ C),

(A + B)⊗ (C + D) = A⊗ C + A⊗D + B ⊗ C + B ⊗D,

(A⊗B)(C ⊗D) = (AC ⊗BD),

where we assume that all the matrix operations appearing in each iden-tity can be performed. Note that here and in the sequel, in order tosimplify notation, we indicate by (AC ⊗ BD) the Kronecker product((AC)⊗ (BD)).The transpose of a Kronecker product is: (A⊗ B)T = AT ⊗ BT . Givena matrix A ∈ Sn, with spectrum σ(A) = {λ1, . . . , λn} and a matrix


B ∈ Sm, with spectrum σ(B) = {µ1, . . . , µm}, the spectrum of A⊗B isgiven by:

σ(A⊗B) = {λiµj : i = 1, . . . , n; j = 1, . . . , m}.Furthermore, in the Frobenius norm, we have that

‖A⊗B‖ = ‖A‖‖B‖.

2 SDP formulation and relaxations of maxcut

Let G(V, E) be a weighted undirected graph, with n = |V | nodes andweights wij for (i, j) ∈ E. Let A ∈ Sn be the weighted adjacency matrix

aij ={

wij (i, j) ∈ E0 otherwise

.

The max cut problem consists in finding a partition of the set of nodesV of the weighted undirected graph G so as to maximize the sum of theweights on the edges that have one end in each side of the partition.Let the vector x ∈ {−1, 1}n represent any cut in the graph, i.e. thesets {i ∈ 1, . . . , n : xi = +1} and {i ∈ 1, . . . , n : xi = −1} constitute apartition of the sets of nodes. Then the weight of the cut induced by thepartition is given by ∑

i<j

aij(1− xixj)

2.

Let L := Diag(Ae)−A denote the Laplacian matrix associated with thegraph. Then it is straightforward to check that

14xT Lx =

∑

i<j

aij(1− xixj)

2,

and hence the max cut problem can be formulated as

max14xT Lx (2)

x2i = 1, i = 1, . . . , n.

By letting Q = 14A, we can solve equivalently the problem

min xT Qx (3)x2

i = 1, i = 1, . . . , n.


Indeed, if x∗MC is an optimal solution of problem (3), and we denote by

z∗MC = x∗MCT Qx∗MC,

the value of the optimal cut is given by

mC = −z∗MC +12

∑

i<j

aij .

It is possible to reformulate the max cut problem as an SDP problem witha rank constraint (see for example [33], [16]). Indeed, let us introducethe following rank one matrix:

X = xxT ,

whose generic elements are Xij = xixj . It is easy to see that

xT Qx =n∑

i=1

n∑

j=1

qijxixj =n∑

i=1

n∑

j=1

qijXij = trace (QX).

Moreover, as a rank one matrix X º 0 can always be written in the formxxT for some x ∈ IRn, we can rewrite problem (3) as

min trace (QX)diag(X) = e (4)rank (X) = 1X º 0, X ∈ Sn,

where diag(X) = e replaces the original constraints x2i = 1 for i =

1, . . . , n.A well known SDP relaxation of max cut (see, for instance, [9], [17],[26], [30]) is obtained by removing from formulation (4) the rank oneconstraint, so that we get the SDP problem:

min trace (QX)diag(X) = eX º 0, X ∈ Sn.

(SDPMC)

We recall (see for example [23]) that a matrix X ∈ Sn is positive semidef-inite if and only if there exists a set of vectors S = {v1, . . . , vn}, vi ∈ IRn

such that X is the Gram matrix of S with respect to the Euclidean in-ner product, i.e. Xij = vT

i vj . Thus it is possible to make the change of


variables X = V V T , where V = [v1 . . . vn]T ∈ IRn×n so that the semidef-initeness constraint can be removed and Problem (SDPMC) is equivalentto the following problem:

min trace (QV V T )diag(V V T ) = e, V ∈ IRn×n. (5)

The equivalence between problem (5) and problem (SDPMC) means thatthere is one-to-one correspondence among global solutions.We observe that the relaxation (5) can be derived directly from Problem(3) by enlarging the space of variables. In fact, if we replace the variablexi with a vector vi ∈ IRn we get exactly Problem (5). However, derivingthis relaxation by SDP, gives the important additional information thatthis problem can be solved in polynomial time (see [19], [29], [30]).Problem (5) has n2 variables, but it is possible to show that the actualnumber of variables needed to obtain an optimal solution of Problem(SDPMC) is much smaller.Let X ∗SDP be the set of all the optimal solutions of Problem (SDPMC)and define the integer rmin as

rmin = minX∈X∗SDP

rank (X).

It is well known (see for example Theorem 4.5.8 in [23]) that a n × nmatrix X º 0 of rank r is congruent to a diagonal matrix consisting ofall zeros and r ones, namely

X = ( V1 V2 )(

Ir 0r×(n−r)

0(n−r)×r 0(n−r)×(n−r)

)(V T

1

V T2

)= V1V

T1 ,

where V1 is a n × r matrix. Thus, X is the Gram matrix in IRr of theset of rows of V1. Hence if the value rmin were known, the dimensionof the matrix V in Problem (5) could be reduced by using the aboveresult. Indeed, let v ∈ IRnr be the vector v = (vT

1 , . . . , vTn )T . Then, we

can consider the following quadratic problem with quadratic constraints

min qr(v) :=n∑

i=1

n∑

j=1

qijvTi vj

‖vi‖2 = 1, i = 1, . . . , n, vi ∈ IRr. (6)

For any r ≥ rmin a global solution of Problem (6) gives a global solutionof Problem (5) and hence of Problem (SDPMC). The value of rmin is notknown, but an upper bound on rmin can be easily computed. In fact, in[1] and [31] a result for SDP problems has been proved that, specializedto the max cut relaxation (SDPMC), can be stated as follows.


Proposition 1 There exists an X ∈ Sn optimal solution of (SDPMC)with rank r satisfying the inequality

r(r + 1)/2 ≤ n.

This result implies that rmin ≤ r where

r = max{k ∈ N : k(k + 1)/2 ≤ n} =⌊√

1 + 8n− 12

⌋. (7)

Thus, Problem (6) gives a global solution of Problem (SDPMC) for allr ≥ r, and r can be evaluated. This result has already been exploited in[17] and [7].To summarize, let X∗

r ∈ X ∗SDP be an optimal solution of Problem (SDPMC)with rank r ≥ rmin, and denote by z∗SDP the optimal objective functionvalue , i.e. z∗SDP = trace (QX∗

r ). Let q∗r denote the optimal objectivefunction value of Problem (6) where r ≥ rmin.Problem (6) is equivalent to Problem (SDPMC), in the sense that everyglobal solution v∗ ∈ Rnr of Problem (6) gives a global solution X∗ ofProblem (SDPMC) where X∗

ij = v∗Ti v∗j and we have z∗SDP = q∗r . We canconclude that for all r ≥ rmin, it holds

z∗SDP = q∗n = . . . = q∗r = . . . = q∗r

= . . . = q∗rmin. (8)

Moreover, when r = 1 we get a problem equivalent to Problem (4), sothat q∗1 ≡ z∗MC, and for increasing values of r, we get non increasingvalues of the optimal value (the feasible region is enlarging). Hence wecan write

z∗SDP ≡ q∗n ≡ q∗r≡ q∗rmin

≤ q∗r ≤ q∗1 ≡ z∗MC. (9)

Problem (6) is a non convex problem with quadratic objective functionand quadratic constraints that can be recast in compact NLP form usingKronecker products. In fact, letting ei ∈ IRn, we can write the vectorv ∈ IRnr as

v =n∑

i=1

(ei ⊗ vi) =

v1...

vn

so that we havevi = (ei ⊗ Ir)T v.

Therefore we can write the objective function of Problem (6) as:n∑

i=1

n∑

j=1

qijvTi vj =

n∑

i=1

n∑

j=1

qij

((ei ⊗ Ir)T v

)T(ej ⊗ Ir)T v

=n∑

i=1

n∑

j=1

vT (qijeieTj ⊗ Ir)v = vT (Q⊗ Ir)v,


while the constraints can be written as

‖vi‖2 = vT (Eii ⊗ Ir) v.

Thus, we obtain the nonlinear programming problem

min vT (Q⊗ Ir)v = qr(v)

vT (Eii ⊗ Ir) v = 1 i = 1, . . . , n,(NLPr)

that is the problem we will focus on in the rest of the paper.

3 Optimality conditions

In this section we study both local and global optimality conditions forproblem (NLPr).The Lagrangian function for Problem (NLPr) is, for an arbitrary fixedvalue r ≥ 1,

L(v, λ) = vT (Q⊗ Ir) v +n∑

i=1

λi

(vT (Eii ⊗ Ir) v − 1

)

= vT [(Q + Λ)⊗ Ir]v − λT e(10)

where λ = (λ1, . . . , λn)T and we set Λ = Diag{λ}. We recall the defini-tion of a stationary point of Problem (NLPr).

