# American Institute of Mathematical Sciences

January  2021, 17(1): 81-99. doi: 10.3934/jimo.2019100

## Robust stochastic optimization with convex risk measures: A discretized subgradient scheme

 1 School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Fiance, China 2 School of Mathematical Science, Chongqing Normal University, China 3 Faculty of Science and Engineering, Curtin University, Perth, Australia

* Corresponding author

Received  March 2018 Revised  July 2018 Published  January 2021 Early access  September 2019

Fund Project: This work is partially supported by Grants 11401384, 11671029, 71631008 and B16002 of National Natural Science Foundation of China and by Grant DP160102819 of Australian Research Council

We study the distributionally robust stochastic optimization problem within a general framework of risk measures, in which the ambiguity set is described by a spectrum of practically used probability distribution constraints such as bounds on mean-deviation and entropic value-at-risk. We show that a subgradient of the objective function can be obtained by solving a finite-dimensional optimization problem, which facilitates subgradient-type algorithms for solving the robust stochastic optimization problem. We develop an algorithm for two-stage robust stochastic programming with conditional value at risk measure. A numerical example is presented to show the effectiveness of the proposed method.

Citation: Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100
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##### References:
parameters of the test problem
 $w$(Wrench) $p$(Plier) $x$: Steel A(lbs.) 1.5 1 $y$: Steel B(lbs.) 1 2 Molding Machine (hours) 1 1 Assembly Machine (hours) .3 .5 Contribution to Earnings (＄/1000 units) 130 100
 $w$(Wrench) $p$(Plier) $x$: Steel A(lbs.) 1.5 1 $y$: Steel B(lbs.) 1 2 Molding Machine (hours) 1 1 Assembly Machine (hours) .3 .5 Contribution to Earnings (＄/1000 units) 130 100
combined distribution of $h$
 $h_2$ $h_1$ 21000 25000 8000 0.25 0.25 10000 0.25 0.25
 $h_2$ $h_1$ 21000 25000 8000 0.25 0.25 10000 0.25 0.25
possible values of h
 $i$ 1 2 3 4 $h_{1i}$ 21000 21000 25000 25000 $h_{2i}$ 8000 10000 8000 10000
 $i$ 1 2 3 4 $h_{1i}$ 21000 21000 25000 25000 $h_{2i}$ 8000 10000 8000 10000
production plans under various scenarios
 $i$ 1 2 3 4 $w_i$ 7988 9969 8000 9969 $p_i$ 77 10 27 10
 $i$ 1 2 3 4 $w_i$ 7988 9969 8000 9969 $p_i$ 77 10 27 10
worst-case distribution of $h$
 Pro 0.5 0.0134 0.0539 0.0006 0.4321 $h_1$ 22355 22216 21008 22239 24021 $h_2$ 8000 10000 10000 10000 10000
 Pro 0.5 0.0134 0.0539 0.0006 0.4321 $h_1$ 22355 22216 21008 22239 24021 $h_2$ 8000 10000 10000 10000 10000
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