Showing papers on "Quadratically constrained quadratic program published in 2009"

Practical Methods of Optimization

Abstract: Preface Table of Notation Part 1: Unconstrained Optimization Introduction Structure of Methods Newton-like Methods Conjugate Direction Methods Restricted Step Methods Sums of Squares and Nonlinear Equations Part 2: Constrained Optimization Introduction Linear Programming The Theory of Constrained Optimization Quadratic Programming General Linearly Constrained Optimization Nonlinear Programming Other Optimization Problems Non-Smooth Optimization References Subject Index.

GloptiPoly 3: moments, optimization and semidefinite programming

Abstract: We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming.

Quadratic Assignment Problem.

Abstract: The quadratic assignment problem (QAP) was introduced by Koopmans and Beckmann in 1957 as a mathematical model for the location of a set of indivisible economical activities [113]. Consider the problem of allocating a set of facilities to a set of locations, with the cost being a function of the distance and flow between the facilities, plus costs associated with a facility being placed at a certain location. The objective is to assign each facility to a location such that the total cost is minimized. Specifically, we are given three n x n input matrices with real elements F = (f ij ), D = (d kl ) and B = (b ik ), where f ij is the flow between the facility i and facility j, d kl is the distance between the location k and location l, and b ik is the cost of placing facility i at location k. The Koopmans-Beckmann version of the QAP can be formulated as follows: Let n be the number of facilities and locations and denote by N the set N = {1, 2,..., n}.

Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming

Abstract: We consider relaxations for nonconvex quadratically constrained quadratic programming (QCQP) based on semidefinite programming (SDP) and the reformulation-linearization technique (RLT). From a theoretical standpoint we show that the addition of a semidefiniteness condition removes a substantial portion of the feasible region corresponding to product terms in the RLT relaxation. On test problems we show that the use of SDP and RLT constraints together can produce bounds that are substantially better than either technique used alone. For highly symmetric problems we also consider the effect of symmetry-breaking based on tightened bounds on variables and/or order constraints.

Algorithm for cardinality-constrained quadratic optimization

Abstract: This paper describes an algorithm for cardinality-constrained quadratic optimization problems, which are convex quadratic programming problems with a limit on the number of non-zeros in the optimal solution. In particular, we consider problems of subset selection in regression and portfolio selection in asset management and propose branch-and-bound based algorithms that take advantage of the special structure of these problems. We compare our tailored methods against CPLEX's quadratic mixed-integer solver and conclude that the proposed algorithms have practical advantages for the special class of problems we consider.

Scenario-based model predictive control of stochastic constrained linear systems

Abstract: In this paper we propose a stochastic model predictive control (MPC) formulation based on scenario generation for linear systems affected by discrete multiplicative disturbances. By separating the problems of (1) stochastic performance, and (2) stochastic stabilization and robust constraints fulfillment of the closed-loop system, we aim at obtaining a less conservative control action with respect to classical robust MPC schemes, still enforcing convergence and feasibility properties for the controlled system. Stochastic performance is addressed for very general classes of stochastic disturbance processes, although discretized in the probability space, by adopting ideas from multi-stage stochastic optimization. Stochastic stability and recursive feasibility are enforced through linear matrix inequality (LMI) problems, which are solved off-line; stochastic performance is optimized by an on-line MPC problem which is formulated as a convex quadratically constrained quadratic program (QCQP) and solved in a receding horizon fashion. The performance achieved by the proposed approach is shown in simulation and compared to the one obtained by standard robust and deterministic MPC schemes.

Regularization Methods for Semidefinite Programming

Abstract: We introduce a new class of algorithms for solving linear semidefinite programming (SDP) problems. Our approach is based on classical tools from convex optimization such as quadratic regularization and augmented Lagrangian techniques. We study the theoretical properties and we show that practical implementations behave very well on some instances of SDP having a large number of constraints. We also show that the “boundary point method” from Povh, Rendl, and Wiegele [Computing, 78 (2006), pp. 277-286] is an instance of this class.

Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem

Abstract: We consider semidefinite programming relaxations of the quadratic assignment problem, and show how to exploit group symmetry in the problem data. Thus we are able to compute the best known lower bounds for several instances of quadratic assignment problems from the problem library: (Burkard et al. in J Global Optim 10:291–403, 1997).

Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs

Abstract: This article addresses the generation of strong polyhedral relaxations for nonconvex, quadratically constrained quadratic programs (QCQPs). Using the convex envelope of multilinear functions as our starting point, we develop a polyhedral relaxation for QCQP, along with a cutting plane algorithm for its implementation. Our relaxations are multiterm, i.e. they are derived from the convex envelope of the sum of multiple bilinear terms of quadratic constraints, thereby providing tighter bounds than the standard termwise relaxation of the bilinear functions. Computational results demonstrate the usefulness of the proposed cutting planes.

A hybrid optimization approach to index tracking

Abstract: Index tracking consists in reproducing the performance of a stock-market index by investing in a subset of the stocks included in the index. A hybrid strategy that combines an evolutionary algorithm with quadratic programming is designed to solve this NP-hard problem: Given a subset of assets, quadratic programming yields the optimal tracking portfolio that invests only in the selected assets. The combinatorial problem of identifying the appropriate assets is solved by a genetic algorithm that uses the output of the quadratic optimization as fitness function. This hybrid approach allows the identification of quasi-optimal tracking portfolios at a reduced computational cost.

Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound

Abstract: We consider a recent branch-and-bound algorithm of the authors for nonconvex quadratic programming. The algorithm is characterized by its use of semidefinite relaxations within a finite branching scheme. In this paper, we specialize the algorithm to the box-constrained case and study its implementation, which is shown to be a state-of-the-art method for globally solving box-constrained nonconvex quadratic programs.

Alternative Theorems for Quadratic Inequality Systems and Global Quadratic Optimization

Abstract: We establish alternative theorems for quadratic inequality systems. Consequently, we obtain Lagrange multiplier characterizations of global optimality for classes of nonconvex quadratic optimization problems. We present a generalization of Dine's theorem to a system of two homogeneous quadratic functions with a regular cone. The class of regular cones are cones $K$ for which $(K\cup-K)$ is a subspace. As a consequence, we establish a generalization of the powerful $S$-lemma, which paves the way to obtain a complete characterization of global optimality for a general quadratic optimization model problem involving a system of equality constraints in addition to a single quadratic inequality constraint. We then present an alternative theorem for a system of three nonhomogeneous inequalities by way of establishing the convexity of the joint-range of three homogeneous quadratic functions using a regular cone. This yields Lagrange multiplier characterizations of global optimality for classes of trust-region type problems with two inequality constraints. Finally, we establish an alternative theorem for systems involving an arbitrary finite number of quadratic inequalities involving Z-matrices, which are matrices with nonpositive off diagonal elements, and present necessary and sufficient conditions for global optimality for classes of nonconvex inequality constrained quadratic optimization problems.

Optimal Design of Nonlinear-Phase FIR Filters With Prescribed Phase Error

Abstract: Constrained least-squares design and constrained Chebyshev design of one- and two-dimensional nonlinear-phase FIR filters with prescribed phase error are considered in this paper by a unified semi-infinite positive-definite quadratic programming approach. In order to obtain unique optimal solutions, we propose to impose constraints on the complex approximation error and the phase error. By introducing a sigmoid phase-error constraint bound function, the group-delay error can be greatly reduced. A Goldfarb-Idnani based algorithm is presented to solve the semi-infinite positive-definite quadratic program resulting from the constrained least-squares design problem, and then applied after some modifications to the constrained Chebyshev design problem, which is proved in this paper to be equivalent also to a semi-infinite positive-definite quadratic program. Through design examples, the proposed method is compared with several existing methods. Simulation results demonstrate the effectiveness and efficiency of the proposed method.

A quadratic approximation for structural topology optimization

Abstract: In topology optimization, it is customary to use reciprocal-like approximations, which result in monotonically decreasing approximate objective functions. In this paper, we demonstrate that efficient quadratic approximations for topology optimization can also be derived, if the approximate Hessian terms are chosen with care. To demonstrate this, we construct a dual SAO algorithm for topology optimization based on a strictly convex, diagonal quadratic approximation to the objective function. Although the approximation is purely quadratic, it does contain essential elements of reciprocal-like approximations: for self-adjoint problems, our approximation is identical to the quadratic or second-order Taylor series approximation to the exponential approximation. We present both a single-point and a two-point variant of the new quadratic approximation.

A Sequential Convex Semidefinite Programming Algorithm with an Application to Multiple-Load Free Material Optimization

Abstract: A new method for the efficient solution of a class of convex semidefinite programming (SDP) problems is introduced. The method extends the sequential convex programming (SCP) concept to optimization problems with matrix variables. The basic idea of the new method is to approximate the original optimization problem by a sequence of subproblems, in which nonlinear functions (defined in matrix variables) are approximated by block separable convex functions. The subproblems are semidefinite programs with a favorable structure which can be efficiently solved by existing SDP software. The new method is shown to be globally convergent. The article is concluded by a series of numerical experiments with free material optimization problems demonstrating the effectiveness of the generalized SCP approach.

