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Calmness in stochastic programming – exact penalization and sample approximation techniques

Martin Branda

Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics

Stochastic Programming and Approximation 21 February 2013, Prague

M.Branda (Charles University) SPaA 2013 1 / 55

Contents

1 Exact penalty method and calmness 2 Stochastic programming formulations 3 Relations between formulations 4 Sample approximations using Monte-Carlo techniques 5 References

M.Branda (Charles University) SPaA 2013 2 / 55

Exact penalty method and calmness

Contents

1 Exact penalty method and calmness 2 Stochastic programming formulations 3 Relations between formulations 4 Sample approximations using Monte-Carlo techniques 5 References

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Exact penalty method and calmness

Basis of the exact penalty method

I.I. Eremin (1966). Penalty method in convex programming. Soviet Math. Dokl. 8, 459–462 W.I. Zangwill (1967). Nonlinear programming via penalty

• functions. Management Sci. 13, 344–358

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Exact penalty method and calmness

Nonlinear programming problems and penalty problem

Nonlinear programming problem min

x∈X f (x)

s.t. gj(x) ≤ 0, j = 1, . . . , m, where f , gj : Rn → R, X ⊆ Rn. Corresponding penalty problem min

x∈X f (x) + N · α(x),

where α(x) =

m

• j=1

|gj(x)|p

+, p ∈ N.

M.Branda (Charles University) SPaA 2013 5 / 55

Exact penalty method and calmness

Nonlinear programming problems and penalty problem

Nonlinear programming problem min

x∈X f (x)

s.t. gj(x) ≤ 0, j = 1, . . . , m, where f , gj : Rn → R, X ⊆ Rn. Corresponding penalty problem min

x∈X f (x) + N · α(x),

where α(x) =

m

• j=1

|gj(x)|p

+, p ∈ N.

M.Branda (Charles University) SPaA 2013 5 / 55

Exact penalty method and calmness

Exterior penalty method

Bazaraa et al. (2006), Theorem 9.2.2 (also for equality constraints): Proposition Let f , gj be continuous, X = ∅, the underlying problem have a feasible

• solution. Assume that for each N there is a solution xN ∈ X of the penalty

problem and {xN} is contained in a compact subset of X. Then min

x∈X{f (x) : gj(x) ≤ 0, j = 1, . . . , m} = sup N≥0

θ(N) = lim

N→∞ θ(N),

where θ(N) = min

x∈X f (x) + N · α(x).

Furthermore, the limit of any convergent subsequence of {xN} is an

• ptimal solution to the original problem, and N · α(xN) → 0 as N → ∞.

Employed by Ermoliev et al. (2001), Branda (2012a, 2012b, 2013), Branda and Dupaˇ cov´ a (2012).

M.Branda (Charles University) SPaA 2013 6 / 55

Exact penalty method and calmness

Exact absolute value penalty method

Remark If for some N > 0 it holds α(xN) = 0, then xN is the optimal solution of the NLP, see Bazaraa et al. (2006), Corollary 9.2.2. It is a question, how to ensure this situations known as “exact penalization” in general. Bazaraa et al. (2006), Theorem 9.3.1 (also for equality constraints): Proposition Let (x∗, v∗) ∈ Rn × Rm

+ be a KKT point. Moreover, suppose that f , gj are

convex functions. Then for N ≥ maxj v∗

j , x∗ minimizes also the penalized

• bjective with p = 1 (L1 penalty).

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Exact penalty method and calmness

Exact absolute value penalty method

Remark If for some N > 0 it holds α(xN) = 0, then xN is the optimal solution of the NLP, see Bazaraa et al. (2006), Corollary 9.2.2. It is a question, how to ensure this situations known as “exact penalization” in general. Bazaraa et al. (2006), Theorem 9.3.1 (also for equality constraints): Proposition Let (x∗, v∗) ∈ Rn × Rm

+ be a KKT point. Moreover, suppose that f , gj are

convex functions. Then for N ≥ maxj v∗

j , x∗ minimizes also the penalized

• bjective with p = 1 (L1 penalty).

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Exact penalty method and calmness

Calmness

Calm problems Calm set-valued mappings (graphs) Calm functions

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Exact penalty method and calmness

A general mathematical program

A general relaxed mathematical program min f (x) s.t. F(x) + u ∈ Λ f : Rn → R, F : Rn → Rm, Λ ⊆ Rm closed, u ∈ Rm. Underlying problem for u = 0. We denote dΛ(x) = minx′∈Λ x − x′.

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Exact penalty method and calmness

Calm problems

Burke (1991a), Definition 1.1: Definition Let x∗ be feasible for the unperturbed problem. Then the problem is said to be calm at x∗ if there exist constant ˜ N ≥ 0 (modulus) and ǫ > 0 (radius) such that for all (x, u) ∈ Rn × Rm satisfying x ∈ Bǫ(x∗) and F(x) + u ∈ Λ, one has f (x) + ˜ N u ≥ f (x∗). Note that then x∗ is necessarily a local solution to the unperturbed problem.

