Standard Quadratic Optimization Problem Research Articles

Symmetry in RLT-type relaxations for the quadratic assignment and standard quadratic optimization problems

The reformulation–linearization technique (RLT), introduced in [Sherali, H. D., Adams. W. P. (1990). A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3(3), 411–430], provides a way to compute a hierarchy of linear programming bounds on the optimal values of NP-hard combinatorial optimization problems. In this paper we show that, in the presence of suitable algebraic symmetry in the original problem data, it is sometimes possible to compute level two RLT bounds with additional linear matrix inequality constraints. As an illustration of our methodology, we compute the best-known bounds for certain graph partitioning problems on strongly regular graphs.

A Quadratically Constrained Quadratic Optimization Model for Completely Positive Cone Programming

We propose a class of quadratic optimization problems which can be reformulated by completely positive cone programming with the same optimal values. 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 inhomogeneous equality where a quadratic form is set to be a positive constant. For the equality constraints, “a hierarchy of copositvity” condition is assumed. This model is a generalization of the standard quadratic optimization problem of minimizing a quadratic form over the standard simplex, and covers many of the existing quadratic optimization problems studied for exact copositive cone and completely positive cone programming relaxations. In particular, it generalizes the recent results on quadratic optimization problems by Burer and the set-semidefinite representation by Eichfelder and Povh [Optim. Lett., 7 (2013), pp. 1373--1386].

Tightening a copositive relaxation for standard quadratic optimization problems

We focus in this paper the problem of improving the semidefinite programming (SDP) relaxations for the standard quadratic optimization problem (standard QP in short) that concerns with minimizing a quadratic form over a simplex. We first analyze the duality gap between the standard QP and one of its SDP relaxations known as "strengthened Shor's relaxation". To estimate the duality gap, we utilize the duality information of the SDP relaxation to construct a graph G ?. The estimation can be then reduced to a two-phase problem of enumerating first all the minimal vertex covers of G ? and solving next a family of second-order cone programming problems. When there is a nonzero duality gap, this duality gap estimation can lead to a strictly tighter lower bound than the strengthened Shor's SDP bound. With the duality gap estimation improving scheme, we develop further a heuristic algorithm for obtaining a good approximate solution for standard QP.

Steepest descent preconditioning for nonlinear GMRES optimization

SUMMARYSteepest descent preconditioning is considered for the recently proposed nonlinear generalized minimal residual (N‐GMRES) optimization algorithm for unconstrained nonlinear optimization. Two steepest descent preconditioning variants are proposed. The first employs a line search, whereas the second employs a predefined small step. A simple global convergence proof is provided for the N‐GMRES optimization algorithm with the first steepest descent preconditioner (with line search), under mild standard conditions on the objective function and the line search processes. Steepest descent preconditioning for N‐GMRES optimization is also motivated by relating it to standard non‐preconditioned GMRES for linear systems in the case of a standard quadratic optimization problem with symmetric positive definite operator. Numerical tests on a variety of model problems show that the N‐GMRES optimization algorithm is able to very significantly accelerate convergence of stand‐alone steepest descent optimization. Moreover, performance of steepest‐descent preconditioned N‐GMRES is shown to be competitive with standard nonlinear conjugate gradient and limited‐memory Broyden–Fletcher–Goldfarb–Shanno methods for the model problems considered. These results serve to theoretically and numerically establish steepest‐descent preconditioned N‐GMRES as a general optimization method for unconstrained nonlinear optimization, with performance that appears promising compared with established techniques. In addition, it is argued that the real potential of the N‐GMRES optimization framework lies in the fact that it can make use of problem‐dependent nonlinear preconditioners that are more powerful than steepest descent (or, equivalently, N‐GMRES can be used as a simple wrapper around any other iterative optimization process to seek acceleration of that process), and this potential is illustrated with a further application example. Copyright © 2012 John Wiley & Sons, Ltd.

Sparse solutions to random standard quadratic optimization problems

The standard quadratic optimization problem (StQP) refers to the problem of minimizing a quadratic form over the standard simplex. Such a problem arises from numerous applications and is known to be NP-hard. In this paper we focus on a special scenario of the StQP where all the elements of the data matrix Q are independently identically distributed and follow a certain distribution such as uniform or exponential distribution. We show that the probability that such a random StQP has a global optimal solution with k nonzero elements decays exponentially in k. Numerical evaluation of our theoretical finding is discussed as well.

