convex duality in math finance - homepages at wmuhomepages.wmich.edu/~zhu/mf/lecture1a.pdf ·...
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
CONVEX DUALITY IN MATH FINANCE1. Constrained Optimization and Lagrange Multipliers
Peter Carr and Qiji Zhu
Morgen Stanley/NYU andWestern Michigan University
September 2, 2015
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Goal
• Increasing concave utilities and convex risk measures arecommon in financial problems.
• Moreover, no-arbitrage often implies the price of manyfinancial derivatives are convex in their parameters.
• Hence convex analysis methods are intrinsically involved inmany financial problems.
• We intend to provide a perspective of many important issuesin mathematical finance based on convex duality theory.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Lagrange multiplier rules (LMR)
We start with Lagrange multiplier rules because
• Many financial problems can naturally be formulated asconstrained optimization problems.
• Lagrange multipliers are convenient tools for solving thoseproblems.
• They also bridge those financial problems to convex duality.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Variational approach
For financial applications it is convenient to take the variationalanalysis view of the Lagrange multiplier. Define
v(y) = inff(x) : g(x) = y
and assume x0 is a solution to v(0).Then
x0 ∈ argmin[f(x)− v(g(x))]
Assuming all involved are smooth then
f ′(x0)− v′(g(x0))g′(x0) = 0
revealing λ = −v′(g(x0)) as a Lagrange multiplier.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Why variational approach
• Explains the Lagrange multipliers as shadow prices.
• Leads naturally to duality.
• Reveals that in dealing with minimization problems what’sessential is a lower horizontal support rather than tangent.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Economic meaning
D. Gale, gives the following economic explanation and provided arigorous proof for convex problems in 1967.
• View y as constrains in resources and −f as output in aneconomy.
• The Lagrange multiplier λ = −v′(g(x0)) then reflects themarginal gain of the output function with respect to theresource constraints.
• Following this observation, if we penalize the resourceutilization with the Lagrange multiplier (shadow price) thenthe constrained optimization problem can be converted to anunconstrained one.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Dual problem
For simplicity consider the linear form of the above economicoutput problem with resource constraint:
maxx
⟨c, x⟩ s.t. Ax = b, x ≥ 0.
Consider the flip side of the problem: what is the fair price(vector) p to buy the resources. Seller is willing to sell only whenA⊤p ≥ c. So we get the dual problem
minp
⟨p, b⟩ s.t. A⊤p ≥ c.
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Dual problem
Clearly for any feasible pair (x, p) for the primal-dual problem wehave weak duality
⟨p, b⟩ = ⟨p,Ax⟩ = ⟨A⊤p, x⟩ ≥ ⟨c, x⟩.
If equality holds at (x, p) then it is easy to check that x solves theprimal and p solves the dual and is a Lagrange multiplier of theprimal.
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Support vs tangent
• In general, v is neither convex nor smooth so we cannot usethe usual Fermat’s rule to get necessary optimality conditions.
• But this also get people to think and to realize a horizontalsupport from below is what really needed.
• This is one of the most important observation that leads tomuch of the modern development of non-smooth andvariational analysis.
• Related framework (e.g. subdifferential, conjugate) andtechniques (of handling them) need to be developed though.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Spaces
• Let X be finite dimensional complete normed vector spaces –Banach spaces.(Examples: RN , RV (Ω,F , P )).
• We denote Br(x) := y ∈ X | ∥y − x∥ ≤ r the closed ballaround x with radius r.
• We use X∗ to denote the dual of X which is also a Banachspace.
• The pairing between x∗ ∈ X∗ and x ∈ X is denoted by⟨x∗, x⟩.
• Let ≤K be the partial order induced by a closed cone K ⊂ X.
• The polar cone of K is defined byK+ := x∗ ∈ X∗ : ⟨x∗, x⟩ ≥ 0, ∀x ∈ K
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Functions
We consider extended valued functionf : X 7→ R ∪ +∞ andoften assume lower semicontinuous condition (lsc).
Lower semicontinuity
We say f is lsc at x ∈ X if
lim infy→x
f(y) ≥ f(x).
We say f is lsc if it is lsc at every point in X.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Functions
A convenient characterization of lsc is
Characterization of lower semicontinuity
An extended valued function f : X 7→ R ∪ +∞ is lsc iff itsepigraph
epi f := (x, r) ∈ X × R | f(x) ≤ r
is closed.
We will often explore the relationship between functions and theirrelated sets and vice versa.
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Sup functions
Lower semicontinuity of the sup
If fα : X 7→ R ∪ +∞ is a family of lsc functions then so issupα fα.
Proof:epi sup
αfα =
∩α
epi fα.