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Exact penalty method and calmness

Exact penalization

Burke (1991a), Theorem 1.1: Proposition Let x∗ be feasible for the unperturbed problem, i.e. with u = 0. Then the unperturbed problem is calm at x∗ with modulus ˜ N and radius ǫ > 0 if and only if x∗ is a local minimum of the function f (x) + N · dΛ(F(x))

• ver the neighbourhood Bǫ(x∗) for all N ≥ ˜

N. Penalty function dΛ(x) = minx′∈Λ x − x′.

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Exact penalty method and calmness

Lipschitz-like properties of set-valued mappings

Set valued mapping (multifunction) Z : Y ⇒ X between metric spaces X and Y . Local Lipschitz property at y: ∃L, ε > 0 dZ(y1)(x) ≤ L · d(y1, y2), ∀x ∈ Z(y2), ∀y1, y2 ∈ Bε(y). Aubin property at y: ∃L, ε > 0 dZ(y1)(x) ≤ L · d(y1, y2), ∀x ∈ Z(y2) ∩ Bε(x), ∀y1, y2 ∈ Bε(y), where x ∈ Z(y). Local upper Lipschitz property at y: ∃L, ε > 0 dZ(y)(x) ≤ L · d(y, y), ∀x ∈ Z(y), ∀y ∈ Bε(y). Calmness at (y, x) ∈ Gph Z: ∃L, ε > 0 dZ(y)(x) ≤ L · d(y, y), ∀x ∈ Z(y) ∩ Bε(x), ∀y ∈ Bε(y).

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Exact penalty method and calmness

Lipschitz-like properties of set-valued mappings

Set valued mapping (multifunction) Z : Y ⇒ X between metric spaces X and Y . Local Lipschitz property at y: ∃L, ε > 0 dZ(y1)(x) ≤ L · d(y1, y2), ∀x ∈ Z(y2), ∀y1, y2 ∈ Bε(y). Aubin property at y: ∃L, ε > 0 dZ(y1)(x) ≤ L · d(y1, y2), ∀x ∈ Z(y2) ∩ Bε(x), ∀y1, y2 ∈ Bε(y), where x ∈ Z(y). Local upper Lipschitz property at y: ∃L, ε > 0 dZ(y)(x) ≤ L · d(y, y), ∀x ∈ Z(y), ∀y ∈ Bε(y). Calmness at (y, x) ∈ Gph Z: ∃L, ε > 0 dZ(y)(x) ≤ L · d(y, y), ∀x ∈ Z(y) ∩ Bε(x), ∀y ∈ Bε(y).

M.Branda (Charles University) SPaA 2013 12 / 55

Exact penalty method and calmness

Lipschitz-like properties of set-valued mappings

Set valued mapping (multifunction) Z : Y ⇒ X between metric spaces X and Y . Local Lipschitz property at y: ∃L, ε > 0 dZ(y1)(x) ≤ L · d(y1, y2), ∀x ∈ Z(y2), ∀y1, y2 ∈ Bε(y). Aubin property at y: ∃L, ε > 0 dZ(y1)(x) ≤ L · d(y1, y2), ∀x ∈ Z(y2) ∩ Bε(x), ∀y1, y2 ∈ Bε(y), where x ∈ Z(y). Local upper Lipschitz property at y: ∃L, ε > 0 dZ(y)(x) ≤ L · d(y, y), ∀x ∈ Z(y), ∀y ∈ Bε(y). Calmness at (y, x) ∈ Gph Z: ∃L, ε > 0 dZ(y)(x) ≤ L · d(y, y), ∀x ∈ Z(y) ∩ Bε(x), ∀y ∈ Bε(y).

M.Branda (Charles University) SPaA 2013 12 / 55

Exact penalty method and calmness

Lipschitz-like properties of set-valued mappings

Set valued mapping (multifunction) Z : Y ⇒ X between metric spaces X and Y . Local Lipschitz property at y: ∃L, ε > 0 dZ(y1)(x) ≤ L · d(y1, y2), ∀x ∈ Z(y2), ∀y1, y2 ∈ Bε(y). Aubin property at y: ∃L, ε > 0 dZ(y1)(x) ≤ L · d(y1, y2), ∀x ∈ Z(y2) ∩ Bε(x), ∀y1, y2 ∈ Bε(y), where x ∈ Z(y). Local upper Lipschitz property at y: ∃L, ε > 0 dZ(y)(x) ≤ L · d(y, y), ∀x ∈ Z(y), ∀y ∈ Bε(y). Calmness at (y, x) ∈ Gph Z: ∃L, ε > 0 dZ(y)(x) ≤ L · d(y, y), ∀x ∈ Z(y) ∩ Bε(x), ∀y ∈ Bε(y).

M.Branda (Charles University) SPaA 2013 12 / 55

Exact penalty method and calmness

Perturbed constraint set

Let F : X → Y be a continuous mapping, Λ ⊆ Y be a closed set. Denote by M(u) = {x ∈ X : F(x) + u ∈ Λ} the perturbation of the (original) constraint set M(0) = F −1(Λ).

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Exact penalty method and calmness

Set-valued mapping calmness implies problem calmness

Hoheisel et al. (2010), Proposition 3.5: Proposition Let x∗ be a local minimizer such that M is calm at (0, x∗). Then the

• riginal problem is calm at x∗.

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Exact penalty method and calmness

Sufficient conditions for calmness

global: M is calm at each point of its graph whenever this graph is polyhedral, cf. Rockafellar, Wets (2003), Example 9.57. local: Generalized Mangasarian-Fromowitz constraint qualification (GMFCQ) for continuously differentiable f and F: F ′(x∗)λ = 0 & λ ∈ NΛ(F(x∗)) ⇒ λ = 0. Limiting normal cone to Π at a is defined by NΠ(a) = lim supa′→a ˆ NΠ(a), where Π ⊆ Rp, a ∈ cl Π. Fr´ echet normal cone to Π at a ˆ NΠ(a) =

• ξ ∈ Rp : lim supa′→a

ξ, a′ − a a − a′

• further generalization of GMFCQ using kernels of coderivatives ...