On the accuracy of uniform polyhedral approximations of the copositive cone

We consider linear optimization problems over the cone of copositive matrices. Such conic optimization problems, called copositive programs, arise from the reformulation of a wide variety of difficult optimization problems. We propose a hierarchy of increasingly better outer polyhedral approximations to the copositive cone. We establish that the sequence of approximations is exact in the limit. By combining our outer polyhedral approximations with the inner polyhedral approximations due to de Klerk and Pasechnik [SIAM J. Optim. 12 (2002), pp. 875–892], we obtain a sequence of increasingly sharper lower and upper bounds on the optimal value of a copositive program. Under primal and dual regularity assumptions, we establish that both sequences converge to the optimal value. For standard quadratic optimization problems, we derive tight bounds on the gap between the upper and lower bounds. We provide closed-form expressions of the bounds for the maximum stable set problem. Our computational results shed light on the quality of the bounds on randomly generated instances.

Vibration Control of a Flexible Beam Using Model Predictive Control

Model predictive control is applied to suppress the vibration of a flexible link with piezoelectric actuators and strain gage transducer. The state-space dynamic model of the system was derived by using finite element method and experimental modal test. On the basis of the model, model predictive controller is designed taking into account the uncertain disturbance and measurement noise. The discrete prediction model is derived from the state-space equation of the system, and the future output is obtained from the model. The uncertain external disturbance and measurement noise are white noise signal, the Kalman filter estimator is designed to estimate the state variables of the system. A standard quadratic programming optimization problem is formed where the performance index function minimizes a quadratic performance function that trades off controller performance and control effort. The constraints are the control input voltage and its change rate. Finally, the optimization problem is solved to obtain the optimal control output. A MIMO control system is built using dSPACE DS1103 platform, and experimental tests are performed. The performances of the controller are verified experimentally. The results of experiment show the effectiveness of the controller.

Vibration Control of a Flexible Beam Using Model Predictive Control

Model predictive control is applied to suppress the vibration of a flexible link with piezoelectric actuators and strain gage transducer. The state-space dynamic model of the system was derived by using finite element method and experimental modal test. On the basis of the model, model predictive controller is designed taking into account the uncertain disturbance and measurement noise. The discrete prediction model is derived from the state-space equation of the system, and the future output is obtained from the model. The uncertain external disturbance and measurement noise are white noise signal, the Kalman filter estimator is designed to estimate the state variables of the system. A standard quadratic programming optimization problem is formed where the performance index function minimizes a quadratic performance function that trades off controller performance and control effort. The constraints are the control input voltage and its change rate. Finally, the optimization problem is solved to obtain the optimal control output. A MIMO control system is built using dSPACE DS1103 platform, and experimental tests are performed. The performances of the controller are verified experimentally. The results of experiment show the effectiveness of the controller.

Separable standard quadratic optimization problems

A standard quadratic optimization problems (StQP) asks for the minimal value of a quadratic form over the standard simplex. StQPs form a central NP-hard class in quadratic optimization and have numerous practical applications. In this note we study the case of a separable objective function and propose an algorithm of worst-case complexity \({\mathcal{O}(n\log n)}\) . Some extensions to multi-StQPs and l 1−ball constrained problems are also addressed briefly.

Unconstrained formulation of standard quadratic optimization problems

A standard quadratic optimization problem (StQP) consists of finding the largest or smallest value of a (possibly indefinite) quadratic form over the standard simplex which is the intersection of a hyperplane with the positive orthant. This NP-hard problem has several immediate real-world applications like the Maximum Clique Problem, and it also occurs in a natural way as a subproblem in quadratic programming with linear constraints. We propose unconstrained reformulations of StQPs, by using different approaches. We test our method on clique problems from the DIMACS challenge.