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
lsc functions related to sets
Let S ⊂ X be closed. Then, all the following functions are lsc• Indicator function:
ιS(x) =
0 x ∈ S
+∞ x ∈ S.
• The negative of the characteristic function:
χS(x) =
1 x ∈ S
0 x ∈ S.
• Distance function:
dS(x) = inf∥x− y∥ : y ∈ S
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
GoalVariational approachThe meaning of Lagrange MultipliersDualitySupport vs tangentNotation
Mappings
Similar concept also extend to mapping f : X 7→ Y when theimage space Y has the partial order ≤K generated by a closedconvex cone K ⊂ Y .
Lower semicontinuity
We say f is lsc at x ∈ X if
epi f := (x, y) ∈ X × Y | f(x) ≤K y
is closed.
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Constrained optimization problemLagrange multipliers
Constrained optimization problem
Let X,Y and Z be finite dimensional Banach spaces and let ≤K
be a partial order in Y induced by a closed convex cone K ⊂ Y .Consider constrained optimization problem
v(y, z) = inff(x) : g(x) ≤K y, h(x) = z, x ∈ C, (1)
where y ∈ Y , z ∈ Z, f, g are lsc, h is continuous and C ⊂ X isclosed. Denote S(y, z) the (possibly empty) solution set of (1).We are interested in (y, z) = (0, 0) but need to imbed the problemin a larger class of problems.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Constrained optimization problemLagrange multipliers
Lagrange multiplier
Lagrange multiplier
We say that λ is a Lagrange multiplier for problem v(0, 0) if(i) (nonnegativity) λ ∈ K+ × Z∗ and,(ii) (unconstrained optimum)
f(x) + ⟨λ, (g(x), h(x))⟩ ≥ v(0, 0).
We denote the set of Lagrange multipliers for problem v(0, 0) by
Λ(0, 0).
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
SubdifferentialGeometryExamples
Subdifferential
As noted above, v may not be differentiable. Subdifferential is asubstitute for the derivative.
Subdifferential
The subdifferential of a lower semi-continuous function ϕ atx ∈ dom ϕ is defined by
∂ϕ(x) = x∗ ∈ X∗ : ϕ(y)− ϕ(x) ≥ ⟨x∗, y − x⟩.
Subdifferential was initially defined for convex functions but worksfor general function as well.
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
SubdifferentialGeometryExamples
Geometry
Derivative has two main usage
1. Derivative act as a linear approximation.
2. In extreme problems, derivative =0 signals horizontal supportfrom below or cap from above.
The idea here is for minimization problem, support from belowonly is enough which leads to subdifferential and does not requiresmoothness.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
SubdifferentialGeometryExamples
Examples
• If f ∈ C1(X) then ∂f(x) = f ′(x).• ∂∥ · ∥(0) = B1(0) = [−1, 1].
• ∂(·)+(0) = [0, 1].
• ∂(·)−(0) = [−1, 0].
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
Lagrange multiplier theorem (LMT)
Characterization of Lagrange multiplier
Λ(0, 0) = −∂v(0, 0).
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
Part 1: Λ(0, 0) ⊇ −∂v(0, 0)
• Suppose that λ ∈ −∂v(0, 0).
• y → v(y, 0) is non-increasing w.r.t. ≤K .
• Thus, for any y ∈ K,
0 ≥ v(y, 0)− v(0, 0) ≥ ⟨−λ, (y, 0)⟩
so that λ ∈ K+ × Z∗.(Property (i))
• Property (ii) follows from, for all x ∈ C,
f(x) + ⟨λ, (g(x), h(x))⟩≥ v(g(x), h(x)) + ⟨λ, (g(x), h(x))⟩ ≥ v(0, 0)
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
Part 2: Λ(0, 0) ⊆ −∂v(0, 0)
Suppose λ satisfies conditions (i) and (ii). Then we have, for anyx ∈ C, g(x) ≤K y and h(x) = z,
f(x) + ⟨λ, (y, z)⟩ ≥ f(x) + ⟨λ, (g(x), h(x))⟩ ≥ v(0, 0).
Taking the infimum with constraints x ∈ C, g(x) ≤K y andh(x) = z, we arrive at
v(y, z) + ⟨λ, (y, z)⟩ ≥ v(0, 0).
Therefore, −λ ∈ ∂v(0, 0).
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
When S(0, 0) = ∅
Lagrange multiplier theorem with complementary slackness
−λ ∈ ∂v(0, 0) and x0 ∈ S(0, 0) if and only if
(i) (nonnegativity) λ ∈ K+ × Z∗;
(ii) (complementary slackness) ⟨λ, (g(x0), h(x0))⟩ = 0;
(iii) (unconstrained optimum) function
x 7→ f(x) + ⟨λ, (g(x), h(x))⟩
attains minimum at x0.