Definition 1 (Stationary point) A point v ∈ IRnr is a stationarypoint of Problem (NLPr) if there exists a Lagrange multiplier λ ∈ IRn

such that (v, λ) ∈ IRnr × IRn satisfies

vT (Eii ⊗ Ir) v = 1, i = 1, . . . , n

∇vL(v, λ) = 0,

where ∇vL(v, λ) ≡ 2[(Q + Λ)⊗ Ir]v.

In the following proposition we state the well known necessary optimalityconditions for Problem (NLPr).

Proposition 2 (First order necessary conditions) Let v ∈ IRnr bea local minimizer of Problem (NLPr). Then there exists unique λ ∈ IRn

such that (v, λ) ∈ IRnr × IRn satisfies:

[( Q + Λ )⊗Ir] v = 0vT (Eii ⊗ Ir) v = 1, i = 1, . . . , n.

(11)


Proof. As the constraints of Problem (NLPr) satisfy the linear inde-pendence constraints qualification, then v is a stationary point and theLagrange multiplier is unique. By Definition 1 of stationary point wehave (11).

The second order necessary conditions for Problem (NLPr) are given inthe next proposition.

Proposition 3 (Second order necessary conditions) Let v ∈ IRnr

be a local minimizer of Problem (NLPr). Then there exists λ ∈ IRn suchthat (v, λ) ∈ IRnr × IRn satisfies (11) and

zT[(

Q + Λ)⊗ Ir

]z ≥ 0

for every z ∈ IRnr such that vT (Eii ⊗ Ir) z = 0 for i = 1, . . . , n.

Now, we state some useful properties of the Lagrange multipliers at astationary point.

Proposition 4 Let v ∈ IRnr be a stationary point of Problem (NLPr).Then we have:

λi = −vT (EiiQ⊗ Ir) v, i = 1, . . . , n, (12)

andn∑

i=1

λi = −qr(v). (13)

Proof. Let v be a stationary pair of Problem (NLPr) so that

12∇vL(v, λ) ≡ (Q⊗ Ir) v +

n∑

j=1

λj (Ejj ⊗ Ir) v = 0. (14)

Premultiplying both sides of (14) by vT (Eii ⊗ Ir) we can write:

vT (Eii ⊗ Ir) (Q⊗ Ir) v +n∑

j=1

λj vT (Eii ⊗ Ir) (Ejj ⊗ Ir) v = 0,

whence, using the properties of Kronecker products and noting thatEiiEjj = 0n×n for i 6= j and EiiEii = Eii, we get

vT (EiiQ⊗ Ir) v + λivT (Eii ⊗ Ir) v = 0.


Therefore, as vT (Eii ⊗ Ir) v = 1, we obtain expression (12) for themultipliers. Finally, summing up all the multipliers, and noting that

n∑

i=1

Eii = In we obtain

∑ni=1 λi = −vT

∑ni=1 (EiiQ⊗ Ir) v

= −vT [(∑n

i=1 Eii) Q⊗ Ir] v = −vT (Q⊗ Ir) v,

which yields (13).

We pointed out in Section 2 that for suitable values of r any global so-lution of the non convex Problem (NLPr) gives a global solution of theconvex Problem (SDPMC). Duality theory applied to Problem (SDPMC)and the connections between Problem (SDPMC) and Problem (NLPr)give us a straightforward global necessary and sufficient optimality con-dition. Letting u ∈ IRn, consider the standard Lagrangian dual of Prob-lem (SDPMC):

max eT u

Q−Diag(u) º 0. (15)

Denote by u∗ ∈ IRn an optimal solution of Problem (15), and let

eT u∗ = z∗DUAL.

Since Slater’s constraint qualification holds both for the primal and thedual problem, it is well known that there is no duality gap. Therefore,X∗ and u∗ are optimal solutions of the primal problem (SDPMC) and ofits dual (15) respectively if and only if:

z∗SDP = trace (QX∗) = eT u∗ = z∗DUAL

diag(X∗) = eX∗ º 0

Q−Diag(u∗) º 0.

(16)

By posing ui = −yi for i = 1, . . . , n we can write problem (15) as

min eT yQ + Y º 0,

where Y = Diag(y).The next proposition gives a necessary and sufficient global optimalitycondition for Problem (NLPr) that simply derives from the optimalityconditions (16) and from the relation (8).


We remark that, from now on, for sake of simplicity, we adopt the follow-ing terminology: whenever we say that a point v∗ ∈ IRnr solves Problem(SDPMC) we mean that X∗ = V ∗V ∗T , where V ∗ = (v∗1 . . . v∗n)T , is anoptimal solution of Problem (SDPMC), namely

V ∗V ∗T ∈ X ∗SDP, and qr(v∗) = z∗SDP.

This implies, by definition, that r ≥ rmin.

Proposition 5 A point v∗ ∈ IRnr is a global minimizer of Problem(NLPr) that solves Problem (SDPMC) if and only if there exists a y∗ ∈IRn such that (v∗, y∗) satisfies

−eT y∗ = qr(v∗)Q + Y ∗ º 0v∗T (Eii ⊗ Ir) v∗ = 1, i = 1, . . . , n.

(17)

Proof The proof easily follows from the primal dual SDP optimalityconditions (16), where we set u∗ = −y∗, and from relation (8) thattogether give for all r ≥ rmin

q∗r = z∗SDP = z∗DUAL = −eT y∗ = qr(v∗).

We note that Proposition 5 gives a necessary and sufficient condition ofglobal optimality in terms of primal-dual relationships, so that checkingthis condition requires the solution of the dual problem (15), which is anSDP problem as well. However, we show that the solution u∗ of the dualproblem actually is obtained by the Lagrange multiplier λ∗ associatedwith the solution v∗ of Problem (NLPr). This yields a necessary andsufficient global optimality condition that is similar in structure to theone valid for the trust region problem [28], in that it requires the Hessianof the Lagrangian to be positive semidefinite at a stationary point.

Proposition 6 A point v∗ ∈ IRnr is a global minimizer of Problem(NLPr) that solves Problem (SDPMC) if and only if there exists a λ∗ ∈IRn such that

[(Q + Λ∗)⊗ Ir] v∗ = 0Q + Λ∗ º 0v∗T (Eii ⊗ Ir) v∗ = 1, i = 1, . . . , n.

(18)

Proof First assume that (18) are satisfied. By (13), we have qr(v∗) =−eT λ∗. The vector u∗ = −λ∗ is feasible for the dual problem (15), and


hence u∗ is optimal for the dual. Therefore, the primal dual optimalityconditions (16) and (8) together give

q∗r = z∗SDP = z∗DUAL = −eT λ∗ = qr(v∗).

As for the necessity part, the first order necessary conditions state thata unique λ∗ ∈ IRn exists such that

[(Q + Λ∗)⊗ Ir] v∗ = 0. (19)

Moreover, since by assumption v∗ ∈ IRnr solves Problem (SDPMC), weget by Proposition 5 that there exist y∗ ∈ IRn such that

−eT y∗ = qr(v∗)Q + Y ∗ º 0,

and hence:−eT y∗ = v∗T (Q⊗ Ir) v∗. (20)

Since v∗T (Eii ⊗ Ir)v∗ = 1, i = 1, . . . , n, we can write

eT y∗ =n∑

i=1

v∗T (y∗i Eii ⊗ Ir)v∗ = v∗T (Y ∗ ⊗ Ir)v∗

that summed up with (20) gives

v∗T [(Q + Y ∗)⊗ Ir] v∗ = 0.

Since Q + Y ∗ is positive semidefinite, it follows that

[(Q + Y ∗)⊗ Ir] v∗ = 0.

The unicity of the multiplier λ∗ satisfying (19) implies that λ∗ = y∗, andhence

Q + Λ∗ = Q + Y ∗ º 0,

so that (18) holds.

The advantage of the condition stated in Proposition 5 is that this con-dition can be computationally checked without solving the dual problem(15), since it requires only the knowledge of the Lagrange multiplierassociated to the point v∗.We point out once more that problem (NLPr) is a non convex optimiza-tion problem so that necessary and sufficient global optimality conditionsare usually not available. However, we were able to derive the abovecondition, thanks to the strict connection between problem (NLPr) for


r ≥ rmin and the convex problem (SDPMC), which indicates some sortof hidden convexity. Hidden convexity of other quadratic problems withquadratic constraints has been exploited to derive global optimality con-ditions for non convex problems [3, 28, 35]. We remark that, for r < rmin,conditions (18) do not apply and global optimality conditions are notavailable.A different sufficient global optimality condition has been proved in [7]by Burer and Monteiro for a more general class of SDP problems. Inparticular, for r < n they prove the following result (here specialized toProblem (SDPMC)) that gives a sufficient condition of global optimality.

Proposition 7 (Proposition 4 in [7]) Let v∗ ∈ IRnr, with r < n, bea local minimum point of Problem (NLPr). Let v ∈ IRn(r+1) be a pointwith components vi ∈ IRr+1 such that

vi =(

v∗i0

).

If v is a local minimum of Problem (NLPr+1), then v∗ is a global mini-mum point of Problem (NLPr) that solves Problem (SDPMC).

