An approximation algorithm for the partial covering 0-1 integer program

The partial covering 0-1 integer program (PCIP) is a relaxed problem of the covering 0-1 integer program (CIP) such that some fixed number of constraints may not be satisfied. This type of relaxation is also discussed in the partial set multi-cover problem (PSMCP) and the partial set cover problem (PSCP). In this paper, we propose … Read more

Geometric descent method for convex composite minimization

In this paper, we extend the geometric descent method recently proposed by Bubeck, Lee and Singh to tackle nonsmooth and strongly convex composite problems. We prove that our proposed algorithm, dubbed geometric proximal gradient method (GeoPG), converges with a linear rate $(1-1/\sqrt{\kappa})$ and thus achieves the optimal rate among first-order methods, where $\kappa$ is the … Read more

The (minimum) rank of typical fooling-set matrices

A fooling-set matrix is a square matrix with nonzero diagonal, but at least one in every pair of diagonally opposite entries is 0. Dietzfelbinger et al. ’96 proved that the rank of such a matrix is at least $\sqrt n$, for a matrix of order $n$. We ask for the typical minimum rank of a … Read more

Fooling Sets and the Spanning Tree Polytope

In the study of extensions of polytopes of combinatorial optimization problems, a notorious open question is that for the size of the smallest extended formulation of the Minimum Spanning Tree problem on a complete graph with n nodes. The best known lower bound is \Omega(n^2), the best known upper bound is O(n^3). In this note … Read more

Complexity Analysis of a Trust Funnel Algorithm for Equality Constrained Optimization

A method is proposed for solving equality constrained nonlinear optimization problems involving twice continuously differentiable functions. The method employs a trust funnel approach consisting of two phases: a first phase to locate an $\epsilon$-feasible point and a second phase to seek optimality while maintaining at least $\epsilon$-feasibility. A two-phase approach of this kind based on … Read more

The Selective Traveling Salesman Problem with Draught Limits

This paper introduces the Selective Traveling Salesman Problem with Draught Limits, an extension of Traveling Salesman Problem with Draught Limits, wherein the goal is to design maximal profit tour respecting draught limit constraints of the visited ports. We propose a mixed integer linear programming formulation for this problem. The proposed mixed integer program is used … Read more