Elsevier

Operations Research Letters

Volume 40, Issue 5, September 2012, Pages 325-328
Operations Research Letters

A second-order cone programming approach for linear programs with joint probabilistic constraints

https://doi.org/10.1016/j.orl.2012.06.008Get rights and content

Abstract

This paper deals with a special case of Linear Programs with joint Probabilistic Constraints (LPPC) with normally distributed coefficients and independent matrix vector rows. Through the piecewise linear approximation and the piecewise tangent approximation, we approximate the stochastic linear programs with two second-order cone programming (SOCP for short) problems. Furthermore, the optimal values of the two SOCP problems are a lower and upper bound of the original problem respectively. Finally, numerical experiments are given on randomly generated data.

Section snippets

Introductions

In this paper, we focus on the following linear program with joint probabilistic or chance constraints: mincTx(LPPC)s.t.Pr{TxD}1αxX where XR+n is a polyhedron, cRn, D=(D1,,DK)RK, T=[T1,,TK]T is a K×n random matrix, where Tk,k=1,,K, is a random vector in Rn, and α is a prespecified confidence parameter. The chance constraint in (1) specifies that all constraints are to be jointly satisfied with a given probability rather than satisfied individually [8], [9].

Chance-constrained

Normally distributed LPPC

We consider a special class of LLPC where Tk,k=1,,K are multivariate normally distributed independent row vectors with known mean vector μk=(μk1,,μkn) and covariance matrix Σk.

Since multivariate normally distributed vectors Tk,k=1,,K, are independent, we can derive a deterministic reformulation of the special case of LPPC as follows: mincTxs.t.μkTx+F1(pyk)Σk1/2xDk,k=1,,K(NLPPC)k=1Kyk=1yk0xX where p=1α and F1() is the inverse of F which is the standard normal cumulative

Approximation of NLPPC

In this section, we show that an approximated solution of problem (2) can be obtained by using a piecewise tangent and piecewise linear approximation techniques to approximate F1(pyk). These approximations give rise to two SOCP problems.

Numerical study

We consider a stochastic variant of the resource constrained shortest path problem (RCSP, for short) where the arc used resources are independent random variables whose distributions are normal with known mean vectors and covariance matrices. When the resource constraints are modeled by a joint chance constraint, it is easy to see that the relaxed RCSP is an NLPPC problem. The RCSP consists in finding the shortest path between two nodes s and t in a given graph under arcs resource constraints 

Acknowledgments

The research of Jianqiang Cheng was supported by a China Scholarship Council (CSC) grant.

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