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Exact penalty method and calmness

Clarke’s exact penalty method

Clarke (1983), Proposition 6.3.1, Proposition 6.4.2, Proposition 6.4.3: Exact L1 penalization of inequality constraints can be obtained under the following assumptions (which imply “calmness” of the problem): inequality constraints gj are convex, X is convex bounded, Slater condition (existence of a strictly feasible point),

• bjective function f is Lipschitz on X.

Employed by Meskarial et al. (2012), Sun et al. (2013).

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Exact penalty method and calmness M.Branda (Charles University) SPaA 2013 17 / 55

Stochastic programming formulations

Contents

1 Exact penalty method and calmness 2 Stochastic programming formulations 3 Relations between formulations 4 Sample approximations using Monte-Carlo techniques 5 References

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Stochastic programming formulations

Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇔ with penalty ⇔ chance constrained problem

• bjective

problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check

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Stochastic programming formulations

Optimization problem with a random factor

Program with a random factor ξ min

x∈X {f (x) : gi(x, ξ) ≤ 0, i = 1, . . . , k} ,

where gi, i = 0, . . . , k, are real functions on Rn × Rn′, X ⊆ Rn and ξ ∈ Rn′ is a realization of a n′-dimensional random vector defined on the probability space (Ω, F, P). If P is known, we can use chance constraints to deal with the random constraints...

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Stochastic programming formulations

Chance constrained problem (CCP)

Chance constrained problem ψǫ = minx∈X f (x), s.t. P

• g11(x, ξ) ≤ 0, . . . , g1k1(x, ξ) ≤ 0
• ≥

1 − ε1, . . . P

• gm1(x, ξ) ≤ 0, . . . , gmkm(x, ξ) ≤ 0
• ≥

1 − εm, with optimal solution xǫ, where we denoted ǫ = (ε1, . . . , εm) with levels εj ∈ (0, 1). The formulation covers the joint (k1 > 1 and m = 1) as well as the individual (kj = 1 and m > 1) chance constrained problems as special cases.

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Stochastic programming formulations

Solving chance constrained problems

In general, the feasible region is not convex even if the functions are convex, it is even not easy to check feasibility because it leads to computations of multivariate integrals. Hence, we will try to reformulate the chance constrained problem using penalty functions.

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Stochastic programming formulations

Penalty functions

Penalty functions ϑj : Rm → R+, j = 1, . . . , m, are continuous nondecreasing, equal to 0 on Rm

− and positive otherwise, e.g.

ϑ1,p(u) =

k

• i=1
• [ui]+p, p ∈ N

ϑ2(u) = max

1≤i≤k[ui]+,

= min

• t ≥ 0 : ui − t ≤ 0, i = 1, . . . , k}

where u ∈ Rm. Let pj denote the penalized constraints pj(x, ξ) = ϑj(gj1(x, ξ), . . . , gjkj(x, ξ)), ∀j.

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Stochastic programming formulations

Penalty function problems

Problem with penalties in the objective function ϕN = min

x∈X

• f (x) + N ·

m

• j=1

E[pj(x, ξ)]

• with an optimal solution xN.

Problem with generalized integrated chance constraints ϕICC

L

= min

x∈X

• f (x) : s.t. E[pj(x, ξ)] ≤ Lj, j = 1, . . . , m
• for some prescribed bounds Lj ≥ 0, L = (L1, . . . , Lm)′, with an optimal

solution xICC

L

(originally defined using ϑ2, cf. Klein Haneveld (1986)).

M.Branda (Charles University) SPaA 2013 24 / 55

Stochastic programming formulations

Penalty function problems

Problem with penalties in the objective function ϕN = min

x∈X

• f (x) + N ·

m

• j=1

E[pj(x, ξ)]

• with an optimal solution xN.

Problem with generalized integrated chance constraints ϕICC

L

= min

x∈X

• f (x) : s.t. E[pj(x, ξ)] ≤ Lj, j = 1, . . . , m
• for some prescribed bounds Lj ≥ 0, L = (L1, . . . , Lm)′, with an optimal

solution xICC

L

(originally defined using ϑ2, cf. Klein Haneveld (1986)).

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Stochastic programming formulations

History and applications of the penalty approach in SP

Pr´ ekopa (1973): CPP and penalization Dupaˇ cov´ a et al (1991): Water management (empirical study) Ermoliev et al (2000): Managing exposure to catastrophic risks (asymptotic equivalence with particular penalty) Branda and Dupaˇ cov´ a (2008, 2012): Contamination technique for CCP (asymptotic equivalence using general penalty functions) ˇ Zampachov´ a (2009): Beam design (reliability problem with partial differential equations - nonlinear - significant reduction of computational time) Branda (2009, 2012a): Value at Risk optimization with transaction costs and integer allocations (general penalty functions and several CC) M.B (2011): Blending problem (asymptotic equivalence with generalized integrated chance constraints)

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Relations between formulations

Contents

1 Exact penalty method and calmness 2 Stochastic programming formulations 3 Relations between formulations 4 Sample approximations using Monte-Carlo techniques 5 References

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Relations between formulations

Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇐ ⇒ with penalty ⇔ chance constrained problem

• bjective

problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check

M.Branda (Charles University) SPaA 2013 27 / 55

Relations between formulations

CCP under finite discrete distribution (FDD)

Let the distribution of random vector ξ be discrete with finite number of realizations ξs, s = 1, . . . , S with known probabilities 0 < ps ≤ 1, S

s=1 ps = 1. The chance constrained problem can be then formulated as

ϕCCP

ǫ

= minx∈X f (x) s.t. S

s=1 psI

• g1(x, ξs) ≤ 0, . . . , gk(x, ξs) ≤ 0
• ≥

1 − ε, (1) where I denotes the indicator function which is equal to one if the condition is satisfied, and 0 otherwise.

