Modern Nonconvex Nondifferentiable Optimization

Cui, Ying; Pang, Jong-Shi

doi:10.1137/1.9781611976748

Cited by 29 publications

(37 citation statements)

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“…One of the difficulties in applying ADMM to our application is the non-existence of the analytical form of part of the objective. Recently, difference-of-convex functions have been systematically studied in [12]. In particular interest to this paper, the recourse function of linearly bi-parameterized two-stage problems with quadratic recourse is shown to have convexity-concavity property [27].…”

Section: Nonsmooth Optimizationmentioning

confidence: 99%

A simplified nonsmooth nonconvex bundle method with applications to security-constrained ACOPF problems

Wang¹,

Petra²

2022

Preprint

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Section: Nonsmooth Optimizationmentioning

confidence: 99%

A simplified nonsmooth nonconvex bundle method with applications to security-constrained ACOPF problems

Wang¹,

Petra²

2022

Preprint

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“…Abstracting assumption (Z) in Section 2 for the functionals {Z } ∈[L] and assumption (Θ) for the functions θ rst/rlx , we make the following blanket assumptions on the functions in (14). Thus the assumptions below on c k (•, •; γ) are satisfied for c rlx k (•, •; γ) and c rst k (•, •; γ) that define the feasible sets in problems ( 15) and ( 16).…”

Section: The Expectation Constrained Spmentioning

confidence: 99%

“…whose feasible set we denote X(Z N ; γ). This empirical problem is the key computational workhorse for solving the expectation problem (14).…”

Section: Blanket Assumptions On (14)mentioning

confidence: 99%

“…Throughout this paper, the class of functions in point (d) above plays a central role, although the probability function P(Z (•, z) ≥ 0) may be discontinuous in general, and not of the dc kind even if it is continuous. By a result of [54], piecewise affine functions, which constitute the most basic class of difference-of-convex functions, can be expressed as the difference of two pointwise maxima of finitely many affine functions; see [14,Subsection 4.4.1] for details. Extending this basic result, a related development is the paper [51] which shows that every upper semicontinuous function is the limit of a hypo-convergent sequence of piecewise affine functions.…”

Section: Introductionmentioning

confidence: 99%

“…Extending this basic result, a related development is the paper [51] which shows that every upper semicontinuous function is the limit of a hypo-convergent sequence of piecewise affine functions. Compared with the linear or convex random functionals considered in the existing literature of CCPs, our overall modeling framework with nonconvex and nondifferentiable random functionals together with probabilities of conjunctive and/or disjunctive random functional inequalities accommodates broader applications in operations research and statistics such as piecewise statistical estimation models [14,34] and in optimal control such as optimal path planning models [6,10].…”

Section: Introductionmentioning

confidence: 99%

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Nonconvex and Nonsmooth Approaches for Affine Chance-Constrained Stochastic Programs

Cui¹,

Liu²,

Pang³

2022

Preprint

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Chance-constrained programs (CCPs) constitute a difficult class of stochastic programs due to its possible nondifferentiability and nonconvexity even with simple linear random functionals. Existing approaches for solving the CCPs mainly deal with convex random functionals within the probability function. In the present paper, we consider two generalizations of the class of chance constraints commonly studied in the literature; one generalization involves probabilities of disjunctive nonconvex functional events and the other generalization involves mixed-signed affine combinations of the resulting probabilities; together, we coin the term affine chance constraint (ACC) system for these generalized chance constraints. Our proposed treatment of such an ACC system involves the fusion of several individually known ideas: (a) parameterized upper and lower approximations of the indicator function in the expectation formulation of probability; (b) external (i.e., fixed) versus internal (i.e., sequential) sampling-based approximation of the expectation operator; (c) constraint penalization as relaxations of feasibility; and (d) convexification of nonconvexity and nondifferentiability via surrogation. The integration of these techniques for solving the affine chance-constrained stochastic program (ACC-SP) is the main contribution of this paper. Indeed, combined together, these ideas lead to several algorithmic strategies with various degrees of practicality and computational efforts for the nonconvex ACC-SP. In an external sampling scheme, a given sample batch (presumably large) is applied to a penalty formulation of a fixed-accuracy approximation of the chance constraints of the problem via their expectation formulation. This results in a sample average approximation scheme, whose almost-sure convergence under a directional derivative condition to a Clarke stationary solution of the expectation constrained-SP as the sample sizes tend to infinity is established. In contrast, sequential sampling, along with surrogation leads to a sequential convex programming based algorithm whose asymptotic convergence for fixed-and diminishing-accuracy approximations of the indicator function can be established under prescribed increments of the sample sizes.

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Second-Order Numerical Variational Analysis

Mordukhovich

2024

Springer Series in Operations Research and Financial Engineering

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Modern Nonconvex Nondifferentiable Optimization

Cited by 29 publications

References 0 publications

A simplified nonsmooth nonconvex bundle method with applications to security-constrained ACOPF problems

A simplified nonsmooth nonconvex bundle method with applications to security-constrained ACOPF problems

Nonconvex and Nonsmooth Approaches for Affine Chance-Constrained Stochastic Programs

Second-Order Numerical Variational Analysis

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