Global optimization for a class of fractional programming problems

Abstract: This paper presents a canonical dual approach to minimizing the sum of a quadratic function and the ratio of two quadratic functions, which is a type of non-convex optimization problem subject to an elliptic constraint. We first relax the fractional structure by introducing a family of parametric subproblems. Under proper conditions on the "problem-defining" matrices associated with the three quadratic functions, we show that the canonical dual of each subproblem becomes a one-dimensional concave maximization problem that exhibits no duality gap. Since the infimum of the optima of the parameterized subproblems leads to a solution to the original problem, we then derive some optimality conditions and existence conditions for finding a global minimizer of the original problem. Some numerical results using the quasi-Newton and line search methods are presented to illustrate our approach.

Distributed Beamforming in Relay-Assisted Multiuser Communications

Abstract: This paper considers a communication network with multiple pairs of source and destination, assisted by multiple relays. It is assumed that perfect channel state information (CSI) is available at the relays. In a two-stage AF protocol, all the sources broadcast their signals to all the relays in the first stage. The received signal at each relay is processed by a beamforming weight and then re-broadcasted to all the destinations at the same time with other relays in the second stage. The focus is to find the optimal beamforming weights to meet a given set of target signal-to-interference-and-noise ratio (SINR) at the destinations, while minimizing the total transmitted power at the relays. We show that this problem can be formulated as a nonconvex quadratically constrained quadratic program (QCQP). Through relaxations, the problem can be solved efficiently by convex programming.

On filter-successive linearization methods for nonlinear semidefinite programming

Abstract: In this paper we present a filter-successive linearization method with trust region for solutions of nonlinear semidefinite programming. Such a method is based on the concept of filter for nonlinear programming introduced by Fletcher and Leyffer in 2002. We describe the new algorithm and prove its global convergence under weaker assumptions. Some numerical results are reported and show that the new method is potentially efficient.

B-453 User's Manual for SparseCoLO: Conversion Methods for SPARSE COnic-form Linear Optimization Problems

Abstract: SparseCoLO is a Matlab package for implementing the four conversion methods, proposed by Kim, Kojima, Mevissen, and Yamashita, via positive semidefinite matrix completion for an optimization problem with matrix inequalities satisfying a sparse chordal graph structure. It is based on a general description of optimization problem including both primal and dual form of linear, semidefinite, and second-order cone programs with equality/inequality constraints. Among the four conversion methods, two methods utilize the domain-space sparsity of a semidefinite matrix variable and the two other methods the range-space sparsity of a linear matrix inequality (LMI) constraint of the given problem. SparseCoLO can be used as a preprocessor to reduce the size of the given problem before applying semidefinite programming solvers. The website for this package is http://www.is.titech.ac.jp/∼kojima/SparseCoLO where the package SparseCoLO and this manual can be downloaded.

A Low-Dimensional Semidefinite Relaxation for the Quadratic Assignment Problem

Abstract: The quadratic assignment problem (QAP) is arguably one of the hardest NP-hard discrete optimization problems. Problems of dimension greater than 25 are still considered to be large scale. Current successful solution techniques use branch-and-bound methods, which rely on obtaining strong and inexpensive bounds. In this paper, we introduce a new semidefinite programming (SDP) relaxation for generating bounds for the QAP in the trace formulation. We apply majorization to obtain a relaxation of the orthogonal similarity set of the quadratic part of the objective function. This exploits the matrix structure of QAP and results in a relaxation with much smaller dimension than other current SDP relaxations. We compare the resulting bounds with several other computationally inexpensive bounds such as the convex quadratic programming relaxation (QPB). We find that our method provides stronger bounds on average and is adaptable for branch-and-bound methods.

Complexity Theory: Quadratic Programming.

Abstract: An internal combustion power plant system provides a rotary engine and a rotary fuel/air mixture compressor for the rotary engine on a common driveshaft, coaxially mounting each end and supported between them by a gearbox which synchronizes operation of various ignition and valve and abutment components of the system; compressed fuel/air mixture is supplied to and ignited in a valve-isolated manifold chamber in the rotary engine in successive charges following which each ignited charge is valved radially into one of plural expanding chambers defined by the rotary engine rotor and abutment mechanism, where it urges rotation of the rotor and then exhausts radially; in preferred embodiment of the exhaust actuates a parallel fuel-feed which booster pumps fuel/air mixture into the manifold chamber; detail improvements disclosed include designs of runners, abutments, valving and rotary compressor mechanism.

