Showing papers on "Quadratically constrained quadratic program published in 2012"
17 Dec 2012
TL;DR: Semidefinite programming in polynomial optimization has been studied extensively in the literature as discussed by the authors, including convex algebraic geometry and convex hulls of algebraic sets.
Abstract: This book provides a self-contained, accessible introduction to the mathematical advances and challenges resulting from the use of semidefinite programming in polynomial optimization. This quickly evolving research area with contributions from the diverse fields of convex geometry, algebraic geometry, and optimization is known as convex algebraic geometry. Each chapter addresses a fundamental aspect of convex algebraic geometry. The book begins with an introduction to nonnegative polynomials and sums of squares and their connections to semidefinite programming and quickly advances to several areas at the forefront of current research. These include semidefinite representability of convex sets, duality theory from the point of view of algebraic geometry, and nontraditional topics such as sums of squares of complex forms and noncommutative sums of squares polynomials. Suitable for a class or seminar, with exercises aimed at teaching the topics to beginners, Semidefinite Optimization and Convex Algebraic Geometry serves as a point of entry into the subject for readers from multiple communities such as engineering, mathematics, and computer science. A guide to the necessary background material is available in the appendix. Audience This book can serve as a textbook for graduate-level courses presenting the basic mathematics behind convex algebraic geometry and semidefinite optimization. Readers conducting research in these areas will discover open problems and potential research directions. Contents: List of Notation; Chapter 1: What is Convex Algebraic Geometry?; Chapter 2: Semidefinite Optimization; Chapter 3: Polynomial Optimization, Sums of Squares, and Applications; Chapter 4: Nonnegative Polynomials and Sums of Squares; Chapter 5: Dualities; Chapter 6: Semidefinite Representability; Chapter 7: Convex Hulls of Algebraic Sets; Chapter 8: Free Convexity; Chapter 9: Sums of Hermitian Squares: Old and New; Appendix A: Background Material.
583 citations
01 Jan 2012
TL;DR: This paper presents an introduction to Semidefinite, Conic and Polynomial Optimization, and discusses Relaxations for Some Combinatorial Optimization Problems, and the State of theArt in Conic Optimization Software.
Abstract: Introduction to Semidefinite, Conic and Polynomial Optimization.- The Approach of Moments for Polynomial Equations.- Algebraic Degree in Semidefinite and Polynomial Optimization.- Semidefinite Representation of Convex Sets and Convex Hulls.- Convex Hulls of Algebraic Sets.- Convex Relations and Integrality Gaps.- Relaxations of Combinatorial Problems via Association Schemes.- Copositive Programming.- Invariant Semidefinite Programs.- A "Joint+Marginal" Approach in Optimization.- An Introduction to Formally Real Jordan Algebras and Their Applications in Optimization.- Complementarity Problems Over Symmetric Conics: A Survey of Recent Developments in Several Aspects.- Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns.- Positivity and Optimization: Beyond Polynomials.- Self-Regular Interior-Point Methods for Semidefinite Optimization.- Elementary Optimality Conditions for Nonlinear SDPs.- Recent Progress in Interior-Point Methods: Cutting Plane Algorithms and Warm Starts.- Exploiting Sparsity in SDP Relaxation of Polynomial Optimization Problems.- Block Coordinate Descent Methods for Semidefinite Programming.- Projection Methods in Conic Optimization.- SDP Relaxations for Non-Commutative Polynomial Optimization.- Semidefinite Programming and Constraint Programming.- The State-of-the-Art in Conic Optimization Software.- Latest Developments in SDPA Family for Solving Large-Scale SDPs.- On the Implementation and Usage of SDPT3: A MATLAB Software Package for Semidefinite-Quadratic-Linear Programming, Version 4.0.- PENNON: Software for Linear and Nonlinear Matrix Inequalities.- SDP Relaxations for Some Combinatorial Optimization Problems.- Computational Approaches to Max-Cut.- Global Approaches for Facility Layout and VLSI Floorplanning.- Euclidean Distance Matrices and Applications.- Sparse PCA: Convex Relaxations, Algorithms and Applications.
336 citations
TL;DR: In this paper, the authors derived mixed-integer quadratic, quadratically constrained, and second-order cone programming models of distribution system reconfiguration, which are to date the first formulations of the ac problem that have convex, continuous relaxations.