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Relations between formulations

Asymptotic equivalence and bounds

Branda (2012a), Branda, Dupaˇ cov´ a (2012): Under the following assumptions, the asymptotic equivalence of the CCP and the PPO problems can be shown: Compactness of the fixed set of feasible solutions. Continuity of the objective function, constraints and the probabilistic functions. Existence of integrable majorants. Existence of a permanently feasible solution. THEN for any prescribed ǫ ∈ (0, 1)m there always exists N large enough so that minimization of the penalty objective generates optimal solutions xN which also satisfy the chance constraints with the given ǫ. We can obtain asymtotic bounds on the optimal values.

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Relations between formulations

Asymptotic equivalence and bounds under finite discrete distributions (FDD)

Branda (2013a): Under the following assumptions, the asymptotic equivalence of the CCP and the PPO problems under finite discrete distributions can be shown: Compactness of the fixed set of feasible solutions. Continuity of the objective function and constraints: gi(·, ξs), i = 1, . . . , k are continuous for all s = 1, . . . , S. Existence of a permanently feasible solution: gi(x′, ξs) ≤ 0, i = 1, . . . , k for all s = 1, . . . , S for at least one x′ ∈ X.

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Relations between formulations

Perturbed problem

We consider the following perturbed version of the problem with a random factor under FDD: min

x∈X f (x)

s.t. (2) g1(x, ξs) ≤ u1s, . . . , gk(x, ξs) ≤ uks, s = 1, . . . , S. We define modified L1-norm for a vector u ∈ I RkS as u =

S

• s=1

ps

k

• i=1

|uis|, which is necessary for showing the asymptotic equivalence.

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Relations between formulations

Perturbed problem

We consider the following perturbed version of the problem with a random factor under FDD: min

x∈X f (x)

s.t. (2) g1(x, ξs) ≤ u1s, . . . , gk(x, ξs) ≤ uks, s = 1, . . . , S. We define modified L1-norm for a vector u ∈ I RkS as u =

S

• s=1

ps

k

• i=1

|uis|, which is necessary for showing the asymptotic equivalence.

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Relations between formulations

Definition Let x∗ be feasible for the unperturbed problem, i.e. (2) with uks = 0. Then the problem is said to be calm at x∗ if there exist constant ˜ N (modulus) and ǫ > 0 (radius) such that for all (x, u) ∈ I Rn × I RkS satisfying x ∈ Bǫ(x∗) and gi(x, ξs) ≤ uis, one has f (x) + ˜ N u ≥ f (x∗). Note that then x∗ is necessarily a local solution to the unperturbed problem.

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Relations between formulations

Proposition Let x∗ be feasible for the unperturbed problem, i.e. (2) with uks = 0, i = 1, . . . , k, s = 1, . . . , S. Then the unperturbed problem is calm at x∗ with modulus ˜ N ≥ 0 and radius ǫ > 0 if and only if x∗ is a local minimum

• f the function

f (x) + N

S

• s=1

ps

k

• i=1

|gi(x, ξs)|+

• ver Bǫ(x∗) for all N ≥ ˜

N. The theorems on asymptotic equivalence can be modified for local minimizers, i.e. X is replaced by X ∩ Bǫ(x∗) for some local optimal solution x∗ in the following theorems. Moreover, a special form of the penalty function is necessary: Φ(x, ξs) = k

i=1 |gi(x, ξs)|+.

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Relations between formulations

Proposition Let x∗ be feasible for the unperturbed problem, i.e. (2) with uks = 0, i = 1, . . . , k, s = 1, . . . , S. Then the unperturbed problem is calm at x∗ with modulus ˜ N ≥ 0 and radius ǫ > 0 if and only if x∗ is a local minimum

• f the function

f (x) + N

S

• s=1

ps

k

• i=1

|gi(x, ξs)|+

• ver Bǫ(x∗) for all N ≥ ˜

N. The theorems on asymptotic equivalence can be modified for local minimizers, i.e. X is replaced by X ∩ Bǫ(x∗) for some local optimal solution x∗ in the following theorems. Moreover, a special form of the penalty function is necessary: Φ(x, ξs) = k

i=1 |gi(x, ξs)|+.

