A second-order cone programming approach for linear programs with joint probabilistic constraints
Section snippets
Introductions
In this paper, we focus on the following linear program with joint probabilistic or chance constraints: where is a polyhedron, , , is a random matrix, where , is a random vector in , and 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 are multivariate normally distributed independent row vectors with known mean vector and covariance matrix .
Since multivariate normally distributed vectors , are independent, we can derive a deterministic reformulation of the special case of LPPC as follows: where and is the inverse of 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 . 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 and 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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