Concave Quadratic Cuts for Mixed-Integer Quadratic Problems

The technique of semidefinite programming (SDP) relaxation can be used to obtain a nontrivial bound on the optimal value of a nonconvex quadratically constrained quadratic program (QCQP). We explore concave quadratic inequalities that hold for any vector in the integer lattice ${\bf Z}^n$, and show that adding these inequalities to a mixed-integer nonconvex QCQP can improve the SDP-based bound on the optimal value. This scheme is tested using several numerical problem instances of the max-cut problem and the integer least squares problem.Comment: 24 pages, 1 figur

Performance Bounds for Constrained Linear Min-Max Control

This paper proposes a method to compute lower performance bounds for discrete-time infinite-horizon min-max control problems with input constraints and bounded disturbances. Such bounds can be used as a performance metric for control policies synthesized via suboptimal design techniques. Our approach is motivated by recent work on performance bounds for stochastic constrained optimal control problems using relaxations of the Bellman equation. The central idea of the paper is to find an unconstrained min-max control problem, with negatively weighted disturbances as in H infinity control, that provides the tightest possible lower performance bound on the original problem of interest and whose value function is easily computed. The new method is demonstrated via a numerical example for a system with box constrained input.Comment: 6 pages, in proceedings of the 2013 European Control Conferenc

Continuous DR-submodular Maximization: Structure and Algorithms

DR-submodular continuous functions are important objectives with wide real-world applications spanning MAP inference in determinantal point processes (DPPs), and mean-field inference for probabilistic submodular models, amongst others. DR-submodularity captures a subclass of non-convex functions that enables both exact minimization and approximate maximization in polynomial time. In this work we study the problem of maximizing non-monotone DR-submodular continuous functions under general down-closed convex constraints. We start by investigating geometric properties that underlie such objectives, e.g., a strong relation between (approximately) stationary points and global optimum is proved. These properties are then used to devise two optimization algorithms with provable guarantees. Concretely, we first devise a "two-phase" algorithm with $1/4$ approximation guarantee. This algorithm allows the use of existing methods for finding (approximately) stationary points as a subroutine, thus, harnessing recent progress in non-convex optimization. Then we present a non-monotone Frank-Wolfe variant with $1/e$ approximation guarantee and sublinear convergence rate. Finally, we extend our approach to a broader class of generalized DR-submodular continuous functions, which captures a wider spectrum of applications. Our theoretical findings are validated on synthetic and real-world problem instances.Comment: Published in NIPS 201

Canonical dual method for mixed integer fourth-order polynomial minimization problems with fixed cost terms

We study a canonical duality method to solve a mixed-integer nonconvex fourth-order polynomial minimization problem with fixed cost terms. This constrained nonconvex problem can be transformed into a continuous concave maximization dual problem without duality gap. The global optimality conditions are proposed and the existence and uniqueness criteria are discussed. Application to a decoupled mixed-integer problem is illustrated and analytic solution for a global minimum is obtained under some suitable conditions. Several examples are given to show the method is effective.Comment: 15 pages, 7 table

Global Solutions to Large-Scale Spherical Constrained Quadratic Minimization via Canonical Dual Approach

This paper presents global optimal solutions to a nonconvex quadratic minimization problem over a sphere constraint. The problem is well-known as a trust region subproblem and has been studied extensively for decades. The main challenge is the so called 'hard case', i.e., the problem has multiple solutions on the boundary of the sphere. By canonical duality theory, this challenging problem is able to reformed as an one-dimensional canonical dual problem without duality gap. Sufficient and necessary conditions are obtained by the triality theory, which can be used to identify whether the problem is hard case or not. A perturbation method and the associated algorithms are proposed to solve this hard case problem. Theoretical results and methods are verified by large-size examples

An ellipsoidal branch and bound algorithm for global optimization

A branch and bound algorithm is developed for global optimization. Branching in the algorithm is accomplished by subdividing the feasible set using ellipses. Lower bounds are obtained by replacing the concave part of the objective function by an affine underestimate. A ball approximation algorithm, obtained by generalizing of a scheme of Lin and Han, is used to solve the convex relaxation of the original problem. The ball approximation algorithm is compared to SEDUMI as well as to gradient projection algorithms using randomly generated test problems with a quadratic objective and ellipsoidal constraints.Comment: 19 page

Canonical Solutions to Nonconvex Minimization Problems over Lorentz Cone

This paper presents a canonical dual approach for solving nonconvex quadratic minimization problem. By using the canonical duality theory, nonconvex primal minimization problems over n-dimensional Lorentz cone can be transformed into certain canonical dual problems with only one dual variable, which can be solved by using standard convex minimization methods. Extremality conditions of these solutions are classified by the triality theory. Applications are illustrated.Comment: 11 pages, 5 figure

A Smoothing SQP Framework for a Class of Composite LqL_q Minimization over Polyhedron

The composite $L_q~(0<q<1)$ minimization problem over a general polyhedron has received various applications in machine learning, wireless communications, image restoration, signal reconstruction, etc. This paper aims to provide a theoretical study on this problem. Firstly, we show that for any fixed $0<q<1$, finding the global minimizer of the problem, even its unconstrained counterpart, is strongly NP-hard. Secondly, we derive Karush-Kuhn-Tucker (KKT) optimality conditions for local minimizers of the problem. Thirdly, we propose a smoothing sequential quadratic programming framework for solving this problem. The framework requires a (approximate) solution of a convex quadratic program at each iteration. Finally, we analyze the worst-case iteration complexity of the framework for returning an $\epsilon$-KKT point; i.e., a feasible point that satisfies a perturbed version of the derived KKT optimality conditions. To the best of our knowledge, the proposed framework is the first one with a worst-case iteration complexity guarantee for solving composite $L_q$ minimization over a general polyhedron

Subdeterminants and Concave Integer Quadratic Programming

We consider the NP-hard problem of minimizing a separable concave quadratic function over the integral points in a polyhedron, and we denote by D the largest absolute value of the subdeterminants of the constraint matrix. In this paper we give an algorithm that finds an epsilon-approximate solution for this problem by solving a number of integer linear programs whose constraint matrices have subdeterminants bounded by D in absolute value. The number of these integer linear programs is polynomial in the dimension n, in D and in 1/epsilon, provided that the number k of variables that appear nonlinearly in the objective is fixed. As a corollary, we obtain the first polynomial-time approximation algorithm for separable concave integer quadratic programming with D at most two and k fixed. In the totally unimodular case D=1, we give an improved algorithm that only needs to solve a number of linear programs that is polynomial in 1/epsilon and is independent on n, provided that k is fixed

Robust Budget Allocation via Continuous Submodular Functions

The optimal allocation of resources for maximizing influence, spread of information or coverage, has gained attention in the past years, in particular in machine learning and data mining. But in applications, the parameters of the problem are rarely known exactly, and using wrong parameters can lead to undesirable outcomes. We hence revisit a continuous version of the Budget Allocation or Bipartite Influence Maximization problem introduced by Alon et al. (2012) from a robust optimization perspective, where an adversary may choose the least favorable parameters within a confidence set. The resulting problem is a nonconvex-concave saddle point problem (or game). We show that this nonconvex problem can be solved exactly by leveraging connections to continuous submodular functions, and by solving a constrained submodular minimization problem. Although constrained submodular minimization is hard in general, here, we establish conditions under which such a problem can be solved to arbitrary precision $\epsilon$.Comment: ICML 201