Actually, by looking at the details of the proof of the above result in [7],it emerges that the only assumption needed is that v∗ and v satisfy onlynecessary conditions. We show that this condition is also necessary, andby exploiting this result we can establish another necessary and sufficientcondition that can be computationally checked. Indeed, we can state thefollowing result.

Proposition 8 A point v∗ ∈ IRnr, with r < n, is a global minimumpoint of Problem (NLPr) that solves Problem (SDPMC) if and only if thefollowing conditions hold:

(i) v∗ is a stationary point for Problem (NLPr),

(ii) the point v ∈ IRn(r+1) with components vi ∈ IRr+1 defined as

vi =(

v∗i0

)(21)

is a stationary point for Problem (NLPr+1) satisfying the secondorder necessary optimality conditions.

Proof First of all, we prove sufficiency. Let λ∗ ∈ IRn be the uniqueLagrange multiplier associated to v∗ ∈ IRnr for Problem (NLPr) and let


λ ∈ IRn be the unique Lagrange multiplier associated to v ∈ IRn(r+1),such that (v, λ) satisfies the first order necessary optimality conditions(11) for Problem (NLPr+1). As a first step, we show that λ = λ∗. Infact, it follows from the expression of v, setting q∗ij = (Q + Λ∗)ij :

[(Q + Λ∗)⊗ Ir+1] v =

q∗11Ir 0r

0Tr q∗11

. . .q∗1nIr 0r

0Tr q∗1n

.... . .

...

q∗n1Ir 0r

0Tr q∗n1

. . .q∗nnIr 0r

0Tr q∗nn

v∗10...

v∗n0

=

n∑

j=1

q∗1jv∗j

0...

n∑

j=1

q∗njv∗j

0

.

(22)Since (v∗, λ∗) is a stationary point of Problem (NLPr), it follows from(22) that

[(Q + Λ∗)⊗ Ir+1] v = 0. (23)

As the Lagrange multiplier λ is unique, (23) implies λ = λ∗. Hence v isa stationary point of Problem (NLPr) with Lagrange multiplier λ∗.

Now, for any w = (w1, . . . , wn)T ∈ IRn, let us define the vector z ∈ IRnr

zT = ( 0Tr w1 0T

r w2 · · · 0Tr wn )

which satisfies

vT [Eii ⊗ Ir+1] z = 0, for all i = 1, . . . , n. (24)

Hence, by the second order necessary conditions for Problem (NLPr+1),we must have zT [(Q + Λ∗)⊗ Ir+1] z ≥ 0 and therefore, by the expression


of z we get

0 ≤ zT [(Q + Λ∗)⊗ Ir+1] z = ( 0Tr w1 . . . 0T

r wn )

0rn∑

j=1

q∗1jwj

...

0rn∑

j=1

q∗njwj

=n∑

j=1

n∑

i=1

q∗ijwiwj = wT (Q + Λ∗)w, (25)

where w is any vector in IRn, which implies (Q+Λ∗) º 0. Then the globaloptimality of v∗ follows from relation (13), that says qr(v∗) = −eT λ∗,and from Proposition 5.

Now we prove the necessity part. Let v∗ ∈ IRnr be a global minimumpoint of Problem (NLPr) that solves Problem (SDPMC). Then, thereexists λ∗ ∈ IRn such that (v∗, λ∗) satisfies first order necessary optimalityconditions for Problem (NLPr). Let us define the vector v ∈ IRn(r+1)

with vector components defined by (21), which is obviously feasible forProblem (NLPr+1). We have that

qr+1(v) = qr(v∗) = z∗SDP,

so that, by (8), v is a global minimum point of Problem (NLPr+1).Moreover, by using again (23), it follows from v∗ being a stationary pointof Problem (NLPr) with Lagrange multiplier λ∗ that (v, λ∗) satisfies

[(Q + Λ∗)⊗ Ir+1] v = 0,

namely v is a stationary point for Problem (NLPr+1) with Lagrangemultiplier λ∗. Again, by the uniqueness of the multipliers for Problem(NLPr+1) and by the global optimality of v, we have that (v, λ∗) satisfiesalso the second order optimality conditions of Problem (NLPr+1), andthis completes the proof.

The optimality conditions derived so far can be related to the eigenvaluebounds given in [32] specialized to Problem (SDPMC). Indeed, we ob-serve that Problem (NLPr) is equivalent to the following problem, where


we add the redundant constraint ‖v‖2 = n :

min vT (Q⊗ Ir) vvT (Eii ⊗ Ir) v = 1, i = 1, . . . , n‖v‖2 = n.

(26)

In connection with problem (26) we can define the function

Ψ(λ) = min‖v‖2=n

L(v, λ),

where L(v, λ) is the Lagrangian function of Problem (NLPr) given by(10). We can state the following result.

Proposition 9 For every λ ∈ IRn, and for every v such that ‖vi‖2 = 1for i = 1, . . . , n, the following inequalities hold

nλmin [Q + Λ]− eT λ ≤ z∗SDP ≤ q∗r ≤ qr(v). (27)

Proof Recalling the properties of Rayleigh quotient, and the proper-ties of the spectrum of the Kronecker product of two matrices, we get

Ψ(λ) = nλmin [Q + Λ]− eT λ.

It is easy to see that for every λ ∈ IRn Ψ(λ) ≤ q∗r , and, since Ψ(λ)does not depend on r, we have also Ψ(λ) ≤ z∗SDP. The right side of theinequality follows from the definition of q∗r .

Now, let (v, λ) be a stationary point of Problem (NLPr) and assumethat

λmin(Q + Λ) = 0.

Then, keeping into account again (13), we have

q(v) = −eT λ ≤ q∗r ,

so that we must have q(v) = q∗r = z∗SDP, which is exactly the sufficientcondition stated in Proposition 6.

4 An exact penalty function

In this section, for an arbitrary r ≥ 1 we consider Problem (NLPr) thatis:

min qr(v)

hi(v) = 1, i = 1, . . . , n,


where v ∈ IRnr, andqr(v) ≡ vT (Q⊗ Ir)v (28)

hi(v) ≡ ‖vi‖2. (29)

We show that solving Problem (NLPr) is equivalent to a single uncon-strained minimization of a differentiable merit function P . We define thefunction P as a continuously differentiable exact penalty function, whichis globally exact [10], in the sense that stationary points and minimiz-ers of P in the whole space IRnr correspond, respectively, to stationarypoints and minimizers of Problem (NLPr), for sufficiently small valuesof a penalty parameter ε > 0.A distinguishing feature of our approach consists in exploiting the specialstructure of the equality constrained Problem (NLPr), for constructinga multiplier function λ : IRnr → IRn, which yields an explicit estimateof the Lagrange multiplier vector associated to Problem (NLPr), as acontinuously differentiable function of the problem variables v. Then,following the construction proposed by Fletcher in [12], the function Pis obtained by replacing with λ(v) the multiplier λ appearing in theAugmented Lagrangian of Hestenes and Powell [34, 21]. We also showthat a threshold value for the penalty parameter can be evaluated apriori, in dependence only on the problem data.In order to illustrate the structure of P more in detail, we first definethe multiplier function by letting

λi(v) = −vT (EiiQ⊗ Ir) v, i = 1, . . . , n. (30)

The properties of this function are summarized in the following propo-sition.

Proposition 10

(i) λi ∈ C∞(IRnr), i = 1, . . . , n with gradient

∇λi(v) = − ((EiiQ + QEii)⊗ Ir) v, i = 1, . . . , n. (31)

(ii) If (v, λ) is a stationary point for Problem (NLPr) then we haveλ(v) = λ.

(iii) For every v we have

n∑

i=1

λi(v) = −qr(v).


(iv) For every v we have that

vT (Ekk ⊗ Ir)∇vL(v, λ(v)) = 2λk(v)(‖vk‖2 − 1),

where

∇vL(v, λ(v)) = ∇qr(v) +n∑

i=1

λi(v)∇hi(v) (32)

∇qr(v) = 2(Q⊗ Ir)v, (33)∇hi(v) = 2 (Eii ⊗ Ir) v, i = 1, . . . , n. (34)

Proof Assertion (i) follows immediately from (30), noting that thesymmetric part of (EiiQ⊗ Ir) is given by

((EiiQ + QEii)⊗ Ir) /2.

Assertion (ii) is a direct consequence of (12) of Proposition 2. Assertion(iii) follows from the fact that

n∑

i=1

vT (EiiQ⊗ Ir) v = vT (Q⊗ Ir) v.

Finally to prove assertion (iv) we note that:

vT (Ekk ⊗ Ir)∇vL(v, λ(v)) = vT (Ekk ⊗ Ir)(∇qr(v) +n∑

i=1

λi(v)∇hi(v))

= 2vT (EkkQ⊗ Ir)v

+2n∑

i=1

λi(v)vT (EkkEii ⊗ Ir)v

= −2λk(v) + 2λk(v)‖vk‖2

= 2(‖vk‖2 − 1)λk(v)

where we use the definition (30) of λk(v) and the identity EkkEii = 0n×n

for k 6= i.

