A New Global Optimization Scheme for Quadratic Programs with Low-Rank Nonconvexity

We consider the classical convex constrained nonconvex quadratic programming problem where the Hessian matrix of the objective to be minimized has r negative eigenvalues, denoted by (QPr). Based on a biconvex programming reformulation in a slightly higher dimension, we propose a novel branch-and-bound algorithm to solve (QP1) and show that it returns an [Formula: see text]-approximate solution of (QP1) in at most [Formula: see text] iterations. We further extend the new algorithm to solve the general (QPr) with r > 1. Computational comparison shows the efficiency of our proposed global optimization method for small r. Finally, we extend the explicit relaxation approach for (QP1) to (QPr) with r > 1. Summary of Contribution: Nonconvex quadratic program (QP) is a classical optimization problem in operations research. This paper aims at globally solving the QP where the Hessian matrix of the objective to be minimized has r negative eigenvalues. It is known to be nondeterministic polynomial-time hard even when r = 1. This paper presents a novel algorithm to globally solve the QP for r = 1 and then extends to general r. Numerical results demonstrate the superiority of the proposed algorithm in comparison with state-of-the-art algorithms/software for small r.

A Continuous-Time Distributed Algorithm for Solving a Class of Decomposable Nonconvex Quadratic Programming

In this paper, a continuous-time distributed algorithm is presented to solve a class of decomposable quadratic programming problems. In the quadratic programming, even if the objective function is nonconvex, the algorithm can still perform well under an extra condition combining with the objective, constraint and coupling matrices. Inspired by recent advances in distributed optimization, the proposed continuous-time algorithm described by multi-agent network with consensus is designed and analyzed. In the network, each agent only accesses the local information of its own and from its neighbors, then all the agents in a connected network cooperatively find the optimal solution with consensus.