Abstract: We derive new mixed-integer quadratic, quadratically constrained, and second-order cone programming models of distribution system reconfiguration, which are to date the first formulations of the ac problem that have convex, continuous relaxations. Each model can be reliably and efficiently solved to optimality using standard commercial software. In the course of deriving each model, we obtain original quadratically constrained and second-order cone approximations to power flow in radial networks.
261 citations
Book•
10 Jan 2012
TL;DR: The basic theory of semidefinite programming is developed, one of the known efficient algorithms is presented in detail, and the principles of some others are described, focusing on approximation algorithms.
Abstract: Semidefinite programs constitute one of the largest classes of optimization problems that can be solved with reasonable efficiency - both in theory and practice. They play a key role in a variety of research areas, such as combinatorial optimization, approximation algorithms, computational complexity, graph theory, geometry, real algebraic geometry and quantum computing. This book is an introduction to selected aspects of semidefinite programming and its use in approximation algorithms. It covers the basics but also a significant amount of recent and more advanced material. There are many computational problems, such as MAXCUT, for which one cannot reasonably expect to obtain an exact solution efficiently, and in such case, one has to settle for approximate solutions. For MAXCUT and its relatives, exciting recent results suggest that semidefinite programming is probably the ultimate tool. Indeed, assuming the Unique Games Conjecture, a plausible but as yet unproven hypothesis, it was shown that for these problems, known algorithms based on semidefinite programming deliver the best possible approximation ratios among all polynomial-time algorithms. This book follows the semidefinite side of these developments, presenting some of the main ideas behind approximation algorithms based on semidefinite programming. It develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others. It also includes applications, focusing on approximation algorithms.
152 citations
TL;DR: This paper introduces a new global optimization algorithm that combines two ideas from the literature—finite branching based on the first-order KKT conditions and polyhedral-semidefinite relaxations of completely positive (or copositive) programs.
Abstract: Nonconvex quadratic programming (QP) is an NP-hard problem that optimizes a general quadratic function over linear constraints. This paper introduces a new global optimization algorithm for this problem, which combines two ideas from the literature—finite branching based on the first-order KKT conditions and polyhedral-semidefinite relaxations of completely positive (or copositive) programs. Through a series of computational experiments comparing the new algorithm with existing codes on a diverse set of test instances, we demonstrate that the new algorithm is an attractive method for globally solving nonconvex QP.
122 citations
TL;DR: This work considers convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and ( possibly nonconvergent) quadRatic constraints.
Abstract: We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let $$\mathcal{F }$$ denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on $$\mathcal{F }$$ is dominated by an alternative methodology based on convexifying the range of the quadratic form $$\genfrac(){0.0pt}{}{1}{x}\genfrac(){0.0pt}{}{1}{x}^T$$ for $$x\in \mathcal{F }$$. We next show that the use of "$$\alpha $$BB" underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms. Finally, we show that the use of a large class of D.C. ("difference of convex") underestimators is dominated by a relaxation that combines semidefiniteness with RLT constraints.
120 citations
TL;DR: This survey paper overviews several simulation-based and simulation-free computationally tractable approximations of chance constrained convex programs, primarily, those ofchance constrained linear, conic quadratic and semidefinite programming.
120 citations
TL;DR: Two randomized approximation algorithms are presented for the robust downlink beamforming optimization problem for secondary multicast transmission in a multiple-input multiple-output (MIMO) spectrum sharing cognitive radio (CR) network and one interesting special case is identified.
Abstract: We consider a robust downlink beamforming optimization problem for secondary multicast transmission in a multiple-input multiple-output (MIMO) spectrum sharing cognitive radio (CR) network. The minimization of transmit power is formulated subject to both quality-of-service (QoS) constraints on the secondary receivers and interference temperature constraints on the primary users, under the assumption of imperfect channel state information (CSI). The problem is a nonconvex quadratically constrained quadratic program (QCQP), and in general it is hard to achieve the global optimality. As a compromise, we present two randomized approximation algorithms for the problem via convex optimization techniques. Apart from the general setting of the robust beamforming problem, we identify one interesting special case, the robust problem of which can be solved efficiently. Simulation results are presented to demonstrate the performance gains of the proposed algorithms over an existing robust design.
