We present a primal-dual majorization-minimization method for solving large-scale linear programs. A smooth barrier augmented Lagrangian (SBAL) function with strict convexity for the dual linear program is derived. The majorization-minimization approach is naturally introduced to develop the smoothness and convexity of the SBAL function. Our method only depends on a factorization of the constant matrix … Read more
The hyperbolic augmented Lagrangian algorithm (HALA) is introduced in the area of continuous optimization for solving nonlinear programming problems. Under mild assumptions, such as: convexity, Slater’s qualification and differentiability, the convergence of the proposed algorithm is proved. We also study the duality theory for the case of the hyperbolic augmented Lagrangian function. Finally, in order … Read more
Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well-studied for various classes of graphs. When it comes to random graphs, only the classical binomial random graph \(G_{n,p}\) has been analysed and shown to have largest independent sets of size \(\Theta(\log{n})\) w.h.p. This classical … Read more
We study semidefinite programming (SDP) relaxations for the NP-hard problem of globally optimizing a quadratic function over the Stiefel manifold. We introduce a strengthened relaxation based on two recent ideas in the literature: (i) a tailored SDP for objectives with a block-diagonal Hessian; (ii) and the use of the Kronecker matrix product to construct SDP relaxations. Using synthetic instances on … Read more
Multiobjective discrete optimization (MODO) techniques, including weight space decomposition, have received increasing attention in the last decade. The primary weight space decomposition technique in the literature is defined for the weighted sum utility function, through which sets of weights are assigned to a subset of the nondominated set. Recent work has begun to study the … Read more
Recent years have seen significant advances in quantum/quantum-inspired technologies capable of approximately searching for the ground state of Ising spin Hamiltonians. The promise of leveraging such technologies to accelerate the solution of difficult optimization problems has spurred an increased interest in exploring methods to integrate Ising problems as part of their solution process, with existing … Read more
The question of finding the largest integer contained between two given lists of integers is trivial when integer ordering is interpreted in its usual way. We propose a nontrivial variant wherein each ordering comparison is performed after integers have been mapped under some bijection, and analyze the computational complexity of our combinatorial problem under a … Read more
A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank one, where the rank is computed with respect the underlying hyperbolic polynomial. This is a natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of … Read more
We present multilinear and mixed-integer multilinear programs to find a Nash equilibrium in multi-player strategic-form games. We compare the formulations to common algorithms in Gambit, and conclude that a multilinear feasibility program finds a Nash equilibrium faster than any of the methods we compare it to, including the quantal response equilibrium method, which is recommended … Read more
Recent studies employ collections of binary decision diagrams (BDDs) to solve combinatorial optimization problems. This paper focuses on the problem of optimally aligning two BDDs, i.e., transforming them to enforce a common order of variables while keeping the total size of the diagrams as small as possible. We address this problem, which is known to … Read more