M.Branda (Charles University) SPaA 2013 33 / 55

Relations between formulations

We consider the CCP and PFP problems and assume: (i) gi(x′, ξs) ≤ 0, i = 1, . . . , k for all s = 1, . . . , S for at least one x′ ∈ X. (ii) the corresponding unperturbed problem, i.e. (2) with uks = 0, is calm at its local optimal solutions modulus ˜ N ≥ 0 and radius ǫ > 0. For arbitrary γ ∈ (0, 1), N > 0 and ε ∈ (0, 1) put ε(x) =

S

• s=1

psI

• Φ(x, ξs) > 0
• ,

αN(x) = N

S

• s=1

psΦ(x, ξs), βε(x) = 1 εγ

S

• s=1

psΦ(x, ξs). Then for any prescribed ε ∈ (0, 1) there always exists N ≤ ˜ N large enough so that minimization of the penalty objective generates optimal solutions xN which also satisfy the probabilistic constraint with the given ε.

M.Branda (Charles University) SPaA 2013 34 / 55

Relations between formulations

We consider the CCP and PFP problems and assume: (i) gi(x′, ξs) ≤ 0, i = 1, . . . , k for all s = 1, . . . , S for at least one x′ ∈ X. (ii) the corresponding unperturbed problem, i.e. (2) with uks = 0, is calm at its local optimal solutions modulus ˜ N ≥ 0 and radius ǫ > 0. For arbitrary γ ∈ (0, 1), N > 0 and ε ∈ (0, 1) put ε(x) =

S

• s=1

psI

• Φ(x, ξs) > 0
• ,

αN(x) = N

S

• s=1

psΦ(x, ξs), βε(x) = 1 εγ

S

• s=1

psΦ(x, ξs). Then for any prescribed ε ∈ (0, 1) there always exists N ≤ ˜ N large enough so that minimization of the penalty objective generates optimal solutions xN which also satisfy the probabilistic constraint with the given ε.

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Relations between formulations

Asymptotic bounds on optimal values

Moreover, bounds on the local optimal value ψε of based on the local

• ptimal value ϕN and vice versa can be constructed:

ϕ1/εγ(xN) − βε(xN)(xCCP

ε(xN))

≤ ψε(N) ≤ ϕN − αN(xN), ψε(xN) + αN(xN) ≤ ϕN ≤ ψN−1/γ + βN−1/γ(xCCP

N−1/γ),

(3) with lim

N→˜ N−

αN(xN) = lim

N→˜ N−

ε(xN) = lim

ε→˜ ε+ λε(xCCP ε

) = 0, for any sequences of the optimal solutions xN and xCCP

ε

, where ˜ ε < mins ps.

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Relations between formulations

Asymptotic bounds on optimal values

Remark If we want to obtain exact convergence of the bounds on optimal values, we have to add the following condition. Since N−1/γ in the upper bound

• n ϕN has to converge to ˜

ε, we obtain the condition N ≥ max{˜ N, ˜ ε−γ}.

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Sample approximations using Monte-Carlo techniques

Contents

1 Exact penalty method and calmness 2 Stochastic programming formulations 3 Relations between formulations 4 Sample approximations using Monte-Carlo techniques 5 References

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Sample approximations using Monte-Carlo techniques

Program with a random factor Stochastic ւ ↓ ց 1. programming formulation Chance Problem Integrated constrained ⇔ with penalty ⇔ chance constrained problem

• bjective

problem (CCP) (PPO) (ICC) Sample 2. approximation ↓ ↓ ↓ (S.A.) S.A. CCP S.A. PPO S.A. ICC 3. Solution ↓ ↓ ↓ validation Reliability Reliability Reliability check check check

M.Branda (Charles University) SPaA 2013 38 / 55

Sample approximations using Monte-Carlo techniques

The SAA technique for expected value constrained problems

Wang, Ahmed (2008): one constraint, iid sampling, finite or infinite bounded set of feasible solutions and Lipschitz continuity Branda (2012b): several constraints, iid sampling, finite or infinite bounded set of feasible solutions and Lipschitz continuity (fixed or random modulus) Branda (2013c): several constraints, non-iid sampling, mixed-integer set of feasible solutions and H-calmness Examples: (generalized) integrated chance constraints, risk-shaping with CVaRs, diversification-consistent DEA tests with CVaR deviations and limited diversification etc., cf. Branda (2012c, 2013b, 2013c).

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Sample approximations using Monte-Carlo techniques

The SAA technique for expected value constrained problems

Wang, Ahmed (2008): one constraint, iid sampling, finite or infinite bounded set of feasible solutions and Lipschitz continuity Branda (2012b): several constraints, iid sampling, finite or infinite bounded set of feasible solutions and Lipschitz continuity (fixed or random modulus) Branda (2013c): several constraints, non-iid sampling, mixed-integer set of feasible solutions and H-calmness Examples: (generalized) integrated chance constraints, risk-shaping with CVaRs, diversification-consistent DEA tests with CVaR deviations and limited diversification etc., cf. Branda (2012c, 2013b, 2013c).

M.Branda (Charles University) SPaA 2013 39 / 55

Sample approximations using Monte-Carlo techniques

The SAA technique for expected value constrained problems

Wang, Ahmed (2008): one constraint, iid sampling, finite or infinite bounded set of feasible solutions and Lipschitz continuity Branda (2012b): several constraints, iid sampling, finite or infinite bounded set of feasible solutions and Lipschitz continuity (fixed or random modulus) Branda (2013c): several constraints, non-iid sampling, mixed-integer set of feasible solutions and H-calmness Examples: (generalized) integrated chance constraints, risk-shaping with CVaRs, diversification-consistent DEA tests with CVaR deviations and limited diversification etc., cf. Branda (2012c, 2013b, 2013c).