103 citations
TL;DR: This work considers the problem of fitting a continuous piecewise linear function to a finite set of data points, modeled as a mathematical program with convex objective, and introduces mixed-binary generalizations that allow variability in the regions defining the best-fit function’s domain.
97 citations
TL;DR: It is proved that the reformulation of (MQP) is the best one within a convex reformulation scheme, from the continuous relaxation point of view, and can be solved by a standard solver that uses a branch and bound algorithm.
Abstract: Let (MQP) be a general mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. In this paper, we present a convex reformulation of (MQP), i.e. we reformulate (MQP) into an equivalent program, with a convex objective function. Such a reformulation can be solved by a standard solver that uses a branch and bound algorithm. We prove that our reformulation is the best one within a convex reformulation scheme, from the continuous relaxation point of view. This reformulation, that we call MIQCR (Mixed Integer Quadratic Convex Reformulation), is based on the solution of an SDP relaxation of (MQP). Computational experiences are carried out with instances of (MQP) including one equality constraint or one inequality constraint. The results show that most of the considered instances with up to 40 variables can be solved in 1 h of CPU time by a standard solver.
97 citations
01 Jan 2012
TL;DR: An outer approximation generated by linearization of convex constraints and linear underestimation of nonconvex constraints is used to relax the problem and an overview of the reformulation, separation, and propagation techniques that are used to handle the quadratic constraints efficiently are given.
Abstract: This paper discusses how to build a solver for mixed integer quadratically constrained programs (MIQCPs) by extending a framework for constraint integer programming (CIP). The advantage of this approach is that we can utilize the full power of advanced MILP and CP technologies, in particular for the linear relaxation and the discrete components of the problem. We use an outer approximation generated by linearization of convex constraints and linear underestimation of nonconvex constraints to relax the problem. Further, we give an overview of the reformulation, separation, and propagation techniques that are used to handle the quadratic constraints efficiently. We implemented these methods in the branch-cut-and-price framework SCIP. Computational experiments indicating the potential of the approach and evaluating the impact of the algorithmic components are provided.
TL;DR: This work investigates the use of linear programming tools for solving semidefinite programming relaxations of quadratically constrained quadratic problems and presents classes of valid linear inequalities, including sparsePSD cuts, and principal minors PSD cuts.
Abstract: We investigate the use of linear programming tools for solving semidefinite programming relaxations of quadratically constrained quadratic problems. Classes of valid linear inequalities are presented, including sparse PSD cuts, and principal minors PSD cuts. Computational results based on instances from the literature are presented.
TL;DR: It is shown that any (nonconvex) quadratically constrained quadratic program (QCQP) can be represented as a generalized copositive program, and two representations are provided: one based on the concept of completely positive (CP) matrices over second-order cones, and onebased on CPMatrices over the positive semidefinite cone.
TL;DR: A sequential quadratic programming (SQP) method is presented that aims to overcome some of the drawbacks of contemporary SQP methods by adding an equality constrained phase that promotes fast convergence and improves performance in the presence of ill conditioning.
Abstract: A sequential quadratic programming (SQP) method is presented that aims to overcome some of the drawbacks of contemporary SQP methods It avoids the diculties associated with indefinite quadratic programming subproblems by defining this subproblem to be always convex The novel feature of the approach is the addition of an equality constrained phase that promotes fast convergence and improves performance in the presence of ill conditioning This equality constrained phase uses exact second order information and can be implemented using either a direct solve or an iterative method The paper studies the global and local convergence properties of the new algorithm and presents a set of numerical experiments to illustrate its practical performance
07 Oct 2012
TL;DR: In this article, a polynomial time test for certifying when such a solution is optimal is presented. But the test has no false positives and the results indicate that false negatives are rare, and the algorithm has excellent performance in practice.
Abstract: Triangulation of a three-dimensional point from n≥2 two-dimensional images can be formulated as a quadratically constrained quadratic program We propose an algorithm to extract candidate solutions to this problem from its semidefinite programming relaxations We then describe a sufficient condition and a polynomial time test for certifying when such a solution is optimal This test has no false positives Experiments indicate that false negatives are rare, and the algorithm has excellent performance in practice We explain this phenomenon in terms of the geometry of the triangulation problem
TL;DR: The assumption that the nonconvex (QP1QC) problem can be solved through a dual approach with no duality gap is relaxed to cover more general cases when the two matrices from the objective and the constraint functions can be simultaneously diagonalizable via congruence.