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
Proof
We have already seen that −λ ∈ ∂v(0, 0) is characterized by
f(x) + ⟨λ, (g(x), h(x))⟩ ≥ v(0, 0) (*)
When x0 ∈ S(0, 0), use v(0, 0) = v(g(x0), h(x0)) = f(x0) we get(ii) and (iii). Conversely (ii) and (iii) with (*) clearly implies thatx0 ∈ S(0, 0).
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
Need to look for a local version
If the Lagrange multiplier - as defined before - exists, will result ina global unconstrained problem.
• In general, this is not to be expected and a local version ofthe subdifferential is often important.
• Financial problems are often convex and the above definitionnaturally fits.
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
Hahn-Banach separating hyperplane theorem
Hahn-Banach separating hyperplane theorem
Let X be a Banach space and let C1 and C2 be convex subsets ofX. Suppose that
C2 ∩ intC1 = ∅. (2)
Then there exists λ ∈ X∗\0 such that, for all x ∈ C1 andy ∈ C2,
⟨λ, y⟩ ≥ ⟨λ, x⟩. (3)
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
Proof
In examples below we always assume the existence of LM.WOLG assume 0 ∈ intC1. Then the gauge function of C1,
γC1(x) := inft | x ∈ tC1
has the property that γC1(x) < 1 iff x ∈ int C1 anddom γC1 = X.Clearly
v(0) = minx,y
f(x) | h(x, y) = 0, y ∈ cl C2 ≥ 0, (4)
where f(x) = γC1(x)− 1, h(x, y) = x− y.
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
Proof
Let λ ∈ X∗ be a Lagrange multiplier. Then, for all x ∈ X andy ∈ clC2,
γC1(x)− 1 + ⟨λ, y − x⟩ ≥ v(0) ≥ 0. (5)
or
⟨λ, y⟩ ≥ ⟨λ, x⟩+ 1− γC1(x). (6)
Letting x = 0 we see that λ = 0. Noting that x ∈ C1 implies that1− γC1(x) ≥ 0 we derive the separation theorem.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
• The Lagrange multiplier plays the role of the separatinghyperplane.
• Existence of an optimal solution is neither expected norneeded.
• Derivative information of the function involved is not needed.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
Sandwich theorem
Sandwich theorem
Let X and Y be Banach spaces, let f and g be convex lscfunctions, and let A : X → Y be a linear mapping. Suppose thatf ≥ −g A and (CQ) 0 ∈ int(dom g −Adom f). Then thereexists an affine function α : X → R of the form
α(x) = ⟨A∗y∗, x⟩+ c
satisfyingf ≥ α ≥ −g A.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
Sketch of Proof
Consider minimization problem
v(y, z) = minf(x) + g(Ax+ y)− z (7)
= minf(x) + r : u−Ax = y, g(u)− r ≤ z.
f ≥ −g A implies that v(0, 0) ≥ 0.CQ implies a Lagrange multiplier of the form (y∗, µ) ∈ Y ∗ ×R+
exists.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
Sketch of Proof
By the definition of the multiplier, for x ∈ X, u ∈ Y and r ≥ g(u),
f(x) + r + ⟨y∗, u−Ax⟩+ µ(g(u)− r) ≥ v(0, 0) ≥ 0. (8)
Letting u = Ax′ and r = g(Ax′) in (8) we have, for x, x′ ∈ X,
f(x)− ⟨A∗y∗, x⟩ ≥ −g(Ax′)− ⟨A∗y∗, x′⟩.
Thus,
a := infx[f(x)− ⟨A∗y∗, x⟩] ≥ b := sup
x′[−g(Ax′)− ⟨A∗y∗, x′⟩].
Picking any c ∈ [a, b], the affine function α(x) = ⟨A∗y∗, x⟩+ cseparates f and −g A.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
Lagrange multiplier theoremProofComplementary slacknessExample: Separation theoremExample: Sandwich theorem
• The Lagrange multiplier induces the separating affine function.
• If A = I, g = ιcl C2 , f = γC1 − 1 we recover the separationtheorem.
• Thus, sandwich theorem provide more information (usefullater in FTAP).
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Enter convex analysis
• We have seen that the existence of LM is equivalent to
∂v(0, 0) = ∅.
• In general this is hard to verify.
• However, in the class of convex functions this is relatively easy.
Let’s get into the details.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Convex sets
Convex sets
We say C ⊂ X is convex if for any x, y ∈ C and λ ∈ [0, 1],
λx+ (1− λ)y ∈ C.
Note that the sum and difference of convex sets are convex and sois the intersection of any class of convex sets. However, the unionof two convex sets may not be convex.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Convex functions
Convex functions
We say f : X 7→ R ∪ +∞ is convex if, for any x, y ∈ X andλ ∈ [0, 1],
f(λx+ (1− λ)y) ≤ λf(x) + (1− λ)f(y).
