We follow the popular approach for unconstrained minimization, i.e. we develop a local quadratic model at a current approximate minimizer in conjunction with a trust region. We then minimize this local model in order to nd the next approximate minimizer. Asymptot-ically, nding the local minimizer of the quadratic model is equivalent to applying Newton's method to the stationarity condition. For...

Let (QP ) be a 0-1 quadratic program which consists in minimizing a quadratic function subject to linear constraints. In this paper, we present a general method to solve (QP ) by reformulation of the problem into an equivalent 0-1 program with a convex quadratic objective function, followed by the use of a standard mixed integer quadratic programming solver. Our convexification method, which is...

We consider a proximal operator given by a quadratic function subject to bound constraints and give an optimization algorithm using the alternating direction method of multipliers (ADMM). The algorithm is particularly efficient to solve a collection of proximal operators that share the same quadratic form, or if the quadratic program is the relaxation of a binary quadratic problem.

Abstract This paper presents a canonical duality theory for solving a general nonconvex 1 quadratic minimization problem with nonconvex constraints. By using the canonical dual 2 transformation developed by the first author, the nonconvex primal problem can be con3 verted into a canonical dual problem with zero duality gap. A general analytical solution 4 form is obtained. Both global and local...

This paper introduces a new filtering algorithm for handling systems of quadratic equations and inequations. Such constraints are widely used to model distance relations in numerous application areas ranging from robotics to chemistry. Classical filtering algorithms are based upon local consistencies and thus, are unable to achieve a significant pruning of the domains of the variables occurring...

In this paper we consider optimization problems de ned by a quadratic objective function and a nite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we rst prove a gene...

This paper analyzes linear least squares problems with absolute quadratic constraints. We develop a generalized theory following Bookstein’s conic-fitting and Fitzgibbon’s direct ellipse-specific fitting. Under simple preconditions, it can be shown that a minimum always exists and can be determined by a generalized eigenvalue problem. This problem is numerically reduced to an eigenvalue problem...

Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g., convex programs an...

The theory of integral quadratic constraints (IQCs) allows verification of stability and gain-bound properties of systems containing nonlinear or uncertain elements. Gain bounds often imply exponential stability, but it can be challenging to compute useful numerical bounds on the exponential decay rate. This work presents a generalization of the classical IQC results of Megretski and Rantzer [1...

We consider the problem of approximating the global maximum of a quadratic program (QP) with n variables subject to bound constraints. Based on the results of Goemans and Williamson 4] and Nesterov 6], we show that a 4=7 approximate solution can be obtained in polynomial time.