Probabilistic Models for Linear Programming

We propose and investigate new probabilistic models for linear programming. In contrast to previous models, ours guarantee the existence of optimal solutions and are symmetric under duality. While in some respects our distributions are very special, there is sufficient flexibility to permit an arbitrary degree of primal and/or dual degeneracy, either just at the optimal solution or throughout the feasible region using null variables. Moreover, the precision of the distributions allows us to compute the probability that the feasible region is bounded as well as the distribution of the distance to a constraint hyperplane and that of the components of a vertex. Interest in these measures stems from Karmarkar's algorithm, and we also introduce a model for generating random linear programming problems on a simplex.