Now, by replacing each multiplier estimate λi with the multiplier func-tion λi(v) in the augmented Lagrangian function of Hestenes-Powell, weobtain the merit function:

P (v) = qr(v) +n∑

i=1

λi(v) (hi(v)− 1) +1ε

n∑

i=1

(hi(v)− 1)2, (35)


where qr(v), hi(v) and λi(v) are defined in (28), (29), (30) and ε > 0 isa sufficiently small parameter depending on the data of the problem andon some user defined parameters (see Propositions 13 and 14 below).

We preliminarily state some basic properties of P , which follow directlyfrom its definition.

Proposition 11

(i) P ∈ C∞(IRnr) with gradient:

∇P (v) = ∇vL(v, λ(v))+n∑

i=1

(∇λi(v)+

2ε∇hi(v)

)(hi(v)−1), (36)

where ∇vL(v, λ(v)) is given by (32), ∇λi(v) by (31) and ∇hi(v)by (34).

(ii) for every feasible v we have that P (v) = q(v) and ∇P (v) =

∇vL(v, λ(v))

Now we consider the unconstrained problem

minv∈Rnr

P (v), (37)

and we study the correspondence between the solutions of Problem (37)and the solutions of the constrained Problem (NLPr).

First, we show that the unconstrained minimization of P admits a solu-tion for every sufficiently small value of ε and this constitutes an impor-tant feature for a computational use of the merit function (the proof isin Appendix A).

Proposition 12 For every ε such that

0 < ε < ε =1

Cn,

with

C :=√

r

n∑

i=1

‖Qi‖, (38)

where Qi is the i−th row of Q, the merit function P is coercive on IRnr

and hence it admits a global minimizer.


In order to establish the exactness properties of the merit function P , wepreliminarily prove the following proposition (the proof is in AppendixA).

Proposition 13 Let v0 be a feasible point for Problem (NLPr) and de-fine the level set

L0 = {v ∈ IRnr : P (v) ≤ P (v0) ≡ qr(v0)}.Let 0 < α < 1 and β > 1 be given scalars. Then, for every ε satisfying

0 < ε < ε = min{

ε,(1− α)2

Cn(1 + (α + n)2),

(β − 1)2

nCβ (2 + β(n + 1))

}, (39)

where C is defined in (38), we have that

L0 ⊆ {v ∈ IRnr : α ≤ ‖vi‖2 ≤ β, i = 1, . . . , n}.

Now, we can state the key result for establishing the exactness propertiesof the merit function P (v) (the proof is in the Appendix A).

Proposition 14 Let v ∈ IRnr be a stationary point of P (v) and 0 <α < 1 and β > 1 be given scalars. If ε satisfies

ε < ε∗ = ε∗(r, α, β) = min{

ε,α

nβ‖Q‖}

, (40)

where ε is defined in (39), then (v, λ(v)) is a stationary pair for Problem(NLPr), and we have P (v) = qr(v).

On the basis of the above result, we can state the exactness propertiesof the penalty function P .

Proposition 15 Let ε ∈ (0, ε∗), where ε∗ is defined in (40). A point v ∈IRnr is a stationary point of P (v) if and only if (v, λ(v)) is a stationarypair for Problem (NLPr); furthermore P (v) = qr(v).

Proof Necessity follows from Proposition 14. As for the sufficient part,let (v, λ(v)) be a stationary pair for Problem (NLPr). Since v is feasiblefor Problem (NLPr), it follows from expression (36) that ∇P (v) = 0.

Proposition 16 Let ε ∈ (0, ε∗), where ε∗ is defined in (40). Everyglobal minimizer of Problem (NLPr) is a global minimizer of P (v) andconversely.


Proof By Proposition 12, the penalty function P admits a global min-imizer v ∈ IRnr, which is obviously a stationary point of P and hence byProposition 15 we have that:

P (v) = qr(v).

On the other hand, if v∗ is a global minimizer of Problem (NLPr), itis also a stationary point and hence Proposition 15 implies again thatP (v∗) = qr(v∗). Now, we proceed by contradiction. Assume that aglobal minimizer v of P (v) is not a global minimizer of Problem (NLPr),then there should exists a point v∗, global minimizer of Problem (NLPr),such that

P (v) = qr(v) > qr(v∗) = P (v∗)

and this contradicts the assumption that v is a global minimizer of P .The converse is true by similar arguments.

Proposition 17 Let ε ∈ (0, ε∗), where ε∗ is defined in (40). If v ∈ IRnr

is a local minimizer of the function P (v) then it is a local minimizer ofProblem (NLPr) and λ(v) is the associated Lagrange multiplier.

Proof Since v is a local minimizer of P (v), it is a stationary point ofP (v). Thus, by Proposition 15, the pair (v, λ(v)) is a stationary pointof Problem (NLPr) and hence λ(v) is the unique associated multiplier.Moreover, P (v) = qr(v) and hence, since v is a local minimizer of P ,there exists a neighborhood B(v) of v such that

qr(v) = P (v) ≤ P (v) for all v ∈ B(v).

Therefore, by using (ii) of Proposition 11, we obtain

P (v) = qr(v) ≤ P (v) = qr(v) for all v ∈ B(v)∩{v : hi(v) = 1, i = 1, . . . , n},and hence v is a local minimizer for Problem (NLPr).

Finally, we report some additional results based on a second order anal-ysis. First we note that the penalty function P (v) is twice continuouslydifferentiable with Hessian matrix given by

∇2P (v) = ∇2vL(v, λ(v)) +

n∑

i=1

∇hi(v)∇λi(v)T

+n∑

i=1

(∇λi(v) +

2ε∇hi(v)

)∇T hi(v)

+n∑

i=1

(∇2λi(v) +

2ε∇2hi(v)

)(hi(v)− 1)

(41)


where

∇2λi(v) = − (EiiQ + QEii)⊗ Ir, ∇2hi(v) = 2(Eii ⊗ Ir).

Now we prove the following result.

Proposition 18 Let ε ∈ (0, ε∗), where ε∗ is defined in (40). Assume(v, λ(v)) is a pair satisfying the second order unconstrained necessaryconditions for the merit function P (v), namely ∇P (v) = 0 and ∇2P (v) º0. Then (v, λ(v)) is a stationary pair satisfying the second order neces-sary conditions for Problem (NLPr).

Proof Since v is a stationary point for P (v), then by Proposition 15,(v, λ(v)) is a stationary pair for Problem (NLPr). Hence v is feasible, sothat by the expression of ∇2P (v), we can write for all z ∈ IRnr

zT∇2P (v)z = zT∇2vL(v, λ(v))z+2

n∑

i=1

zT∇λi(v)∇hi(v)T z+2ε

n∑

i=1

‖∇T hi(v)z‖2.

In particular we have for all z ∈ IRnr such that ∇T hi(v)z = 0 for i =1, . . . , n, that

0 ≤ zT∇2P (v)z = zT∇2vL(v, λ(v))z,

which proves the assertion.

5 A globally convergent algorithm

In section 4, we have recast the problem of locating a constrained solutionof problem (NLPr) as the problem of locating an unconstrained solutionof the penalty function P (v). In particular, we showed that there isa one to one correspondence between constrained stationary points ofProblem (NLPr) and unconstrained stationary points of Problem (37)for finite values of the penalty parameter ε. This allows us to use anunconstrained method converging to stationary points of the penaltyfunction P in order to compute stationary points of (NLPr). To bemore precise, we assume that an unconstrained minimization procedureUNC satisfying the following property is available.

Property A Given a continuously differentiable function with compactlevel sets, procedure UNC produces a sequence of points belonging to theinitial level set such that:

(i) it admits at least an accumulation point,

(ii) every accumulation point is a stationary point of the function.


In literature, there are many unconstrained minimization methods satis-fying Property A, see for example [4]. We note that the penalty functionP (v) is continuously differentiable and, by Proposition 12, P (v) has com-pact level sets for ε ∈ (0, ε∗). Hence we can apply procedure UNC tofind a stationary point of the penalty function P (v).We can easily state the following convergence result that does not requireany further assumption.

Proposition 19 Let r be given and consider problem (NLPr). Let v0 ∈IRnr be a feasible point for problem (NLPr) and let ε ∈ (0, ε∗), where ε∗

is defined by (40). Procedure UNC applied to the merit function P (v)produces a sequence {vk} ∈ IRnr such that

(i) {vk} is bounded and it admits at least an accumulation point;

(ii) every accumulation point is a stationary point of of Problem (NLPr);

(iii) if v is an accumulation point then

qr(v) ≤ qr(v0).

Proof Proposition 12 implies that P (v) has compact level sets. There-fore Property A implies that UNC produces a sequence that has at leastan accumulation point and all the accumulation points are stationarypoints of Problem (37). Finally Proposition 15 implies that the sta-tionary points of P are stationary points of Problem (NLPr), and wehave

qr(v) = P (v) ≤ P (v0) = qr(v0).