Semidefinite programming, matrix decomposition, and radar code design

Abstract: In this chapter, we study specific rank-1 decomposition techniques for Hermitian positive semidefinite matrices. Based on the semidefinite programming relaxation method and the decomposition techniques, we identify several classes of quadratically constrained quadratic programming problems that are polynomially solvable. Typically, such problems do not have too many constraints. As an example, we demonstrate how to apply the new techniques to solve an optimal code design problem arising from radar signal processing. Introduction and notation Semidefinite programming (SDP) is a relatively new subject of research in optimization. Its success has caused major excitement in the field. One is referred to Boyd and Vandenberghe [11] for an excellent introduction to SDP and its applications. In this chapter, we shall elaborate on a special application of SDP for solving quadratically constrained quadratic programming (QCQP) problems. The techniques we shall introduce are related to how a positive semidefinite matrix can be decomposed into a sum of rank-1 positive semidefinite matrices, in a specific way that helps to solve nonconvex quadratic optimization with quadratic constraints. The advantage of the method is that the convexity of the original quadratic optimization problem becomes irrelevant; only the number of constraints is important for the method to be effective. We further present a study on how this method helps to solve a radar code design problem. Through this investigation, we aim to make a case that solving nonconvex quadratic optimization by SDP is a viable approach.

A successive SDP-NSDP approach to a robust optimization problem in finance

Abstract: The robustification of trading strategies is of particular interest in financial market applications. In this paper we robustify a portfolio strategy recently introduced in the literature against model errors in the sense of a worst case design. As it turns out, the resulting optimization problem can be solved by a sequence of linear and nonlinear semidefinite programs (SDP/NSDP), where the nonlinearity is introduced by the parameters of a parabolic differential equation. The nonlinear semidefinite program naturally arises in the computation of the worst case constraint violation which is equivalent to an eigenvalue minimization problem. Further we prove convergence for the iterates generated by the sequential SDP-NSDP approach.

A Structure Exploiting Preprocessor for Semidefinite Programs Derived From the Kalman-Yakubovich-Popov Lemma

Abstract: Semidefinite programs derived from the Kalman-Yakubovich-Popov (KYP) lemma are quite common in control and signal processing applications. The programs are often of high dimension which makes them hard or impossible to solve with general-purpose solvers. Here we present a customized preprocessor, KYPD, that utilizes the inherent structure of this particular optimization problem. The key to an efficient implementation is to transform the optimization problem into an equivalent semidefinite program. This equivalent problem has much fewer variables and the matrices in the linear matrix inequality constraints are of low rank. KYPD can use any primal-dual solver for semidefinite programs as an underlying solver.

Free Material Optimization with Fundamental Eigenfrequency Constraints

Abstract: The goal of this paper is to formulate and solve free material optimization problems with constraints on the smallest eigenfrequency of the optimal structure. A natural formulation of this problem as a linear semidefinite program turns out to be numerically intractable. As an alternative, we propose a new approach, which is based on a nonlinear semidefinite low-rank approximation of the semidefinite dual. We introduce an algorithm based on this approach and analyze its convergence properties. The article is concluded by numerical experiments proving the effectiveness of the new approach.

A min-max framework of cascaded classifier with multiple instance learning for computer aided diagnosis

Abstract: The computer aided diagnosis (CAD) problems of detecting potentially diseased structures from medical images are typically distinguished by the following challenging characteristics: extremely unbalanced data between negative and positive classes; stringent real-time requirement of online execution; multiple positive candidates generated for the same malignant structure that are highly correlated and spatially close to each other. To address all these problems, we propose a novel learning formulation to combine cascade classification and multiple instance learning (MIL) in a unified min-max framework, leading to a joint optimization problem which can be converted to a tractable quadratically constrained quadratic program and efficiently solved by block-coordinate optimization algorithms. We apply the proposed approach to the CAD problems of detecting pulmonary embolism and colon cancer from computed tomography images. Experimental results show that our approach significantly reduces the computational cost while yielding comparable detection accuracy to the current state-of-the-art MIL or cascaded classifiers. Although not specifically designed for balanced MIL problems, the proposed method achieves superior performance on balanced MIL benchmark data such as MUSK and image data sets.