Algorithms for stochastic optimization with function or expectation constraints

This paper considers the problem of minimizing an expectation function over a closed convex set, coupled with a function or expectation constraint on either decision variables or problem parameters. We first present a new stochastic approximation (SA) type algorithm, namely the cooperative SA (CSA),...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 76; no. 2; pp. 461 - 498
Main Authors: Lan, Guanghui, Zhou, Zhiqiang
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2020
Springer Nature B.V
Subjects:
Algorithms
Approximation
Constraints
Convergence
Convex and Discrete Geometry
Management Science
Mathematical analysis
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Parameters
Statistics
90C25
90C22
90C06
49M37
Stochastic optimization
Convex programming
Subgradient method
Complexity
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:This paper considers the problem of minimizing an expectation function over a closed convex set, coupled with a function or expectation constraint on either decision variables or problem parameters. We first present a new stochastic approximation (SA) type algorithm, namely the cooperative SA (CSA), to handle problems with the constraint on devision variables. We show that this algorithm exhibits the optimal O ( 1 / ϵ 2 ) rate of convergence, in terms of both optimality gap and constraint violation, when the objective and constraint functions are generally convex, where ϵ denotes the optimality gap and infeasibility. Moreover, we show that this rate of convergence can be improved to O ( 1 / ϵ ) if the objective and constraint functions are strongly convex. We then present a variant of CSA, namely the cooperative stochastic parameter approximation (CSPA) algorithm, to deal with the situation when the constraint is defined over problem parameters and show that it exhibits similar optimal rate of convergence to CSA. It is worth noting that CSA and CSPA are primal methods which do not require the iterations on the dual space and/or the estimation on the size of the dual variables. To the best of our knowledge, this is the first time that such optimal SA methods for solving function or expectation constrained stochastic optimization are presented in the literature.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-020-00179-x