We remark that the penalty function is exact, which implies that a sin-gle unconstrained minimization is needed to get a solution of Problem(NLPr) for a given r. In particular, we would like to solve Problem(NLPr) for values of r guaranteeing the correspondence between globalsolutions of Problem (NLPr) and of Problem (SDPMC). Hence, if thevalue of the parameter rmin were known a priori, we could solve a singleunconstrained problem, that is Problem (37) for r = rmin. However, thevalue rmin is not known, and the only value of r that can be calculatedand that guarantees the correspondence between solutions of (SDPMC)and global solutions of (NLPr), is r as defined in (7). However, thisvalue is usually larger than the actual value needed to obtain a solutionof Problem (SDPMC). The necessary and sufficient condition of Propo-sition 6 allows us to use a smaller value of r, since it gives a condition


to recognize a global solution of Problem (SDPMC). In particular, westart with a small value of r ≥ 2, and we look for a solution v of prob-lem (NLPr). If the pair (v, λ(v)) satisfies the condition Q + Λ(v) º 0of Proposition 6, then v is a global solution of (NLP)r and provides asolution of (SDPMC). If the pair (v, λ(v)) does not satisfy the conditionQ+Λ(v) º 0, we increase r up to a maximum value rp ∈ [r, n], and solveagain Problem (37). Now we can describe more formally our algorithm,that we call EXact Penalty Algorithm (EXPA).

An EXact Penalty Algorithm for the low rank SDPrelaxation (EXPA)

Data. Given the n× n matrix Q.

Inizialization. Find the value r given by (7). Set integers 2 ≤ r1 <r2 < . . . < rp with rp ∈ [r, n].

For j = 1, . . . , p

1. Find a feasible point v0 ∈ IRnrj

; find ε∗ as in (40), and setε < ε∗ .

2. Find a stationary point v ∈ IRnrj

of problem (NLPrj ) by us-ing the unconstrained formulation (37) and Procedure UNC.

3. Compute the minimum eigenvalue λmin(v) of Q + Λ(v).If λmin(v) = 0, then Exit;

End for

Return v ∈ IRnrj

, λ(v), λmin(v) and zB(v) = nλmin(v)− eT λ(v).

The EXPA scheme returns v ∈ IRnrj

, λmin(v), and zB(v). Proposition9 implies that zB(v) ≤ z∗SDP ≤ z∗MC, and hence zB constitutes a validbound for the max cut problem that can be used for example in a branchand bound algorithm. If λmin(v) = 0, then zB(v) = z∗SDP, and a solutionfor (SDPMC) is X∗ = V V T . The algorithm may fail in providing thesolution of Problem (SDPMC) if the maximum rp is reached and thecorresponding λmin(v) 6= 0. Such a failure of the algorithm may occurwhen the point v is a stationary point of Problem (NLPrp) that is not aglobal minimum point. In this case, however, we know that

zB(v) ≤ z∗SDP ≤ qr(v),

and hence the quantity qr(v)− zB(v) provides a measure of the distance


from the solution value z∗SDP. Anyway, we remark that, in practice, thestopping condition λmin(v) = 0 is always satisfied, so that algorithmEXPA always converges to a solution of (SDPMC), as we will illustratein the next section.

6 Numerical Results

In this section, we describe our computational experience with the algo-rithm EXPA. We implemented our algorithm in Fortran 90 and ran allthe experiments on an Intel Pentium 4 CPU 3.20 GHz with Ram of 2,00GB.As unconstrained minimization Procedure UNC we use a Fortran 90 im-plementation of a non monotone Barzilai-Borwein gradient method [18].We chose this local minimization method since non monotone methodsare usually more effective for minimizing penalty functions. Moreover,this method satisfies Property A stated in the previous section.In order to evaluate the minimum eigenvalue of Q + Λ(v) we used theARPACK subroutines dsaupd and dseupd, that can be downloaded fromthe web page

http://www.caam.rice.edu/software/ARPACK/

but any other subroutine able to solve large scale eigenvalue problemscould be used.We implemented two different versions of EXPA and we refer to themas EXPA-BOUND and EXPA-R respectively. In the first version ofthe algorithm, EXPA-BOUND, we implement a code on the basis ofthe scheme described in the previous section. We select the value r1

depending on the dimension of the problem, according to Table 1 (thisrule was established empirically).

n r1

≤ 200 8≥ 250, < 800 10≥ 800, < 1000 15≥ 1000, < 5000 18

≥ 5000 20

Table 1: Values of r1 depending on the dimension of the problem.

As a stopping criterion to check positive semidefiniteness of Q + Λ(v),


we consider|λmin (v) | < 10−2.

We point out that on all the test problems that we solved, this optimalitycriterion is satisfied for j = 1, so that the stationary point of (NLPr1)returned by procedure UNC is actually a global solution of (SDPMC).In the second version of the algorithm, EXPA-R, we set r1 = rp = r.In this case, we do not perform Step 3, that is we do not evaluate theminimum eigenvalue of Q+Λ(v) so that we cannot certify optimality evenif, in practice, we always obtain a global solution of (NLPr) and henceof (SDPMC). We stres the main difference between the two algorithms.EXPA-BOUND looks for solutions of (NLPr) with r << r and checksif the current point is a global minimum or not; hence either it certifiesthat we have found a solution of Problem (SDPMC) or it provides a validbound for max cut. EXPA-R is an heuristic for solving (NLPr) that, inpractice, gives the same optimal value as EXPA-BOUND.In order to evaluate the two versions of EXPA algorithm, we comparethem with two codes which are respectively an exact algorithm and anheuristic for (SDPMC). In particular we consider the Burer and Mon-teiro’s software SDPLR, version 1.02 downloaded from the webpagehttp://dollar.biz.uiowa.edu/~burer/software/SDPLR/index.html,

and the interior point method SDPA proposed in [14, 15, 25, 36], down-loaded from the webpage

http://grid.r.dendai.ac.jp/sdpa.

The algorithm SDPLR is the one we drew inspiration from. It solvesProblem (NLPr), by an augmented Lagrangian approach solving a se-quence of unconstrained minimization problems by means of L-BFGSmethod. The algorithm works very well in practice and it is able tohandle large scale problems but, as already pointed out, convergencetowards stationary points of the augmented Lagrangian cannot be guar-anteed and optimality cannot be certified. On the other hand, SDPA isa primal-dual interior point method that guarantees feasibility at eachiteration and convergence to a global optimum of Problem (SDPMC).However, the code cannot be used for solving very large instances. In-deed, it runs out of memory with problems where n ≥ 3000.As for SDPA and SDPLR, we used their standard stopping criterion.We compare the different codes on the basis of the computational timerequired to reach the same solution. We consider that the methodsconverge to the same point whenever the relative difference between theobjective function values is less than 10−4.

We tested the performance of the methods on three different set of prob-lems. The first set belongs to the SDPLIB collection of semidefinite


programming test problems (hosted by B. Borchers) that can be down-loaded from the web page

http://infohost.nmt.edu/∼sdplib.The smallest problems (mcp set) have been contributed by Katsuki Fu-jisawa ([13]), while the maxG problems were supplied by Steve Benson([2]). More details on these problems are listed in Table B1 in AppendixB.The second set of problems is the torus set that is part of the DIMACSlibrary of mixed semidefinite-quadratic-linear programs. These problemscan be downloaded from the web page

http://dimacs.rutgers.edu/Challenges/Seventh/Instances.

The characteristics of this set of problems are reported in Table B2 inAppendix B.Finally, the third set of problems belongs to the Gset and they are ran-domly generated by a machine-independent graph generator, rudy, cre-ated by Giovanni Rinaldi. These problems can also be downloaded bySamuel Burer’s webpage

http://dollar.biz.uiowa.edu/ burer/software

/SDPLR/miscellaneous.html

The characteristics of these problems are listed in Table B3 in AppendixB.We report in Table 2 the objective function values found by the fourmethods where we set the objective value to *** whenever SDPA runsout of memory. It is clear from the table that although SDPA is a littlebit more precise, all the methods converge to the same objective func-tion value except in the case of problem G65 where, with the standardaccuracy, SDPLR does not converge to the global optimum. In Table 3we report the cpu times in seconds needed by the four methods. In thistable, we report an additional column UNC with the time needed by theunconstrained minimization routine UNC to solve Problem (37) in thecase of EXPA-BOUND. By comparing the values in column UNC andin column EXPA-BOUND, it turns out that, for many problems, mostof the computational time is spent to evaluate λmin(v), that is to certifyoptimality of the stationary point v. It is possible to improve the overallcomputational time by developing a faster way to establish if Q + Λ(v)is positive semidefinite.In order to better read the results, we consider the performance profileof the different methods with respect to the computational time, byfollowing the approach proposed in [11]. In particular, given ns solversin a set S and np problems, for each problem p and each solver s ∈ S


we use the performance ratio

rp,s =tp,s

min{tp,s : s ∈ S} ,

where tp,s is the computing time needed by the solver s in order to solveproblem p. Given this performance ratio, its distribution function is theprobability that a performance ratio rp,s is within a factor τ of the bestpossible ratio, namely

ρs(τ) =1np

size{p ∈ P : rp,s ≤ τ}.