M.Branda (Charles University) SPaA 2013 39 / 55

Sample approximations using Monte-Carlo techniques

Mixed-integer problem with expected value constraints

Let X ⊆ I Rn, Y ⊆ I Zn′ and Z = {(x, y) ∈ X × Y : h1(x, y) ≤ 0, . . . , hk(x, y) ≤ 0} be the deterministic mixed-integer part of the set of feasible solutions with hj(x, y) : I Rn × I Rn′ → I

• R. Let ξ be a random vector on the probability

space (Ξ, A, P), pj(x, y, ξ), j = 1, . . . , m, be real functions on I Rn × I Rn′ × I Rn′′ measurable in ξ for all x ∈ X and y ∈ Y . We assume that the objective function f (x, y) : I Rn × I Rn′ → I R does not depend on the random vector.

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Sample approximations using Monte-Carlo techniques

Mixed-integer problem with expected value constraints

We denote the set of feasible solutions as ZL =

• (x, y) ∈ Z : pj(x, y) := I

E[pj(x, y, ξ)] ≤ Lj, j = 1, . . . , m

• for some prescribed bounds Lj ∈ I

R, L = (L1, . . . , Lm)′. We assume that the levels are chosen in such a way that the set of feasible solutions is nonempty and that the expectations are finite for all (x, y) ∈ Z. Then, the stochastic programming problem with the expected value constraints can be formulated as min

(x,y)∈ZL

f (x, y). (4)

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Sample approximations using Monte-Carlo techniques

Mixed-integer problem with expected value constraints

Let ξ1, . . . , ξS be a Monte-Carlo sample of the underlying distribution of the random vector ξ. We denote the set of feasible solutions of the sample-approximated problem as Z S

L =

• (x, y) ∈ Z : pS

j (x, y) := 1

S

S

• s=1

pj(x, y, ξs) ≤ Lj, j = 1, . . . , m

• .

The sample version of the problem with several expected value constraints is defined as min

(x,y)∈Z S

L

f (x, y). (5) where the levels Lj ∈ I R are allowed to be different from the original levels.

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Sample approximations using Monte-Carlo techniques

Calm functions

p(x, ξ) is said to be be H¨

• lder-calm (H-calm) at x with modulus

Mx(ξ) > 0, and order γx > 0, if there exist a measurable function Mx(ξ) : Ξ → R+ and positive numbers γx, δx such that |p(x, ξ) − p(x′, ξ)| ≤ Mx(ξ)

• x − x′

γx , for all ξ ∈ Ξ and x − x′ ≤ δx. p(x, ξ) is said to be be almost H¨

• lder-calm (almost H-calm) at x

with modulus Mx(ξ) > 0, and order γx > 0, if for any ε > 0, there exist a measurable function Mx(ξ) : Ξ → R+ and positive numbers γx, δx, C and an open set ∆x(ε) ⊂ Ξ such that P(ξ ∈ ∆x(ε)) ≤ Cε and |p(x, ξ) − p(x′, ξ)| ≤ Mx(ξ)

• x − x′

γx , for all ξ ∈ Ξ \ ∆x(ε) and x − x′ ≤ δx.

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Sample approximations using Monte-Carlo techniques

Calm functions

p(x, ξ) is said to be be H¨

• lder-calm (H-calm) at x with modulus

Mx(ξ) > 0, and order γx > 0, if there exist a measurable function Mx(ξ) : Ξ → R+ and positive numbers γx, δx such that |p(x, ξ) − p(x′, ξ)| ≤ Mx(ξ)

• x − x′

γx , for all ξ ∈ Ξ and x − x′ ≤ δx. p(x, ξ) is said to be be almost H¨

• lder-calm (almost H-calm) at x

with modulus Mx(ξ) > 0, and order γx > 0, if for any ε > 0, there exist a measurable function Mx(ξ) : Ξ → R+ and positive numbers γx, δx, C and an open set ∆x(ε) ⊂ Ξ such that P(ξ ∈ ∆x(ε)) ≤ Cε and |p(x, ξ) − p(x′, ξ)| ≤ Mx(ξ)

• x − x′

γx , for all ξ ∈ Ξ \ ∆x(ε) and x − x′ ≤ δx.

M.Branda (Charles University) SPaA 2013 43 / 55

Sample approximations using Monte-Carlo techniques

First we review a result which holds if n = 0 and m = 1, i.e. the set of feasible solutions is finite with one expected value constraint. Then, see the proof of Proposition 1 by Wang and Ahmed (2008) under iid sampling, it holds that P(Z S

L+τ ⊆ ZL)

≥ 1 − |Y | exp

• − S min

y∈Y Iy(τ)

• ,

(6) where |Y | denotes the cardinality of Y , Iy is the large deviation rate functions, i.e. the Fenchel dual to the logarithm of the finite moment generating function of the difference p(y, ξ) − p(y) which is defined as Iy(τ) = sup

t∈I R

• tτ − Ψy(t)
• ,

where Ψy(t) = ln I E

• et(p(y,ξ)−p(y))

, p(y) = I E[p(y, ξ)]. The estimate is based on Cram´ er’s large deviation theory, see, e.g., Dembo and Zeitouni (2010).

