Revisiting Degeneracy, Strict Feasibility, Stability, in Linear Programming

Currently, the simplex method and the interior point method are indisputably the most popular algorithms for solving linear programs. Unlike general conic programs, linear programs, LPs, with a finite optimal value do not require strict feasibility in order to establish strong duality. Hence strict feasibility is seldom a concern, even though strict feasibility is equivalent … Read more

A Strengthened Barvinok-Pataki Bound on SDP Rank

The Barvinok-Pataki bound provides an upper bound on the rank of extreme points of a spectrahedron. This bound depends solely on the number of affine constraints of the problem, i.e., on the algebra of the problem. Specifically, the triangular number of the rank r is upper bounded by the number of affine constraints. We revisit … Read more

Robust Interior Point Method for Quantum Key Distribution Rate Computation

While the security proof method for quantum key distribution, QKD, based on the numerical key rate calculation problem, is powerful in principle, the practicality of the method is limited by computational resources and the efficiency of the underlying algorithm for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model … Read more