This paper proposes a simple facet enumeration algorithm for convex polytopes. The complexity of the algorithm is analyzed. The algorithm is implemented in Matlab. Some polytopes with known H-representations and V-representations are used as test examples. The preliminary numerical test shows the effectiveness and efficiency of the proposed algorithm. Due to the duality between the … Read more
We consider solving (convex) conic linear optimization problems, at the scale where matrix-factorization-free methods are attractive or necessary. The restarted primal-dual hybrid gradient method (rPDHG) — with heuristic enhancements and GPU implementation — has been very successful in solving huge-scale linear optimization problems (LPs). However, its application to more general conic convex optimization problems is … Read more
We present a Julia computational toolbox for linear optimization problems with joint affine chance constraints under elliptically symmetric uncertainty. The toolbox combines a spherical–radial oracle for estimating the joint probability and its gradient with three structured optimization methods: Proximal, Feasible, and Penalty. The oracle supports several elliptically symmetric distributions and is integrated with these methods … Read more
When linearizing binary quadratic programs, the most usual way is to replace bilinear products with additional variables constrained to take on consistent values in any feasible solution. In this setting, column generation is a principally desirable solution technique, for instance because the number of such additional linearization variables may be large while many of them … Read more
The integrality gap of the subtour elimination relaxation for the Traveling Salesman Problem is a longstanding open problem, epitomized by the \(\frac{4}{3}\)-conjecture. Understanding this gap requires a detailed analysis of the extreme points of the subtour elimination polytope. In this work, we introduce a new perspective for studying the integrality gap through what we call … Read more
We address the classical Dantzig – Fulkerson – Johnson formulation of the symmetric metric Traveling Salesman Problem and study the integrality gap of its linear relaxation, namely the Subtour Elimination Problem (SEP). This integrality gap is conjectured to be 4/3. We prove that, when solving a problem on n nodes, if the optimal SEP solution … Read more
This paper presents a Matlab implementation of an arc-search infeasible interior point algorithm for linear programming (LP), which has a proven polynomial bound of O(√nL), the best among all interior-point algorithms for LP. Software architecture and major functions are discussed. Its ease of use is described by a simple example. Crucial strategies are summarized. Quality … Read more
Dadush et al.\ (2024) recently developed a scaling-invariant layered least squares algorithm for linear programming whose complexity depends on the optimal condition measure $\bar{\chi}_A^*$. Their work builds on Vavasis and Ye’s (1996) algorithm whose running time depends only on the constraint matrix $A$ through the condition number $\bar{\chi}_A$. Monteiro-Tsuchiya (2003) defined the optimal condition number … Read more
We systematically investigate the potentials of improving the lower bound obtained with a linear relaxation of the Quadratic Unconstrained Binary Optimization problem by integrating curvature information from an accompanying quadratic convex underestimator via gradient inequalities. On the one hand, we exemplify to which extent this hybrid approach may provide a lower bound that is strictly … Read more
We study the convex hull obtained after deleting prescribed vertices from the binary cube. The analysis separates three regimes according to the number of deleted vertices. When this number is fixed, both the original-space facet count and the linear extension complexity remain linear in the ambient dimension, up to constants depending only on the number … Read more