Strong duality of a conic optimization problem with a single hyperplane and two cone constraints

Strong (Lagrangian) duality of general conic optimization problems (COPs) has long been studied and its profound and complicated results appear in different forms in a wide range of literatures. As a result, characterizing the known and unknown results can sometimes be difficult. The aim of this article is to provide a unified and geometric view … Read more

Generating Cutting Inequalities Successively for Quadratic Optimization Problems in Binary Variables

We propose a successive generation of cutting inequalities for binary quadratic optimization problems. Multiple cutting inequalities are successively generated for the convex hull of the set of the optimal solutions $\subset \{0, 1\}^n$, while the standard cutting inequalities are used for the convex hull of the feasible region. An arbitrary linear inequality with integer coefficients … Read more

MatQapNB User Guide: A branch-and-bound program for QAPs in Matlab with the Newton-Bracketing method

MatQapNB is a MATLAB toolbox that implements a parallel branch-and-bound method using NewtBracket (the Newton bracketing method [4]) for its lower bounding procedure. It can solve small to medium scale Quadratic Assignment Problem (QAP) instances with dimension up to 30. MatQapNB was used in the numerical experiments on QAPs in the recent article “Solving challenging … 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

User manual of NewtBracket: “A Newton-Bracketing method for a simple conic optimization problem” with applications to QOPs in binary variables

We describe the Matlab package NewtBracket for solving a simple conic optimization problem that minimizes a linear objective function subject to a single linear equality constraint and a convex cone constraint. The problem is converted into the problem of finding the largest zero $y^*$ of a continuously differentiable (except at $y^*$) convex function $g : … Read more

A Newton-bracketing method for a simple conic optimization problem

For the Lagrangian-DNN relaxation of quadratic optimization problems (QOPs), we propose a Newton-bracketing method to improve the performance of the bisection-projection method implemented in BBCPOP [to appear in ACM Tran. Softw., 2019]. The relaxation problem is converted into the problem of finding the largest zero $y^*$ of a continuously differentiable (except at $y^*$) convex function … Read more

Doubly nonnegative relaxations are equivalent to completely positive reformulations of quadratic optimization problems with block-clique graph structures

We study the equivalence among a nonconvex QOP, its CPP and DNN relaxations under the assumption that the aggregated and correlative sparsity of the data matrices of the CPP relaxation is represented by a block-clique graph $G$. By exploiting the correlative sparsity, we decompose the CPP relaxation problem into a clique-tree structured family of smaller … Read more

A Geometrical Analysis of a Class of Nonconvex Conic Programs for Convex Conic Reformulations of Quadratic and Polynomial Optimization Problems

We present a geometrical analysis on the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems with a class of geometrically defined nonconvex conic programs and their covexification. The class of nonconvex conic programs is described with a linear objective function in a linear space $V$, and the constraint … Read more

BBCPOP: A Sparse Doubly Nonnegative Relaxation of Polynomial Optimization Problems with Binary, Box and Complementarity Constraints

The software package BBCPOP is a MATLAB implementation of a hierarchy of sparse doubly nonnegative (DNN) relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box and complementarity (BBC) constraints. Given a POP in the class and a relaxation order, BBCPOP constructs a simple conic optimization problem (COP), which serves as a … Read more

User Manual for BBCPOP: A Sparse Doubly Nonnegative Relaxation of Polynomial Optimization Problems with Binary, Box and Complementarity Constraints

BBCPOP proposed in [4] is a MATLAB implementation of a hierarchy of sparse doubly nonnegative (DNN) relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box and complementarity constraints. Given a POP in the class and a relaxation order (or a hierarchy level), BBCPOP constructs a simple conic optimization problem (COP), which … Read more