Randomly generated polytopes for testing mathematical programming algorithms

Randomly generated polytopes are used frequently to test and compare algorithms for a variety of mathematical programming problems. These polytopes are constructed by generating linear inequality constraints with coefficients drawn independently from a distribution such as the uniform or the normal. It is noted that this class of 'random' polytopes has a special property: the angles between the hyperplanes, though dependent on the specific distribution used, tend to be equal when the dimension of the space increases. Obviously this structure of 'random' polytopes may bias test results

Preprocessing and Regularization for Degenerate Semidefinite Programs

This paper presents a backward stable preprocessing technique for (nearly) ill-posed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming rely on primal-dual interior-point, p-d i-p methods. These algorithms require the Slater constraint qualification for both the primal and dual problems. This assumption guarantees the existence of Lagrange multipliers, well-posedness of the problem, and stability of algorithms. However, there are many instances of SDPs where the Slater constraint qualification fails or nearly fails. Our backward stable preprocessing technique is based on applying the Borwein-Wolkowicz facial reduction process to find a finite number, k, of rank-revealing orthogonal rotations of the problem. After an appropriate truncation, this results in a smaller, well-posed, nearby problem that satisfies the Robinson constraint qualification, and one that can be solved by standard SDP solvers. Th