This talk deals with a Linear Programs with joint Probabilistic Constraints (LPPC). The aim is threefold. Firstly, we consider LPPC with normally distributed coefficients and independent matrix vector rows. We propose an approximation scheme based on two SOCP problems, and using piecewise linear and piecewise tangent approximations. We show that the optimal values of the two SOCP problems are a lower and upper bounds of the original problem respectively. Numerical experiments are given on randomly generated data. Secondly, we extend our approach to 0-1 LPPC. The constraint matrix vector rows are assumed to be independent, and the coefficients are normally distributed. Our main results show that this non-convex problem can be approximated by a convex completely positive problem. Moreover, we show that the optimal values of the latter converge to the optimal values of the original problem. Randomly generated instances highlight the efficiency of our approach. Finally, we investigate the problem of LPPC where the rows of the constraint matrix are dependent, the dependence is driven by a convenient Archimedean copula. Further, we assume the distribution of the constraint rows to be elliptically distributed, covering normal, t, or Laplace distributions. Under these and some additional conditions, we prove the convexity of the investigated set of feasible solutions. We also extend our approximation scheme aforementioned for this class of stochastic programming problems. Preliminary numerical results are given.
