Exact Matrix Completion via High-Rank Matrices in Sum-of-Squares Relaxations

We study exact matrix completion from partially available data with hidden connectivity patterns. Exact matrix completion was shown to be possible recently by Cosse and Demanet in 2021 with Lasserre’s relaxation using the trace of the variable matrix as the objective function with given data structured in a chain format. In this study, we introduce … Read more

Further Development in Convex Conic Reformulation of Geometric Nonconvex Conic Optimization Problems

\(\) A geometric nonconvex conic optimization problem (COP) was recently proposed by Kim, Kojima and Toh asa unified framework for convex conic reformulation of a class of quadratic optimization problems and polynomial optimization problems. The nonconvex COP minimizes a linear function over the intersection of a nonconvex cone K, a convex subcone J of the … Read more

Equivalent Sufficient Conditions for Global Optimality of Quadratically Constrained Quadratic Program

\(\) We study the equivalence of several well-known sufficient optimality conditions for a general quadratically constrained quadratic program (QCQP). The conditions are classified in two categories. The first one is for determining an optimal solution and the second one is for finding an optimal value. The first category of conditions includes the existence of a … Read more

The Largest Unsolved QAP Instance Tai256c Can Be Converted into A 256-dimensional Simple BQOP with A Single Cardinality Constraint

Tai256c is the largest unsolved quadratic assignment problem (QAP) instance in QAPLIB; a 1.48\% gap remains between the best known feasible objective value and lower bound of the unknown optimal value. This paper shows that the instance can be converted into a 256 dimensional binary quadratic optimization problem (BQOP) with a single cardinality constraint which … Read more

Solving Challenging Large Scale QAPs

We report our progress on the project for solving larger scale quadratic assignment problems (QAPs). Our main approach to solve large scale NP-hard combinatorial optimization problems such as QAPs is a parallel branch-and-bound method eciently implemented on a powerful computer system using the Ubiquity Generator (UG) framework that can utilize more than 100,000 cores. Lower … Read more