A Stochastic Primal-Dual Method for Optimization with Conditional Value at Risk Constraints

Avinash N. Madavan () and Subhonmesh Bose ()
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Avinash N. Madavan: University of Illinois at Urbana-Champaign
Subhonmesh Bose: University of Illinois at Urbana-Champaign

Journal of Optimization Theory and Applications, 2021, vol. 190, issue 2, No 4, 428-460

Abstract: Abstract We study a first-order primal-dual subgradient method to optimize risk-constrained risk-penalized optimization problems, where risk is modeled via the popular conditional value at risk (CVaR) measure. The algorithm processes independent and identically distributed samples from the underlying uncertainty in an online fashion and produces an $$\eta /\sqrt{K}$$ η / K -approximately feasible and $$\eta /\sqrt{K}$$ η / K -approximately optimal point within K iterations with constant step-size, where $$\eta $$ η increases with tunable risk-parameters of CVaR. We find optimized step sizes using our bounds and precisely characterize the computational cost of risk aversion as revealed by the growth in $$\eta $$ η . Our proposed algorithm makes a simple modification to a typical primal-dual stochastic subgradient algorithm. With this mild change, our analysis surprisingly obviates the need to impose a priori bounds or complex adaptive bounding schemes for dual variables to execute the algorithm as assumed in many prior works. We also draw interesting parallels in sample complexity with that for chance-constrained programs derived in the literature with a very different solution architecture.

Keywords: Primal-dual optimization; Stochastic optimization; Risk-sensitive optimization; Conditional value at risk; 90C15; 90C25; 90C30 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10957-021-01888-x

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