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Nonemptyness of subdifferential
The most useful property of a convex function related to Lagrangemultipliers is
Nonemptyness of subdifferential
Let f : X 7→ R ∪ +∞ be a convex function. Then for anyx ∈ int dom f ,
∂f(x) = ∅.
Prove and strengthening this result will be one of the focuses inthe next lecture.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Epigraph characterizations
Epigraph characterizations
Function f : X 7→ R ∪ +∞ is convex iff epi f is a convex set inX × R.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Subdifferential characterizations
Subdifferential characterizations
Function f : X 7→ R ∪ +∞ is convex iff, for anyx∗ ∈ ∂f(x), y∗ ∈ ∂f(y),
⟨y∗ − x∗, y − x⟩ ≥ 0.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Cyclical monotonicity
In fact, we have the stronger result:
Cyclical monotonicity
Function f : X 7→ R ∪ +∞ is convex iff, for any m pairsx∗i ∈ ∂f(xi), i = 1, 2, . . . ,m,
⟨x∗1, x2 − x1⟩+ ⟨x∗2, x3 − x2⟩+ . . .+ ⟨x∗m, x1 − xm⟩ ≤ 0.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Cyclical monotonicity: Proof
Simply add the following inequalities:
f(x2)− f(x1) ≥ ⟨x∗1, x2 − x1⟩f(x3)− f(x2) ≥ ⟨x∗2, x3 − x2⟩
. . . . . .
f(x1)− f(xm) ≥ ⟨x∗m, x1 − xm⟩.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Corollary
Derivative characterizations
Function f : R 7→ R ∪ +∞ is convex iff f ′ is increasing orf ′′ ≥ 0.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Simple operations
Operations
Suppose functions f, g : X 7→ R ∪ +∞ are convex andh : R → R is increasing then the following functions are convex:
• f + g.
• af, a > 0.
• h f .
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Supremum
Supremum
If fα : X 7→ R ∪ +∞ are convex then so is supα fα.
Key of the Proof:
epi supα
fα =∩
epi fα.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Optimal value function
Optimal value function
Suppose that in problem v(y, z), f is convex, g is ≤K convex andh is affine and C is convex. Then the optimal value function v isconvex.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Proof
Consider (yi, zi) ∈ dom v, i = 1, 2. ∀ε > 0, ∃xiε feasible to theconstraint of v(yi, zi) s.t. f(xiε) < v(yi, zi) + ε, i = 1, 2.Now for any λ ∈ [0, 1], we have
f(λx1ε + (1− λ)x2ε) ≤ λf(x1ε) + (1− λ)f(x2ε) (9)
< λv(y1, z1) + (1− λ)v(y2, z2) + ε.
Since λx1ε +(1−λ)x2ε is feasible for v(λ(y1, z1)+ (1−λ)(y2, z2)),v(λ(y1, z1) + (1− λ)(y2, z2)) ≤ f(λx1ε + (1− λ)x2ε). Combiningwith (9) and letting ε → 0 we arrive at
v(λ(y1, z1) + (1− λ)(y2, z2)) ≤ λv(y1, z1) + (1− λ)v(y2, z2),
that is to say v is convex.Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Convex function related to convex sets
Let C be a closed convex set. Then the following functions areconvex:
• Distance function: dC . (View it as an optimal value function.)
• Support function: σC(x∗) = supx∈C⟨x∗, x⟩. (View it as
supremum of linear functions.)
• Indicator function: ιC .
• Gauge function: γC(x) = inft > 0 : x ∈ tC. (Immitate theproof of convexity of optimal value function.)
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Convex inf-convolution
Define the inf-convolution of f, g by
fg(x) = infy[f(x− y) + g(y)]
.
Convexity of inf-convolution
If f, g are convex then so is fg.
Proof: fg(x) = inff(u) + r : g(y)− r ≤ 0, u+ y = x.
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IntroductionConstrained optimization problem
SubdifferentialLagrange multiplier theoremConvex sets and functions
DefinitionsCharacterizationsOperations perserving convexityOptimal value functionSome useful convex functionsConvexification
Convexification
Often one need to generate a related convex function from one ora family of functions not necessarily convex. The following areseveral common methods suited for different applications:
• Integration of an increasing function.
• Convexification: Let f be an arbitrary function definef∗∗ = supg : g ≤ f and g convex.
• Regularized inf: Even if fα are all convex its inf is notnecessarily convex. But the regularized inf below is alwaysconvex:
ˆinfαfα = supg : g ≤ fα and g convex.
Peter Carr and Qiji Zhu CONVEX DUALITY IN MATH FINANCE