A regularized smoothing method for fully parameterized convex problems with applications to convex and nonconvex two-stage stochastic programming

We present an approach to regularize and approximate solution mappings of parametric convex optimization problems that combines interior penalty (log-barrier) solutions with Tikhonov regularization. Because the regularized mappings are single-valued and smooth under reasonable conditions, they can b...

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Bibliographic Details
Published in:Mathematical programming Vol. 189; no. 1-2; pp. 117 - 149
Main Authors: Borges, Pedro, Sagastizábal, Claudia, Solodov, Mikhail
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2021
Springer Nature B.V
Subjects:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Computational geometry
Continuity (mathematics)
Convexity
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Parameterization
Regularization
Smoothing
Stochastic programming
Theoretical
Upper bounds
Nonconvex stochastic programming
Lipschitz continuity of value functions
65K05
90C25
Two-stage stochastic programming
65K10
90C15
Interior penalty solutions
Tikhonov regularization
46N10
Smoothing techniques
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We present an approach to regularize and approximate solution mappings of parametric convex optimization problems that combines interior penalty (log-barrier) solutions with Tikhonov regularization. Because the regularized mappings are single-valued and smooth under reasonable conditions, they can be used to build a computationally practical smoothing for the associated optimal value function. The value function in question, while resulting from parameterized convex problems, need not be convex. One motivating application of interest is two-stage (possibly nonconvex) stochastic programming. We show that our approach, being computationally implementable, provides locally bounded upper bounds for the subdifferential of the value function of qualified convex problems. As a by-product of our development, we also recover that in the given setting the value function is locally Lipschitz continuous. Numerical experiments are presented for two-stage convex stochastic programming problems, comparing the approach with the bundle method for nonsmooth optimization.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-020-01582-2