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

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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 \(\eta /\sqrt{K}\)-approximately feasible and \(\eta /\sqrt{K}\)-approximately optimal point within K iterations with constant step-size, where \(\eta \) 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 \(\eta \). 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.

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Notes

  1. For the unconstrained problem, variance-reduced stochastic gradient descent methods can efficiently minimize the resulting finite sum as in [19, 38].

  2. requires that we sample \(\omega \) once more for the z-update.

  3. The integrality of K is ignored for notational convenience.

  4. Lemma 2.1 provides sufficient conditions for the existence of such a saddle point.

  5. \(\mathrm{CVaR}\) of any random variable can only vary between the mean and the maximum value that random variable can take.

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Acknowledgements

We thank Eilyan Bitar, Rayadurgam Srikant, Tamer Başar, and Stan Uryasev for helpful discussions. This work was partially supported by the National Science Foundation under grant no. CAREER-2048065, the International Institute of Carbon-Neutral Energy Research (\(\hbox {I}^2\)CNER), and the Power System Engineering Research Center (PSERC).

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  1. University of Illinois at Urbana-Champaign, Urbana, IL, USA

    Avinash N. Madavan & Subhonmesh Bose

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  1. Avinash N. Madavan
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Correspondence to Avinash N. Madavan.

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Communicated by Xiaolu Tan.

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Madavan, A.N., Bose, S. A Stochastic Primal-Dual Method for Optimization with Conditional Value at Risk Constraints. J Optim Theory Appl 190, 428–460 (2021). https://doi.org/10.1007/s10957-021-01888-x

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