Multi-Standard Quadratic Optimization: interior point methods and cone programming reformulation

A Standard Quadratic Optimization Problem (StQP) consists of maximizing a (possibly indefinite) quadratic form over the standard simplex. Likewise, in a multi-StQP we have to maximize a (possibly indefinite) quadratic form over the Cartesian product of several standard simplices (of possibly different dimensions). Among many other applications, multi-StQPs occur in Machine Learning Problems. Several converging monotone interior point methods are established, which differ from the usual ones used in cone programming. Further, we prove an exact cone programming reformulation for establishing rigorous yet affordable bounds and finding improving directions.

Self-adaptive support vector machines: modelling and experiments

Method In this paper, we introduce a bi-level optimization formulation for the model and feature selection problems of support vector machines (SVMs). A bi-level optimization model is proposed to select the best model, where the standard convex quadratic optimization problem of the SVM training is cast as a subproblem.

New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability

A standard quadratic optimization problem (StQP) consists in minimizing a quadratic form over a simplex. Among the problems which can be transformed into a StQP are the general quadratic problem over a polytope, and the maximum clique problem in a graph. In this paper we present several new polynomial-time bounds for StQP ranging from very simple and cheap ones to more complex and tight constructions. The main tools employed in the conception and analysis of most bounds are Semidefinite Programming and decomposition of the objective function into a sum of two quadratic functions, each of which is easy to minimize. We provide a complete diagram of the dominance, incomparability, or equivalence relations among the bounds proposed in this and in previous works. In particular, we show that one of our new bounds dominates all the others. Furthermore, a specialization of such bound dominates Schrijver’s improvement of Lovasz’s θ function bound for the maximum size of a clique in a graph.

A linear programming reformulation of the standard quadratic optimization problem

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note, we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP's whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.

Quartic Formulation of Standard Quadratic Optimization Problems

A standard quadratic optimization problem (StQP) consists of finding the largest or smallest value of a (possibly indefinite) quadratic form over the standard simplex which is the intersection of a hyperplane with the positive orthant. This NP-hard problem has several immediate real-world applications like the Maximum-Clique Problem, and it also occurs in a natural way as a subproblem in quadratic programming with linear constraints. To get rid of the (sign) constraints, we propose a quartic reformulation of StQPs, which is a special case (degree four) of a homogeneous program over the unit sphere. It turns out that while KKT points are not exactly corresponding to each other, there is a one-to-one correspondence between feasible points of the StQP satisfying second-order necessary optimality conditions, to the counterparts in the quartic homogeneous formulation. We supplement this study by showing how exact penalty approaches can be used for finding local solutions satisfying second-order necessary optimality conditions to the quartic problem: we show that the level sets of the penalty function are bounded for a finite value of the penalty parameter which can be fixed in advance, thus establishing exact equivalence of the constrained quartic problem with the unconstrained penalized version.

A Linear Programming Reformulation of the Standard Quadratic Optimization Problem

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NPhard, and contains the maximum stable set problem in graphs as a special case. In this note we show that the SQO problem may be reformulated as an (exponentially sized) linear program.

Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming

The problem of minimizing a (non-convex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matrices via systems of linear inequalities, and, more refined, linear matrix inequalities (LMI's). In particular, we show that our approach leads to a polynomial-time approximation scheme for the standard quadratic optimzation problem. This is an improvement on the previous complexity result by Nesterov who showed that a 2/3-approximation is always possible. Numerical examples from various applications are provided to illustrate our approach.

Branch-and-bound approaches to standard quadratic optimization problems

This paper explores several possibilities for applying branch-and-bound techniques to a central problem class in quadratic programming, the so-called Standard Quadratic Problems (StQPs), which consist of finding a (global) minimizer of a quadratic form over the standard simplex. Since a crucial part of the procedures is based on efficient local optimization, different procedures to obtain local solutions are discussed, and a new class of ascent directions is proposed, for which a convergence result is established. Main emphasis is laid upon a d.c.-based branch-and-bound algorithm, and various strategies for obtaining an efficient d.c. decomposition are discussed.

On Copositive Programming and Standard Quadratic Optimization Problems

A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization FFT where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.

On Standard Quadratic Optimization Problems

A standard quadratic optimization problem (QP) consists of finding (global) maximizers of a quadratic form over the standard simplex. Standard QPs arise quite naturally in copositivity-based proced...