Solving Stochastic Optimization with Expectation Constraints Efficiently by a Stochastic Augmented Lagrangian-Type Algorithm

Liwei Zhang (), Yule Zhang (), Jia Wu () and Xiantao Xiao ()
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Liwei Zhang: School of Mathematical Sciences, Dalian University of Technology, 116024 Dalian, China
Yule Zhang: School of Science, Dalian Maritime University, 116026 Dalian, China
Jia Wu: School of Mathematical Sciences, Dalian University of Technology, 116024 Dalian, China
Xiantao Xiao: School of Mathematical Sciences, Dalian University of Technology, 116024 Dalian, China

INFORMS Journal on Computing, 2022, vol. 34, issue 6, 2989-3006

Abstract: This paper considers the problem of minimizing a convex expectation function with a set of inequality convex expectation constraints. We propose a stochastic augmented Lagrangian-type algorithm—namely, the stochastic linearized proximal method of multipliers—to solve this convex stochastic optimization problem. This algorithm can be roughly viewed as a hybrid of stochastic approximation and the traditional proximal method of multipliers. Under mild conditions, we show that this algorithm exhibits O ( K − 1 / 2 ) expected convergence rates for both objective reduction and constraint violation if parameters in the algorithm are properly chosen, where K denotes the number of iterations. Moreover, we show that, with high probability, the algorithm has a O ( log ( K ) K − 1 / 2 ) constraint violation bound and O ( log 3 / 2 ( K ) K − 1 / 2 ) objective bound. Numerical results demonstrate that the proposed algorithm is efficient.

Keywords: stochastic approximation; linearized proximal method of multipliers; expectation constrained stochastic program; expected convergence rate; high-probability bound (search for similar items in EconPapers)
Date: 2022
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