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

Haodong Yu; Jie Sun

doi:10.3934/jimo.2019100

Article Contents

2021, Volume 17, Issue 1: 81-99. Doi: 10.3934/jimo.2019100

This issue Previous Article A chance-constrained stochastic model predictive control problem with disturbance feedback Next Article A diagonal PRP-type projection method for convex constrained nonlinear monotone equations

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

Haodong Yu^1, and
Jie Sun^2,3, ,

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
^* Corresponding author

Received: February 28, 2018

Revised: June 30, 2018

Early access: September 2019

Published: January 2021

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 90C15; 90C47.

Citation:

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Table 1. 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

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Table 2. combined distribution of $ h $

$ h_2 $	$ h_1 $
$ h_2 $	21000	25000
8000	0.25	0.25
10000	0.25	0.25

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Table 3. possible values of h

$ i $	1	2	3	4
$ h_{1i} $	21000	21000	25000	25000
$ h_{2i} $	8000	10000	8000	10000

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Table 4. production plans under various scenarios

$ i $	1	2	3	4
$ w_i $	7988	9969	8000	9969
$ p_i $	77	10	27	10

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Table 5. 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

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