As a first step, we compare the performance of the two algorithms thatcertify optimality: EXPA-BOUND and SDPA. We select all the testproblems where n ≤ 2000, because SDPA cannot solve larger problems.


graph EXPA-BOUND EXPA-R SDPA LSPDRmcp100 -226.15734 -226.15734 -226.15738 -226.15658mcp124-1 -141.99043 -141.99043 -141.99050 -141.98972mcp124-2 -269.88016 -269.88016 -269.88023 -269.87881mcp124-3 -467.75008 -467.75008 -467.75022 -467.74893mcp124-4 -864.41170 -864.41170 -864.41197 -864.39337mcp250-1 -317.26419 -317.26419 -317.26454 -317.26306mcp250-2 -531.93007 -531.93007 -531.93031 -531.92978mcp250-3 -981.17215 -981.17215 -981.17275 -981.16863mcp250-4 -1681.95692 -1681.95692 -1681.96037 -1681.93548mcp500-1 -598.14735 -598.14735 -598.14865 -598.14487mcp500-2 -1070.05539 -1070.05539 -1070.05696 -1070.04917mcp500-3 -1847.96846 -1847.96846 -1847.97069 -1847.94426mcp500-4 -3566.73600 -3566.73600 -3566.73920 -3566.70166maxG11 -629.13647 -629.12457 -629.16479 -629.12790maxG32 -1567.53673 -1567.55953 -1567.63968 -1567.55170maxG51 -4006.25161 -4006.25127 -4006.25570 -4006.20514maxG55 -11039.44111 -11039.44155 *** -11039.33380maxG60 -15222.19399 -15222.24820 *** -15222.09090

torusg3-8 -457.35501 -457.35207 -483.40951 -457.34398torusg3-15 -3134.52888 -3134.55687 *** -3134.46443toruspm3-8-50 -527.80741 -527.80674 -527.80872 -527.80400toruspm3-15-50 -3475.12125 -3475.09161 *** -3474.92171

G01 -12083.19457 -12083.19525 -12083.19879 -12082.94910G14 -3191.55937 -3191.56400 -3191.56692 -3191.47778G22 -14135.89843 -14135.93922 -14135.94618 -14135.80890G35 -8014.73467 -8014.73041 -8014.74067 -8013.98163G36 -8005.94474 -8005.96062 -8005.96401 -8005.66409G43 -7032.21956 -7032.21677 -7032.22202 -7032.12060G48 -5999.99961 -5999.99377 *** -5999.95947G52 -4009.63475 -4009.63665 -4009.63901 -4009.44213G57 -3885.35728 -3885.29597 *** -3885.16885G58 -20135.86499 -20136.17098 *** -20135.53210G62 -5430.64358 -5430.67518 *** -5430.37047G63 -28243.26686 -28244.38697 *** -28241.31670G64 -10465.70537 -10465.87642 *** -10465.77620G65 -6205.20446 -6205.17941 *** -6204.54366G66 -7076.84074 -7076.91309 *** -7076.69518G67 -7743.99395 -7744.06431 *** -7742.01175G70 -9861.49888 -9861.46800 *** -9861.38499G72 -7808.17675 -7808.16852 *** -7807.85812G77 -11045.01064 -11045.05108 *** -11044.32160G81 -15654.87604 -15655.28128 *** -15652.08250

Table 2: Objective function values found by the different methods


graph UNC EXPA-BOUND EXPA-R SDPA LSPDRmcp100 0.04688 0.07813 0.04688 0.578 0.187mcp124-1 0.0625 0.17188 0.14063 0.937 0.563mcp124-2 0.03125 0.04688 0.07813 1.015 0.281mcp124-3 0.0625 0.64063 0.09375 1.016 0.312mcp124-4 0.09375 0.10938 0.1875 1.046 0.219mcp250-1 0.15625 0.20313 0.29688 6.843 1.515mcp250-2 0.17188 0.20313 0.26563 7.203 3.250mcp250-3 0.17188 1.39063 0.39063 7.437 2.297mcp250-4 0.375 3.92188 0.67188 7.609 2.610mcp500-1 0.42188 1.71875 1.625 52.671 18mcp500-2 0.45313 2.375 1.01563 60.031 14.282mcp500-3 0.48438 7.39063 0.98438 57.812 11.891mcp500-4 0.85938 0.96875 1.64063 59.813 14.531maxG11 1.78125 3.10938 3.15625 273.187 211.625maxG32 6.875 12.54688 25.04688 4130.453 134.938maxG51 4.81250 8.78125 18.82812 542.5 130.204maxG55 25.26562 26.15625 133.8281 *** 210.203maxG60 53.0625 54.26562 193.375 *** 542.187

torusg3-8 0.76563 3.48438 1.25 102.906 44.406torusg3-15 20.4375 23.79688 94.57812 *** 117.594toruspm3-8-50 0.64063 6.85938 1.01563 65.687 12.5toruspm3-15-50 16.57812 25.98438 59.71875 *** 77.656

G01 3.625 4 7.15625 241.594 58.313G14 4.40625 4.70313 5.65625 279.765 110.109G22 9.25 9.42188 32.14062 3924.328 49.625G35 34.54688 38.3125 121.1562 4272.781 73.265G36 49.71875 51.60938 182.9531 4590.453 78.407G43 2.70313 15.65625 6.48438 498.188 88.844G48 8.46875 8.65625 53.95312 *** 95.719G52 5.10938 10.03125 17.23438 541.75 207.313G57 34.04688 44.54688 160.8906 *** 290.5G58 302.1094 310.5156 1276.625 *** 787.593G62 43.78125 79.03125 244.2812 *** 446.5G63 1012.562 1031.859 1936.156 *** 2453.687G64 498.6094 507.4375 1074.609 *** 793.266G65 67.75 100.4688 308.0312 *** 691.032G66 73.04688 110.1562 425.9375 *** 852.031G67 64.15625 93.78125 509.2812 *** 412.313G70 92.5625 367.625 666.75 *** 1020.141G72 80.64062 173.9531 588.5625 *** 1338.297G77 113.7344 154.3594 983.6094 *** 1569.985G81 179.5938 272.3281 1978.078 *** 1691.125

Table 3: Cpu time needed by the different methods


In figures 1 and 2 we report ρs(τ) for both methods. In the first figure,we report ρs(τ) for small values of τ ≤ 20, while in the second figure, wereport ρs(τ) also for larger values.

2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

tau

SDPAEXPA−BOUND

Figure 1: Performance profile for all the test problems with n ≤ 1000with τ ≤ 20


50 100 150 200 250 300 350 400 450 5000

0.2

0.4

0.6

0.8

1

SDPAEXPA−BOUND

Figure 2: Performance profile for all test problems with n ≤ 2000

It clearly emerges that EXPA-BOUND outperforms SDPA in terms ofcomputing time.As a second step, we compare on all the test problems the three methodsEXPA-BOUND, EXPA-R and SDPLR. Indeed, EXPA-R and SDPLRare both heuristics for the solution of (SDPMC) which uses the samevalue of r. In figures 3 and 4 we report ρs(τ) for the three methods.Also in this case, in the first figure we consider the behavior for smallvalues of τ ≤ 20, while in the second we allow τ to get larger.


2 4 6 8 10 12 14 16 18 200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

EXPA−BOUNDSDPLREXPA−R

Figure 3: Performance profile for all the test problems with τ ≤ 20

It emerges from these figures that EXPA-BOUND and EXPA-R has abetter behavior than both SDPLR; further that EXPA-BOUND has aslight better behavior than EXPA-R, although it is quite comparablewith it. We point out that the computational time in EXPA-R is dueto the time of the unconstrained Procedure UNC for solving Problem37 with r = r, whereas in EXPA-BOUND the overall time is the sumof the time of Procedure UNC and of the evaluation of the minimumeigenvalue.


10 20 30 40 50 60 70 80 900.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

EXPA−BOUNDSDPLREXPA−R

Figure 4: Performance profile for all the test problems

7 Concluding Remarks

In this paper we have studied a nonlinear programming reformulation ofthe low-rank semidefinite programming relaxation LRSDP arising fromthe max cut problem and we have developed a globally convergent algo-rithm. We have performed an extensive numerical testing comparing ourcode to other existing codes on a large set of problems with dimension upto 20.000. The results obtained are encouraging, and naturally lead totry to extend the approach to the solution of standard linear semidefiniteprogramming problems. For the LRSDP problem, the penalty functionapproach works quite efficiently, since the structure of the constraintsallows us to define a simple multiplier function. In the general case, how-ever, it could be more effective to define an exact augmented Lagrangian,and this is part of our future research. Moreover, another aspect thatdeserves further investigation in the context of max cut problems couldbe the use of nonlinear techniques in order to determine feasible cuts,starting from the solution of the LRSDP problem, to be used in a branchand bound type algorithm, as it has been done in the paper [5].

Appendix A

Proof of Proposition 12


We must prove that, as ‖v‖ → ∞, the function value P (v) goes toinfinity.Recalling (30), and point (iii) of Proposition 10, we can write the meritfunction (35) in the following form:

P (v) = 2vT (Q⊗ Ir) v−n∑

i=1

[vT (EiiQ⊗ Ir) v

] ‖vi‖2+1ε

n∑

i=1

(‖vi‖2 − 1)2

.

