A robust Lagrangian-DNN method for a class of quadratic optimization problems

The Lagrangian-doubly nonnegative (DNN) relaxation has recently been shown to provide effective lower bounds for a large class of nonconvex quadratic optimization problems (QOPs) using the bisection method combined with first-order methods by Kim, Kojima and Toh in 2016. While the bisection method has demonstrated the computational efficiency, determining the validity of a computed lower … Read more

Lagrangian-Conic Relaxations, Part II: Applications to Polynomial Optimization Problems

We present the moment cone (MC) relaxation and a hierarchy of sparse Lagrangian-SDP relaxations of polynomial optimization problems (POPs) using the unified framework established in Part I. The MC relaxation is derived for a POP of minimizing a polynomial subject to a nonconvex cone constraint and polynomial equality constraints. It is an extension of the … Read more

Lagrangian-Conic Relaxations, Part I: A Unified Framework and Its Applications to Quadratic Optimization Problems

In Part I of a series of study on Lagrangian-conic relaxations, we introduce a unified framework for conic and Lagrangian-conic relaxations of quadratic optimization problems (QOPs) and polynomial optimization problems (POPs). The framework is constructed with a linear conic optimization problem (COP) in a finite dimensional vector space endowed with an inner product, where the … Read more

Extension of Completely Positive Cone Relaxation to Polynomial Optimization

We propose the moment cone relaxation for a class of polynomial optimization problems (POPs) to extend the results on the completely positive cone programming relaxation for the quadratic optimization (QOP) model by Arima, Kim and Kojima. The moment cone relaxation is constructed to take advantage of sparsity of the POPs, so that efficient numerical methods … Read more

Simplified Copositive and Lagrangian Relaxations for Linearly Constrained Quadratic Optimization Problems in Continuous and Binary Variables

For a quadratic optimization problem (QOP) with linear equality constraints in continuous nonnegative variables and binary variables, we propose three relaxations in simplified forms with a parameter $\lambda$: Lagrangian, completely positive, and copositive relaxations. These relaxations are obtained by reducing the QOP to an equivalent QOP with a single quadratic equality constraint in nonnegative variables, … Read more

A Quadratically Constrained Quadratic Optimization Model for Completely Positive Cone Programming

We propose a class of quadratic optimization problems whose exact optimal objective values can be computed by their completely positive cone programming relaxations. The objective function can be any quadratic form. The constraints of each problem are described in terms of quadratic forms with no linear terms, and all constraints are homogeneous equalities, except one … Read more