# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021132
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## Relaxation schemes for the joint linear chance constraint based on probability inequalities

 1 School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

* Corresponding author: Shisen Liu

Received  November 2020 Revised  April 2021 Early access August 2021

This paper is concerned with the joint chance constraint for a system of linear inequalities. We discuss computationally tractble relaxations of this constraint based on various probability inequalities, including Chebyshev inequality, Petrov exponential inequalities, and others. Under the linear decision rule and additional assumptions about first and second order moments of the random vector, we establish several upper bounds for a single chance constraint. This approach is then extended to handle the joint linear constraint. It is shown that the relaxed constraints are second-order cone representable. Numerical test results are presented and the problem of how to choose proper probability inequalities is discussed.

Citation: Yanjun Wang, Shisen Liu. Relaxation schemes for the joint linear chance constraint based on probability inequalities. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021132
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##### References:
Changes of the optimal value in single chance constrained program
Changes of the optimal value in joint chance constrained program
Comparison of different probability inequalities under 0.05 confidence level}
 Inequality Type $\mathbb{E}$($\xi_1, \xi_2$) Var($\xi_1, \xi_2$) bound($\xi_1, \xi_2$) $x^*$ optimal value Chebyshev (0, 0) ($0.1^2, 0.2^2$) - $(0.2627, -0.5084)^T$ 0.7634 Petrov (0, 0) ($0.1^2, 0.2^2$) $[\pm0.3], [\pm0.3]$ $(0.2510, -0.5528)^T$ 0.7287 Hoeffding (0, 0) - $[\pm0.3], [\pm0.3]$ $(0.2040, -0.6678)^T$ 0.6584
 Inequality Type $\mathbb{E}$($\xi_1, \xi_2$) Var($\xi_1, \xi_2$) bound($\xi_1, \xi_2$) $x^*$ optimal value Chebyshev (0, 0) ($0.1^2, 0.2^2$) - $(0.2627, -0.5084)^T$ 0.7634 Petrov (0, 0) ($0.1^2, 0.2^2$) $[\pm0.3], [\pm0.3]$ $(0.2510, -0.5528)^T$ 0.7287 Hoeffding (0, 0) - $[\pm0.3], [\pm0.3]$ $(0.2040, -0.6678)^T$ 0.6584
Comparison of different probability inequalities under 0.05 confidence level}
 Inequality Type $\mathbb{E}$($\xi_1, \xi_2$) Var($\xi_1, \xi_2$) bound($\xi_1, \xi_2$) $x^*$ optimization value Chebyshev (0, 0) ($0.1^2, 0.2^2$) - $(0.0584, -0.1573)^T$ 1.3251 Petrov (0, 0) ($0.1^2, 0.2^2$) $[\pm0.3], [\pm0.3]$ $(-0.1472, -0.5511)^T$ 1.1078 Hoeffding (0, 0) - $[\pm0.3], [\pm0.3]$ $(-0.0824, -0.8908)^T$ 0.7459
 Inequality Type $\mathbb{E}$($\xi_1, \xi_2$) Var($\xi_1, \xi_2$) bound($\xi_1, \xi_2$) $x^*$ optimization value Chebyshev (0, 0) ($0.1^2, 0.2^2$) - $(0.0584, -0.1573)^T$ 1.3251 Petrov (0, 0) ($0.1^2, 0.2^2$) $[\pm0.3], [\pm0.3]$ $(-0.1472, -0.5511)^T$ 1.1078 Hoeffding (0, 0) - $[\pm0.3], [\pm0.3]$ $(-0.0824, -0.8908)^T$ 0.7459
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