Noting that ‖vi‖ ≤ ‖v‖ and ‖Q⊗Ir‖ =√

r‖Q‖, and using (rather crude)majorizations, we get

P (v) ≥ −2‖v‖2‖Q⊗ Ir‖ − ‖v‖4n∑

i=1

‖EiiQ⊗ Ir‖+1ε

n∑

i=1

[‖vi‖2 − 1]2

= −2√

r‖v‖2‖Q‖ − C‖v‖4 +1ε

n∑

i=1

[‖vi‖2 − 1]2

where C is given by (38). Letting

zi = ‖vi‖2 − 1, i = 1, . . . , n, and z = (z1 . . . zn)T ,

so that‖vi‖2 = zi + 1, i = 1, . . . , n

‖v‖2 =n∑

i=1

‖vi‖2 = eT z + n

we obtain:P (v) ≥ φ(z), (42)

where φ is the quadratic function given by

φ(z) = −2√

r‖Q‖ (eT z + n

)− C((eT z)2 + 2neT z + n2) +1ε‖z‖2

= zT

(1εI − CeeT

)z − (

2√

r‖Q‖+ 2nC)eT z − 2

√rn‖Q‖ − n2C.

(43)Now, the function φ is coercive if and only if (1/ε)I −CeeT Â 0, namelyfor all ε such that ε < ε, where ε = 1/(Cn). On the other hand, if‖v‖ → ∞, we obviously have that ‖z‖ → ∞ and therefore, for every0 < ε < ε, we can assert that

lim‖v‖→∞

P (v) ≥ lim‖z‖→∞

φ(z) = ∞.

Thus, P is coercive and has compact level sets and hence the existenceof a global minimizer follows from the continuity of P .


Proof of Proposition 13Since v0 is feasible, we have that P (v0) = qr(v0). Moreover, reasoningas in the proof of Proposition 12, we can write

P (v) ≥ −2√

r‖Q‖‖v‖2 − C‖v‖4 +1ε

n∑

h=1

[‖vh‖2 − 1]2

;

Since√

r‖Q‖ ≤ √r

n∑

i=1

‖Qi‖ = C, we can write

P (v) ≥ −2C‖v‖2 − C‖v‖4 +1ε

n∑

h=1

[‖vh‖2 − 1]2

. (44)

For ε satisfying (39) and 0 < α < 1, first we show that the existence ofsome i such that ‖vi‖2 < α implies that

P (v) > P (v0).

Assuming ‖vi‖2 < α for a given i, set θ = ‖vi‖2 and define a vectorw ∈ Rn−1 with components

wjh= ‖vh‖2 − 1, for h = 1, . . . , n, h 6= i,

so that

eT w =∑

h6=i

(‖vh‖2 − 1),

‖w‖2 =∑

h6=i

(‖vh‖2 − 1)2,

‖v‖2 =∑

h6=i

(wjh+ 1) + θ = eT w + n− 1 + θ,

where e ∈ IRn−1. Using straightforward majorizations and rearranging,from (44) we get:

P (v) ≥ −C(θ+n)2+1ε

(1− θ)2+wT

(1εIn−1 − CeeT

)w−2C(θ+n)eT w

≥ −C(θ + n)2 +1ε

(1− θ)2 + Ψ(w, θ), (45)

where

Ψ(w, θ) =(

1ε− C‖e‖2

)‖w‖2 − 2C(θ + n)eT w

=(

1ε− C(n− 1)

)‖w‖2 − 2C(θ + n)eT w.


It is easily seen that for every ε < ε such that

ε =1

Cn<

1C(n− 1)

the Hessian of the quadratic function Ψ(w, θ) is positive definite; more-over, the minimum point w∗ of Ψ(w, θ) with respect to w is given by

w∗ =εC(θ + n)

1− Cε(n− 1)e.

Then, by computing the minimum value Ψ(w∗, θ), we obtain, for everyw ∈ Rn−1:

Ψ(w, θ) ≥ Ψ(z∗, θ) = −C2ε(θ + n)2(n− 1)1− Cε(n− 1)

. (46)

Since ε < 1/Cn, we have that 1− Cε(n− 1) > Cε, and hence by (46) itfollows

Ψ(w, θ) ≥ −C(θ + n)2(n− 1).

Then, from (45) and recalling the assumption that θ < α we can write:

P (v) ≥ −Cn(α + n)2 +1ε

(1− α)2 . (47)

Thus ε <(1− α)2

Cn(1 + (α + n)2)combined with (47) implies

P (v) ≥ −Cn(α + n)2 +1ε

(1− α)2 > Cn. (48)

Since P (v0) = q(v0), and v0 is feasible, we have that

P (v0) = vT0 (Q⊗ Ir) v0 ≤ ‖Q⊗ Ir‖‖v0‖2 =

√rn‖Q‖ ≤ Cn. (49)

but then (48) implies P (v) > P (v0).

Now we show that if ‖vi‖2 > β > 1 for an index i ∈ {1, . . . , n} thenP (v) > P (v0). Let t ≡ max1≤h≤n ‖vh‖2. First we note that

‖vi‖2 > β > 1 implies that t = max1≤h≤n

‖vh‖2 > β > 1,

and‖v‖2 ≤ n max

1≤h≤n‖vh‖2 = nt.


Let us consider again the inequality (44), hence we can write

P (v) ≥ −2Cnt− Cn2t2 +1ε(t− 1)2

= t2

(−2

Cn

t− Cn2 +

1ε

(1− 1

t

)2)

≥ t2

(−Cn

(2β

+ n

)+

1ε

(1− 1

β

)2)

(50)

Furthermore by (49) we can write

P (v0) ≤ Cn ≤ Cnβ2 < Cnt2.

Thus

ε <(β − 1)2

nCβ (2 + β(n + 1))

combined with (50) implies

P (v) ≥ t2

(−Cn

(2β

+ n

)+

1ε

(1− 1

β

)2)

> Cnt2

but then we have by (49) P (v) > P (v0).

Proof of Proposition 14

By expression (36) of ∇P (v) we can write

vT (Ekk ⊗ Ir)∇P (v) = vT (Ekk ⊗ Ir)∇L(v, λ(v))+

∑ni=1 vT (Ekk ⊗ Ir)

(∇λi(v) + 2

ε∇hi(v))(hi(v)− 1).

From (29) (iv) of Proposition 10, using the properties of Kroneckerproducts, recalling that hk(v) = vT (Ekk ⊗ Ir)v and observing that


EkkEii = 0n×n, for all i 6= k, and EkkEii = Ekk, for i = k, we get

vT (Ekk ⊗ Ir)∇P (v) = 2λk(v)(hk(v)− 1)

−n∑

i=1

(hi(v)− 1) vT [(EkkEiiQ + EkkQEii)⊗ Ir] v

+4ε

n∑

i=1

vT (EkkEii ⊗ Ir)v (hi(v)− 1)

= 2λk(v)(hk(v)− 1)− (hk(v)− 1) vT (EkkQ⊗ Ir) v

+4εhk(v) (hk(v)− 1)−

n∑

i=1

(hi(v)− 1) vT (EkkQEii ⊗ Ir)v

= (3λk(v) +4εhk(v))(hk(v)− 1)

−n∑

i=1

(hi(v)− 1) vT (EkkQEii ⊗ Ir) v.

Since ∇P (v) = 0, from the last equality we get for all k = 1, . . . n

(hk(v)− 1)[3λk(v) +

4εhk(v)

]−

n∑

i=1

(hi(v)− 1)vT [EkkQEii ⊗ Ir] v = 0.

(51)Now, we observe that

vT [EkkQEii ⊗ Ir] v = qkivTk vi

so that, setting

bkk = 3λk(v) +4εhk(v)− qkkhk(v),

bik = bki = −qkivTk vi i 6= k,

zi = hi(v)− 1,

formula (51) becomes

bkkzk −∑

i 6=k

bkizi = 0 k = 1, . . . , n.

Letting B = (bij)i,j=1,...,n ∈ Sn, the equations above can be rewrittenas the homogeneous system

Bz = 0.


Now we show that the choice of ε implies that the matrix B is strictlydiagonally dominant, that is |bkk| > |

∑

i6=k

bki| for all k, namely that

∣∣∣∣3λk(v) +4εhk(v)− qkkhk(v)

∣∣∣∣ >

∣∣∣∣∣∣∑

i 6=k

qkivTk vi

∣∣∣∣∣∣for k = 1, . . . , n.

The above inequality is implied by

4εhk(v)− 3 |λk(v)| −

n∑

i=1

|qkivTk vi| > 0 for k = 1, . . . , n.. (52)

Since

|λk(v)| =∣∣vT (EkkQ⊗ Ir)v

∣∣ ≤ ‖(Ekk ⊗ Ir)v‖‖Q⊗ Irv‖ ≤√

r‖Q‖‖vk‖‖v‖n∑

i=1

|qki|‖vk‖‖vi‖ = ‖vk‖n∑

i=1

|qki|‖vi‖ ≤ n‖vk‖ maxi=1,...,n

|qki|‖vi‖,

condition (52) is implied, in turn, by

4εhk(v)− 3

√r‖Q‖‖vk‖‖v‖ − n‖vk‖ max

i=1,...,n|qki|‖vi‖ > 0. (53)

By Proposition 13, we know that for all α ∈ (0, 1), we can choose ε ≤ εsuch that α ≤ hi(v) = ‖vi‖2 ≤ β for all v ∈ L0. Thus condition (53) isimplied by

4εα− 3

√β‖Q‖√r‖v‖ − nβ max

i=1,...,n|qki| > 0 for k = 1, . . . , n.

