Solving Nonsmooth and Nonconvex Compound Stochastic Programs with Applications to Risk Measure Minimization

Liu, Junyi; Cui, Ying; Pang, Jong-Shi

doi:10.1287/moor.2021.1247

Cited by 8 publications

(4 citation statements)

References 39 publications

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“…The approach of [49] iteratively constructs convex second-order models of the objective and employs a line search for globalization; in [50] similar models of the objective are constructed, but a trust region is employed for globalization, yielding convergence of a subsequence to a Dini stationary point. Difference-of-convex approaches for nonsmooth composite objectives have also been studied in recent works [15,41].…”

Section: Literature Reviewmentioning

confidence: 99%

Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

Larson,

Menickelly,

Zhou

2020

Preprint

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We propose a manifold sampling algorithm for minimizing a nonsmooth composition f = h • F , where we assume h is nonsmooth and may be inexpensively computed in closed form and F is smooth but its Jacobian may not be available. We additionally assume that the composition h • F defines a continuous selection. Manifold sampling algorithms can be classified as model-based derivative-free methods, in that models of F are combined with particularly sampled information about h to yield local models for use within a trust-region framework. We demonstrate that cluster points of the sequence of iterates generated by the manifold sampling algorithm are Clarke stationary. We consider the tractability of three particular subproblems generated by the manifold sampling algorithm and the extent to which inexact solutions to these subproblems may be tolerated. Numerical results demonstrate that manifold sampling as a derivative-free algorithm is competitive with state-of-the-art algorithms for nonsmooth optimization that utilize first-order information about f .

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Section: Literature Reviewmentioning

confidence: 99%

Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

Larson,

Menickelly,

Zhou

2020

Preprint

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“…These properties make it useful in a variety of applications. For instance, bPOE has recently been used to describe risk faced by financial institutions by Norton (2019), for solving financial optimization problems by Liu et al (2022), and in a variety of engineering problems by Kouri and Shapiro (2018), Chaudhuri et al (2022), andZrazhevsky et al (2023).…”

Section: Introductionmentioning

confidence: 99%

Derivative of Reduced Cumulative Distribution Function and Applications

Maritato,

Uryasev

2023

JRFM

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The reduced cumulative distribution function (rCDF) is the maximal lower bound for the cumulative distribution function (CDF). It is equivalent to the inverse of the conditional value at risk (CVaR), or one minus the buffered probability of exceedance (bPOE). This paper introduces the reduced probability density function (rPDF), the derivative of rCDF. We first explore the relation between rCDF and other risk measures. Then we describe three means of calculating rPDF for a distribution, depending on what is known about the distribution. For functions with a closed-form formula for bPOE, we derive closed-form formulae for rPDF. Further, we describe formulae for rPDF based on a numerical bPOE when there is a closed-form formula for CVaR but no closed-form formula for bPOE. Finally, we give a method for numerically calculating rPDF for an empirical distribution, and compare the results with other methods for known distributions. We conducted a case study and used rPDF for sensitivity analysis and parameter estimation with a method similar to the maximum likelihood method.

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“…While computability remains a main concern, the actual computation of the solution is not for the external sampling scheme. In contrast, in an internal, or sequential sampling method [4,26,33,63], samples are gradually accumulated as the iteration proceeds in order to potentially improve the approximation of the expectation operator. By taking advantage of the early stage of the algorithm, the computational cost of subsequent iterations can be reduced.…”

Section: Introductionmentioning

confidence: 99%

“…Subgradient-based majorization: This approach has its origin from the early days of deterministic difference-of-convex (dc) programming [32]; it is most recently extended in the study of compound stochastic programs with multiple expectation functions [33]. The approach provides a generalization to the choice of a single index in defining the sets A g (x, z) and/or A h (x, z); it has the computational advantage of avoiding the pointwise-minimum surrogation when these sets are not singletons.…”

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confidence: 99%

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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Solving Nonsmooth and Nonconvex Compound Stochastic Programs with Applications to Risk Measure Minimization

Cited by 8 publications

References 39 publications

Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

Derivative of Reduced Cumulative Distribution Function and Applications

Nonconvex and Nonsmooth Approaches for Affine Chance-Constrained Stochastic Programs

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