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Sample approximations using Monte-Carlo techniques

The estimate remains valid also for non-iid sampling if the G¨ artner-Ellis theorem is used and a condition on the convergence of logarithmic moment generating functions is added. For every y ∈ Y and t ∈ I R, denote ΨS

y (t) = ln I

E

• et(pS(y)−p(y))

. Then the assumption of G¨ artner-Ellis theorem holds if Ψy(t) = lim

S→∞

ΨS

y (St)

S (7) exists as an extended real number for all t ∈ I R and Ψy(t) < ∞ for t close to 0, c.f. Theorem 2.3.6 in Dembo and Zeitouni (2010). This conditions is trivially fulfilled for iid samples. Moreover, it can be verified for finite state Markov chains, see Chapter 3 in Dembo and Zeitouni (2010). See also Drew, Homem-de-Mello (2012) for Latin Hypercube Sampling.

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Sample approximations using Monte-Carlo techniques

Branda (2013c): Let (i) Y ⊆ I Rn′ be finite, and X ⊆ I Rn be bounded, i.e. D = sup{x − x′∞ : x, x′ ∈ X} be a finite diameter, (ii) p(x, y, ξ) be uniformly H-calm in x ∈ X for each y ∈ Y , moduli M(ξ) > 0, and order γ > 0: |p(x, y, ξ) − p(x′, y, ξ)| ≤ M(ξ)

• x − x′

γ , ∀x, x′ ∈ X, ∀ξ ∈ Ξ, with M = I E[M(ξ)] < ∞, (iii) the logarithmic moment generating functions Ψxy(t) of p(x, y, ξ) − p(x, y) be finite around 0 and Ψxy(t) = lim

S→∞

ΨS

xy(St)

S , for all t ∈ I R and for all (x, y) ∈ Z. (iv) the logarithmic moment generating function ΨM(t) of M(ξ) − M be finite around 0 and ΨM(t) = lim

S→∞

ΨS

M(St)

S , ∀t ∈ I R.

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Sample approximations using Monte-Carlo techniques

Branda (2013c): Let (i) Y ⊆ I Rn′ be finite, and X ⊆ I Rn be bounded, i.e. D = sup{x − x′∞ : x, x′ ∈ X} be a finite diameter, (ii) p(x, y, ξ) be uniformly H-calm in x ∈ X for each y ∈ Y , moduli M(ξ) > 0, and order γ > 0: |p(x, y, ξ) − p(x′, y, ξ)| ≤ M(ξ)

• x − x′

γ , ∀x, x′ ∈ X, ∀ξ ∈ Ξ, with M = I E[M(ξ)] < ∞, (iii) the logarithmic moment generating functions Ψxy(t) of p(x, y, ξ) − p(x, y) be finite around 0 and Ψxy(t) = lim

S→∞

ΨS

xy(St)

S , for all t ∈ I R and for all (x, y) ∈ Z. (iv) the logarithmic moment generating function ΨM(t) of M(ξ) − M be finite around 0 and ΨM(t) = lim

S→∞

ΨS

M(St)

S , ∀t ∈ I R.

M.Branda (Charles University) SPaA 2013 46 / 55

Sample approximations using Monte-Carlo techniques

Branda (2013c): Let (i) Y ⊆ I Rn′ be finite, and X ⊆ I Rn be bounded, i.e. D = sup{x − x′∞ : x, x′ ∈ X} be a finite diameter, (ii) p(x, y, ξ) be uniformly H-calm in x ∈ X for each y ∈ Y , moduli M(ξ) > 0, and order γ > 0: |p(x, y, ξ) − p(x′, y, ξ)| ≤ M(ξ)

• x − x′

γ , ∀x, x′ ∈ X, ∀ξ ∈ Ξ, with M = I E[M(ξ)] < ∞, (iii) the logarithmic moment generating functions Ψxy(t) of p(x, y, ξ) − p(x, y) be finite around 0 and Ψxy(t) = lim

S→∞

ΨS

xy(St)

S , for all t ∈ I R and for all (x, y) ∈ Z. (iv) the logarithmic moment generating function ΨM(t) of M(ξ) − M be finite around 0 and ΨM(t) = lim

S→∞

ΨS

M(St)

S , ∀t ∈ I R.

M.Branda (Charles University) SPaA 2013 46 / 55

Sample approximations using Monte-Carlo techniques

Then, for τ > 0 small, (a) the probability that the set of feasible solutions is contained in the relaxed sample-approximated set of feasible solutions increases exponentially with increasing sample size, and it holds P(ZL ⊆ Z S

L+τ) ≥ 1 −

• 1 + |Y |Dn(4M + τ)n/γ

τ n/γ

• exp
• − Sd(τ)
• ,

where σ2

xy

= Var[p(x, y, ξ) − p(x, y)], σ2

M

= Var[M(ξ) − M], υ =

• τ

4M + τ 1/γ , d(τ) = min

• min

(x,y)∈Zυ

τ 2 8σ2

xy

, τ 2 8σ2

M

• .

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Sample approximations using Monte-Carlo techniques

(b) we can get an estimate for the sample size which is necessary to ensure that the original feasibility set is contained in the relaxed sample-approximated feasibility set with a high probability, equal to 1 − δ, δ ∈ (0, 1): S ≥ 1 d(τ)

• ln 1

δ + ln

• 1 + |Y |Dn(4M + τ)n/γ

τ n/γ

• .