As

‖v‖ =

(n∑

i=1

‖vi‖2)1/2

≤(

n∑

i=1

β

)1/2

=√

nβ,

we can finally write

4εα− 3

√nrβ‖Q‖ − nβ max

i=1,...,n|qki| ≥

4εα− 3nβ‖Q‖ − nβ‖Q‖ > 0 for k = 1, . . . , n.

Hence for ε satisfying (40), the above inequality is satisfied, so thatB is diagonally dominant, and this implies that the unique solution of


system Bz = 0 is z = 0, namely hi(v) = 1 which in turns implies thatv is feasible for Problem (NLPr). Hence from ∇P (v) = 0, recalling (ii)of Proposition 11, we get ∇vL(v, λ(v)) = 0 which gives the first orderoptimality conditions and P (v) = q(v).

Appendix B

In this section we report the tables containg the characteristics of thetest problems used.

graph |V | |E|mcp100 100 269mcp124-1 124 149mcp124-2 124 318mcp124-3 124 620mcp124-4 124 1271mcp250-1 250 331mcp250-2 250 612mcp250-3 250 1283mcp250-4 250 2421mcp500-1 500 625mcp500-2 500 1223mcp500-3 500 2355mcp500-4 500 5120maxG11 800 1600maxG32 2000 4000maxG51 1000 5909maxG55 5000 12498maxG60 7000 17148

Table B1: The SDPLIB max cut problems

graph |V | |E|torusg3-8 512 1536torusg3-15 3375 10125toruspm3-8-50 512 1536toruspm15-50 3375 10125

Table B2: The torus max cut problems


graph |V | |E|G01 800 19176G14 800 4694G22 2000 19990G35 2000 11778G36 2000 11766G43 1000 9990G48 3000 6000G52 1000 5916G57 5000 10000G58 5000 29570G63 7000 41459G64 7000 41459G65 8000 16000G66 9000 18000G67 10000 20000G70 10000 9999G72 10000 20000G77 14000 28000G81 20000 40000

Table B3: The Gset max cut problems

References

[1] A. Barvinok. Problems of distance geometry and convex propertiesof quadratic maps. Discrete Computational Geometry, 13:189–202(1995).

[2] S. J. Benson, Y. Ye, and X. Zhang. Solving Large-Scale SparseSemidefinite Programs for Combinatorial Optimization. SIAMJournal on Optimization, 10(2):443–461 (2000).

[3] A. Ben-T and M. Teboulle. Hidden convexity in some nonconvexquadratically constrained quadratic programming. MathematicalProgramming 72:51-63, (1996).

[4] Dimitri P. Bertsekas. Nonlinear Programming, Athena Scientific,1999.


[5] S. Burer, R. D. C. Monteiro and Y. Zhang. Rank-Two RelaxationHeuristics for max cut and Other Binary Quadratic Programs.SIAMJournal on Optimization, 12(2):503–521 (2002).

[6] S. Burer and R.D.C. Monteiro. A projected gradient algortihmfor solving the Maxcut SDP relaxation. Optimization Methods andSoftware, 15:175–200 (2001).

[7] S. Burer and R.D.C. Monteiro. A nonlinear programming algorithmfor solving semidefinite programs via low-rank factorization. Math.Programming, Ser. B 95:329–357 (2003).

[8] S. Burer and R.D.C. Monteiro. Local Minima and Convergence inLow-Rank Semidefinite Programming. Mathematical Programming,Ser. A, 103:427-444 (2005).

[9] C. Delorme and S. Poljak. Laplacian eigenvalues and the maximumcut problem. Mathematical Programming, 62(3):557–574 (1993).

[10] G. Di Pillo and L. Grippo.Exact Penalty Functions in ConstrainedOptimization Problems. SIAM Journal on Control and Optimiza-tion, 27(6):1333–1360 (1989).

[11] E.D. Dolan and J.J. More. Benchmarking optimization softwarewith performance profile. Mthematical Programming, Ser. A,91:201–213 (2002).

[12] R. Fletcher. A new approach for variable metric algorithms. Com-puter J., 13:317-322, 1970.

[13] K. Fujisawa, M. Fukuda, M. Kojima, and K. Nakata. NumericalEvaluation of SDPA (Semidefinite Programming Algorithm) HighPerformance Optimization, H.Frenk, K. Roos, T. Terlaky and S.Zhang eds., Kluwer Academic Press, 1999, pp.267-301.

[14] K. Fujisawa, M. Kojima, and K. Nakata. Exploiting sparsity inprimal-dual interior-point methods for semidefinite programming,Mathematical Programming B, 79, 235-253 (1997).

[15] K. Fujisawa, M. Kojima, K. Nakata, and M. Yamashita. SDPA(SemiDefinite Programming Algorithm) User’s manual — version6.2.0 Research Report B-308, Dept. Math. & Comp. Sciences, TokyoInstitute of Technology, December 1995, Revised September 2004.

[16] M.X. Goemans. Semidefinite Programming in Combinatorial Opti-mization. Mathematical Programming, 79, 143–161 (1997).


[17] M.X. Goemans and D.P. Williamson. Improved approximation algo-rithms for maximum cut and satisfiability problems using semidef-inite programming. J. Assoc. Comput. Mach., 42(6):1115–1145(1995).

[18] L. Grippo, M. Sciandrone. Nonmonotone globalization techniquesfor the Barzilai-Borwein gradient method. Computational Optimiza-tion and Applications, 23:143–169 (2002).

[19] M. Grotschel, L. Lovasz, and A. Schrijver. The ellipsoid methodand its consequences in combinatorial optimization. Combinatorica,1:169–197 (1981).

[20] C. Helmberg and F. Rendl. A spectral bundle method for semidef-inite programming. SIAM J. on Optimization, 3:673–696 (1999).

[21] M. Hestenes, Multiplier and gradient methods. Journal of Optimiza-tion Theory and Application 4 303–320 (1969).

[22] S. Homer and M. Peinado. Desing and performance of parallel andDistributed Approximation Algortihm for the Maxcut. J. of Paralleland Distributed Computing, 46:48–61 (1997).

[23] R. A. Horn and C. R. Johnson. Matrix analysis, Cambridge Uni-versity Press, Cambridge, 1985.

[24] R. A. Horn and C. R. Johnson. Topics in matrix analysis, Cam-bridge University Press, New York, 1986.

[25] M. Kojima, S. Shindoh, and S. Hara. Interior-point methods for themonotone semidefinite linear complementarity problem in symmet-ric matrices,SIAM Journal on Optimization 7, 86-125 (1997).

[26] M. Laurent, S. Poljak, and F. Rendl. Connections between semidef-inite relaxations of the maxcut and stable set problems. Mathemat-ical Programming, 77:225–246 (1997).

[27] R. D. C. Monteiro.First- and second-order methods for semidefiniteprogramming.Mathematical Programming 97:209–244 (2003).

[28] J. More, Generalization of the trust region problem, OptimizationMethods and Software, 2 (1993), pp. 189–209.

[29] Yu. Nesterov and A. Nemirovskii. Self-concordant functions andpolynomial time methods in convex programming. Central Eco-nomical and Mathematical Institute, U.S.S.R. Academy of Dcience,Moscow (1990).


[30] Yu. Nesterov and A. Nemirovskii. Interiorpoint polynomial algo-rithms in convex programming. Society for Industrial and AppliedMathematics (SIAM), Philadelphia, PA (1994).

[31] G. Pataki. On the rank of extreme matrices in semidefinite pro-grams and the multiplicity of optimal eigenvalues. Mathematics ofOperations Research, 23:339358 (1998).

[32] S. Poljak, F. Rendl, and H. Wolkowicz. A recipe for semidefiniterelaxation for 0-1 quadratic programming. Journal of Global Opti-mization, 7:51–73 (1995).

[33] S. Poljak and Z. Tuza. Maximum cuts and largest bipartite sub-graphs. In W. Cook, L. Lov’asz, and P. Seymour, editors, Combi-natorial Optimization, DIMACS series in Disc. Math. and Theor.Comp. Sci. AMS (1995).

[34] M. J. D. Powell. A method for nonlinear constraints in minimizationproblem. In Optimization, R. Fletcher (ed.), pp. 283–298, AcademicPress, New York, 1969.

[35] R. Stern and H. Wolkowicz. Indefinite trust region subproblemsand nonsymmetric eigenvalue perturbations, SIAM Journal on Op-timization, 5(2):286–313 (1995).

[36] M. Yamashita, K. Fujisawa, and M. Kojima. Implementation andevaluation of SDPA 6.0 (SemiDefinite Programming Algorithm 6.0),Optimization Methods and Software 18, 491-505 (2003).

an unconstrained minimization method for solving low rank

Documents