Abstract: This paper extends and completes the discussion by Xing et al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted) about the quadratic programming over one quadratic constraint (QP1QC). In particular, we relax the assumption to cover more general cases when the two matrices from the objective and the constraint functions can be simultaneously diagonalizable via congruence. Under such an assumption, the nonconvex (QP1QC) problem can be solved through a dual approach with no duality gap. This is unusual for general nonconvex programming but we can explain by showing that (QP1QC) is indeed equivalent to a linearly constrained convex problem, which happens to be dual of the dual of itself. Another type of hidden convexity can be also found in the boundarification technique developed in Xing et al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted).
25 Jun 2012
TL;DR: The proposed algorithm can be viewed as a generalization of Young's algorithm and analysis techniques for positive linear programs to the semidefinite programming setting and the total work of the algorithm is nearly-linear in the number of non-zero entries in the factorization.
Abstract: This paper studies the problem of finding a (1+e)-approximate solution to positive semidefinite programs. These are semidefinite programs in which all matrices in the constraints and objective are positive semidefinite and all scalars are non-negative. At FOCS'11, Jain and Yao gave an NC algorithm that requires O(t 1/e13 log13 m log n) iterations on input n constraint matrices of dimension m-by-m, where each iteration performs at least Ω(mω) work since it involves computing the spectral decomposition. We present a simpler NC parallel algorithm that on input with n constraint matrices, requires O(1/e4 log4 n log(1/e)) iterations, each of which involves only simple matrix operations and computing the trace of the product of a matrix exponential and a positive semidefinite matrix. Further, given a positive SDP in a factorized form, the total work of our algorithm is nearly-linear in the number of non-zero entries in the factorization. Our algorithm can be viewed as a generalization of Young's algorithm and analysis techniques for positive linear programs (Young, FOCS'01) to the semidefinite programming setting.
TL;DR: In this paper, a semidefinite programming approach for the solution of scheduling problems that have disjoint feasible regions is presented. The approach is based on formulating the problem as a mathematical program with vanishing constraints.
Abstract: This paper presents a semidefinite programming approach for the solution of scheduling problems that have disjoint feasible regions. The approach is based on formulating the problem as a mathematical program with vanishing constraints. The solution is obtained using Shor's semidefinite relaxation scheme combined with a rank constraint which is enforced via convex iteration.
TL;DR: This work explores how to minimize the spread of infection via quarantining with limited resources, and shows how this problem can be posed as a non-convex quadratically constrained quadratic program (QCQP), and used to derive a link removal algorithm.
Abstract: The structure of the contact network through which a disease spreads may influence the optimal use of resources for epidemic control. In this work, we explore how to minimize the spread of infection via quarantining with limited resources. In particular, we examine which links should be removed from the contact network, given a constraint on the number of removable links, such that the number of nodes which are no longer at risk for infection is maximized. We show how this problem can be posed as a non-convex quadratically constrained quadratic program (QCQP), and we use this formulation to derive a link removal algorithm. The performance of our QCQP-based algorithm is validated on small Erdős–Renyi and small-world random graphs, and then tested on larger, more realistic networks, including a real-world network of injection drug use. We show that our approach achieves near optimal performance and out-performs other intuitive link removal algorithms, such as removing links in order of edge centrality.
01 Jan 2012
TL;DR: This work focuses on the special case where the cone is chosen as the cone of positive semidefinite matrices for which the resulting optimization problem is called a semideFinite optimization problem.
Abstract: Conic optimization refers to the problem of optimizing a linear function over the intersection of an affine space and a closed convex cone. We focus particularly on the special case where the cone is chosen as the cone of positive semidefinite matrices for which the resulting optimization problem is called a semidefinite optimization problem.
TL;DR: In this article, the authors study how to solve semidefinite programming (SDP) relaxations for large-scale polynomial optimization problems and show that with regularization methods, significantly bigger problems could be solved on a regular computer, which is almost impossible with interior point methods.