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Sample approximations using Monte-Carlo techniques

We can compare our estimate for γ = 1 with the result of Wang and Ahmed (2008) that is also valid for our problem with iid sampling, but it does not take into account the structure of the set of feasible solutions. In

• ur case, H-calmness (Lipschitz continuity) is necessary only with respect

to the continuous variables and the diameter of the set relates to the real bounded part only. To apply Wang and Ahmed (2008) estimate, Lipschitz modulus ˜ M(ξ) > 0 of function p(·, ·, ξ) is necessary, i.e. for ξ ∈ Ξ |p(x, y, ξ) − p(x′, y′, ξ)| ≤ ˜ M(ξ)

• (x, y) − (x′, y′)
• ∞ , ∀(x, y), (x′, y′) ∈ Z.

The Lipschitz constants are then incorporated into the rate estimate d(τ). Higher the variance of the random Lipschitz modulus is, lower the rate of convergence and higher the sample size estimate are obtained.

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Sample approximations using Monte-Carlo techniques

Moreover, the difference can be also identified in the estimate of the size

• f the finite set, which is necessary to approximate the set of feasible
• solutions. In our case, it is

|Y | D(4M + τ) τ n . In the proof of Proposition 2 by Wang and Ahmed (2008), it was estimated by ˜ D(4 ˜ M + τ) τ n+n′ , where the diameter ˜ D is computed with respect to the continuous and discrete part of the set of feasible solutions ˜ D = sup{

• (x, y) − (x′, y′)
• ∞ : (x, y), (x′, y′) ∈ Z}

and ˜ M = I E ˜ M(ξ).

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Sample approximations using Monte-Carlo techniques

Branda (2013c): Extension to several expected value constraints, comparison of the estimates on a problem with CVaR risk-shaping.

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References

Contents

1 Exact penalty method and calmness 2 Stochastic programming formulations 3 Relations between formulations 4 Sample approximations using Monte-Carlo techniques 5 References

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References

• M. Branda (2012a). Chance constrained problems: penalty reformulation and

performance of sample approximation technique. Kybernetika 48(1) 105–122.

• M. Branda (2012b). Sample approximation technique for mixed-integer

stochastic programming problems with several chance constraints. Operations Research Letters 40(3) 207–211.

• M. Branda (2012c). Stochastic programming problems with generalized

integrated chance constraints. Optimization 61(8), 949–968.

• M. Branda (2013a). On relations between chance constrained and penalty

function problems under discrete distributions. Accepted to Mathematical Methods of Operations Research. DOI: 10.1007/s00186-013-0428-7

• M. Branda (2013b). Diversification-consistent data envelopment analysis with

general deviation measures. European Journal of Operational Research 226(3), 626–635.

• M. Branda (2013c). Sample approximation technique for mixed-integer

stochastic programming problems with several expected value constraints. Accepted to Optimization Letters. DOI: 10.1007/s11590-013-0642-5

• M. Branda, J. Dupaˇ

cov´ a (2012). Approximations and contamination bounds for probabilistic programs. Annals of Operations Research 193(1) 3-19.

M.Branda (Charles University) SPaA 2013 53 / 55

References

J.V. Burke (1991). Calmness and exact penalization. SIAM Journal of Control and Optimization 29, 493–497. F.H. Clarke (1983). Optimization and Nonsmooth Analysis, Wiley, New York S.S. Drew, T. Homem-de-Mello (2012). Some Large Deviations Results for Latin Hypercube Sampling. Methodology and Computing in Applied Probability 14, 203–232.

• J. Dupaˇ

cov´ a, A. Gaivoronski, Z. Kos, T. Szantai (1991). Stochastic programming in water management: A case study and a comparison of solution techniques. European Journal of Operational Research 52, 28–44. Y.M. Ermoliev, T.Y. Ermolieva, G.J. Macdonald, V.I. Norkin (2000). Stochastic

• ptimization of insurance portfolios for managing exposure to catastrophic
• risks. Annals of Operations Research 99, 207–225.

W.K. Klein Haneveld (1986). Duality in stochastic linear and dynamic

• programming. Lecture Notes in economics and mathematical systems, vol. 274,

Springer.

• R. Henrion, J.V. Outrata (2005). Calmness of constraint systems with
• applications. Mathematical Programming, Ser. B 104, 437–464.

M.Branda (Charles University) SPaA 2013 54 / 55

References

• T. Hoheisel, Ch. Kanzowa, J.V. Outrata (2010). Exact penalty results for

mathematical programs with vanishing constraints. Nonlinear Analysis 72, 2514–2526.

• J. Luedtke, S. Ahmed (2008). A sample approximation approach for
• ptimization with probabilistic constraints. SIAM Journal on Optimization 19,

674–699.

• R. Meskarian, H. Xu, J. Fliege (2012). Numerical Methods for Stochastic

Programs with Second Order Dominance Constraints with Applications to Portfolio Optimization. European Journal of Operational Research 216, 376–385.

• A. Pr´

ekopa (1973). Contributions to the theory of stochastic programming. Mathematical Programming 4, 202–221.

• H. Sun, H. Xu, R. Meskarian, Y. Wang (2013). Exact Penalization, Level

Function Method and Modified Cutting-Plane Method for Stochastic Programs with Second Order Stochastic Dominance Constraints. To appear in SIAM Journal on Optimization.

• W. Wang, S. Ahmed (2008). Sample average approximation of expected value

constrained stochastic programs. Operations Research Letters 36, 515—519.

• E. ˇ

Zampachov´ a (2009). Approximations in stochastic optimization and their

• applications. Ph.D. thesis, Brno University of Technology.

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