This paper presents the Lagrangian duality theory for mixed-integer semidefinite programming (MISDP). We derive the Lagrangian dual problem and prove that the resulting Lagrangian dual bound dominates the bound obtained from the continuous relaxation of the MISDP problem. We present a hierarchy of Lagrangian dual bounds by exploiting the theory of integer positive semidefinite matrices … Read more
We study the lift-and-project rank of the stable set polytope of graphs with respect to the Lovász–Schrijver SDP operator \( \text{LS}_+ \) applied to the fractional stable set polytope. In particular, we show that for every positive integer \( \ell \), the smallest possible graph with \( \text{LS}_+ \)-rank \( \ell \) contains \( 3\ell … Read more
We consider the problem of computing the optimal solution and objective of a linear program under linearly changing linear constraints. The problem studied is given by $\min c^t x \text{ s.t } Ax + \lambda Dx \leq b$ where $\lambda$ belongs to a set of predefined values $\Lambda$. Based on the information given by a … Read more
Inspired by the impact of the Goemans-Williamson algorithm on combinatorial optimization, we construct an analogous relax-then-sample strategy for low-rank optimization problems. First, for orthogonally constrained quadratic optimization problems, we derive a semidefinite relaxation and a randomized rounding scheme, which obtains provably near-optimal solutions, mimicking the blueprint from Goemans and Williamson for the Max-Cut problem. We … Read more
The restarted primal-dual hybrid gradient method (rPDHG) is a first-order method that has recently received significant attention for its computational effectiveness in solving linear program (LP) problems. Despite its impressive practical performance, the theoretical iteration bounds for rPDHG can be exponentially poor. To shrink this gap between theory and practice, we show that rPDHG achieves … Read more
We study proximity (resp. integrality gap), that is, the distance (resp. difference) between the optimal solutions (resp. optimal values) of convex integer programs (IP) and the optimal solutions (resp. optimal values) of their continuous relaxations. We show that these values can be upper bounded in terms of the recession cone of the feasible region of … Read more
A convex cone K is said to be homogeneous if its group of automorphisms acts transitively on its relative interior. Important examples of homogeneous cones include symmetric cones and cones of positive semidefinite (PSD) matrices that follow a sparsity pattern given by a homogeneous chordal graph. Our goal in this paper is to elucidate the … Read more
The traveling salesman problem is a widely studied classical combinatorial problem for which there are several integer linear formulations. In this work, we consider the Miller-Tucker-Zemlin (MTZ), Desrochers-Laporte (DL) and Single Commodity Flow (SCF) formulations. We argue that the choice of some parameters of these formulations is arbitrary and, therefore, there are families of formulations … Read more
In this paper, we reveal a new characterization of the super-efficiency model for Data Envelopment Analysis (DEA). In DEA, the efficiency of each decision making unit (DMU) is measured by the ratio the weighted sum of outputs divided by the weighted sum of inputs.In order to measure efficiency of a DMU, ${\rm DMU}_j$, say, in CCR … Read more
The primal-dual interior point method (IPM) is widely regarded as the most efficient IPM variant for linear optimization. In this paper, we demonstrate that the improved stability of the pure primal IPM can allow speedups relative to a primal-dual solver, particularly as the IPM approaches convergence. The stability of the primal scaling matrix makes it … Read more