Abstract: We study how to solve semidefinite programming (SDP) relaxations for large-scale polynomial optimization. When interior-point methods are used, typically only small or moderately large problems could be solved. This paper studies regularization methods for solving polynomial optimization problems. We describe these methods for semidefinite optimization with block structures and then apply them to solve large-scale polynomial optimization problems. The performance is tested on various numerical examples. With regularization methods, significantly bigger problems could be solved on a regular computer, which is almost impossible with interior point methods.
TL;DR: This work presents a fuzzy relation quadratic programming model with a quadRatic objective function subject to the max-product fuzzy relation inequality constraints and proposes an algorithm to solve it.
01 Dec 2012
TL;DR: A smooth vector field is proposed that is constructed such that its trajectories converge to the saddle point of the Lagrangian function associated to the convex optimization problem.
Abstract: In this paper we consider the class of convex optimization problems with affine inequality constraints and focus hereby on the class of quadratic programs. We propose a smooth vector field that is constructed such that its trajectories converge to the saddle point of the Lagrangian function associated to the convex optimization problem. We establish global asymptotic stability as well as exponential stability under mild assumptions for different variants of the vector field and propose a continuous-time Nesterov method.
TL;DR: This paper considers the multi-way clustering problem based on graph partitioning models by the Ratio cut and Normalized cut and forms the problem using new quadratic models using Spectral relaxations, new semidefinite programming relaxations and linearization techniques.
Abstract: In this paper, we consider the multi-way clustering problem based on graph partitioning models by the Ratio cut and Normalized cut. We formulate the problem using new quadratic models. Spectral relaxations, new semidefinite programming relaxations and linearization techniques are used to solve these problems. It has been shown that our proposed methods can obtain improved solutions. We also adapt our proposed techniques to the bipartite graph partitioning problem for biclustering.
TL;DR: A branch-and-bound algorithm that applies a novel bounding procedure, based on recent semidefinite programming techniques, that is less tight than the standard semideFinite bounds, but cheaper to get.
Abstract: This paper deals with the computation of exact solutions of a classical NP-hard problem in combinatorial optimization, the $$k$$-cluster problem. This problem consists in finding a heaviest subgraph with $$k$$ nodes in an edge weighted graph. We present a branch-and-bound algorithm that applies a novel bounding procedure, based on recent semidefinite programming techniques. We use new semidefinite bounds that are less tight than the standard semidefinite bounds, but cheaper to get. The experiments show that this approach is competitive with the best existing ones.
TL;DR: This paper establishes exact SDP relaxations for classes of nonlinear semidefinite programming problems with SOS-convex polynomials and derives numerically checkable conditions, completely characterizing minimizers of these classes of problems.
TL;DR: Certain rich geometric properties hidden behind quadratic 0–1 programming are explored and new lower bounding methods and variable fixation techniques are derived by investigating geometric features of the ellipse contour of a convex quadRatic function.
Abstract: We explore in this paper certain rich geometric properties hidden behind quadratic 0---1 programming. Especially, we derive new lower bounding methods and variable fixation techniques for quadratic 0---1 optimization problems by investigating geometric features of the ellipse contour of a (perturbed) convex quadratic function. These findings further lead to some new optimality conditions for quadratic 0---1 programming. Integrating these novel solution schemes into a proposed solution algorithm of a branch-and-bound type, we obtain promising preliminary computational results.
TL;DR: Numerical results indicate that the computational complexity of the proposed method scales well when applied to a finite horizon discrete-time optimal control problem with linear dynamics, quadratic cost, and linear inequality constraints, which arises in model predictive control applications.
Abstract: A method is proposed for reducing the cost of computing search directions in an interior point method for a quadratic program. The KKT system is partitioned and modified, based on the ratios of the slack variables and dual variables associated with the inequality constraints, to produce a smaller, approximate linear system. Analytical and numerical results are included that suggest the distribution of eigenvalues of the new, approximate system matrix is improved, which makes it more amenable to being solved with an iterative linear solver. For this purpose, new preconditioners are also presented to allow iterative methods, such as MINRES, to be used. Numerical results indicate that the computational complexity of the proposed method scales well when applied to a finite horizon discrete-time optimal control problem with linear dynamics, quadratic cost, and linear inequality constraints, which arises in model predictive control applications.