This paper is concerned with the analysis and comparison of semidefinite programming (SDP) relaxations for the satisfiability (SAT) problem. Our presentation is focussed on the special case of 3-SAT, but the ideas pre...
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This paper is concerned with the analysis and comparison of semidefinite programming (SDP) relaxations for the satisfiability (SAT) problem. Our presentation is focussed on the special case of 3-SAT, but the ideas presented can in principle be extended to any instance of SAT specified by a set of boolean variables and a propositional formula in conjunctive normal form. We propose a new SDP relaxation for 3-SAT and prove some of its theoretical properties. We also show that, together with two SDP relaxations previously proposed in the literature, the new relaxation completes a trio of linearly sized relaxations with increasing rank-based guarantees for proving satisfiability. A comparison of the relative practical performances of the SDP relaxations shows that, among these three relaxations, the new relaxation provides the best tradeoff between theoretical strength and practical performance within an enumerative algorithm.
We present a new semidefinite programming formulation of sum-of-squares representations of nonnegative polynomials, cosine polynomials, and trigonometric polynomials of one variable. The parametrization is based on di...
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We present a new semidefinite programming formulation of sum-of-squares representations of nonnegative polynomials, cosine polynomials, and trigonometric polynomials of one variable. The parametrization is based on discrete transforms (specifically, the discrete Fourier, cosine, and polynomial transforms) and has a simple structure that can be exploited by straightforward modi. cations of standard interior-point algorithms.
The purpose of this paper is to develop certain geometric results concerning the feasible regions of semidefinite Programs, called here Spectrahedra. We first develop a characterization for the faces of spectrahedra. ...
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The purpose of this paper is to develop certain geometric results concerning the feasible regions of semidefinite Programs, called here Spectrahedra. We first develop a characterization for the faces of spectrahedra. More specifically, given a point x in a spectrahedron, we derive an expression for the minimal face containing x. Among other things, this is shown to yield characterizations for extreme points and extreme rays of spectrahedra. We then introduce the notion of an algebraic polar of a spectrahedron, and present its relation to the usual geometric polar.
Consider the problem of signal detection via multiple distributed noisy sensors. We study a linear decision fusion rule of [Z. Quan, S. Cui, and A. H. Sayed, "Optimal Linear Cooperation for Spectrum Sensing in Co...
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Consider the problem of signal detection via multiple distributed noisy sensors. We study a linear decision fusion rule of [Z. Quan, S. Cui, and A. H. Sayed, "Optimal Linear Cooperation for Spectrum Sensing in Cognitive Radio Networks," IEEE J. Sel. Topics Signal Process., vol. 2, no. 1, pp. 28-40, Feb. 2008] to combine the local statistics from individual sensors into a global statistic for binary hypothesis testing. The objective is to maximize the probability of detection subject to an upper limit on the probability of false alarm. We propose a more efficient solution that employs a divide-and-conquer strategy to divide the decision optimization problem into two subproblems. Each subproblem is a nonconvex program with a quadratic constraint. Through a judicious reformulation and by employing a special matrix decomposition technique, we show that the two nonconvex subproblems can be solved by semidefinite programs in a globally optimal fashion. Hence, we can obtain the optimal linear fusion rule for the distributed detection problem. Compared with the likelihood-ratio test approach, optimal linear fusion can achieve comparable performance with considerable design flexibility and reduced complexity.
In this paper, we propose a second-order corrector interior-point algorithm for semidefinite programming (SDP). This algorithm is based on the wide neighborhood. The complexity bound is O(root nL) for the Nesterov-Tod...
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In this paper, we propose a second-order corrector interior-point algorithm for semidefinite programming (SDP). This algorithm is based on the wide neighborhood. The complexity bound is O(root nL) for the Nesterov-Todd direction, which coincides with the best known complexity results for SDP. To our best knowledge, this is the first wide neighborhood second-order corrector algorithm with the same complexity as small neighborhood interior-point methods for SDP. Some numerical results are provided as well.
Optimal experiment design (OED) for parameter estimation in nonlinear dynamic (bio)chemical processes is studied in this work. To reduce the uncertainty in an experiment, a suitable measure of the Fisher information m...
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Optimal experiment design (OED) for parameter estimation in nonlinear dynamic (bio)chemical processes is studied in this work. To reduce the uncertainty in an experiment, a suitable measure of the Fisher information matrix or variance-covariance matrix has to be optimized. In this work, novel optimization algorithms based on sequential semidefinite programming (SDP) are proposed. The sequential SDP approach has specific advantages over sequential quadratic programming in the context of OED. First of all, it guarantees on a matrix level a decrease of the uncertainty in the parameter estimation procedure by introducing a linear matrix inequality. Second, it allows an easy formulation of E-optimal designs in a direct optimal control optimization scheme. Finally, a third advantage of SDP is that problems involving the inverse of a matrix can be easily reformulated. The proposed techniques are illustrated in the design of experiments for a fed-batch bioreactor and a microbial kinetics case study. (c) 2014 American Institute of Chemical Engineers AIChE J, 60: 1728-1739, 2014
In this two-part study, we discuss possible extensions of the main ideas and methods of constrained DC optimization to the case of nonlinear semidefinite programming problems and more general nonlinear cone constraine...
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In this two-part study, we discuss possible extensions of the main ideas and methods of constrained DC optimization to the case of nonlinear semidefinite programming problems and more general nonlinear cone constrained optimization problems. In the first paper, we analyse two different approaches to the definition of DC matrix-valued functions (namely, order-theoretic and componentwise), study some properties of convex and DC matrix-valued mappings and demonstrate how to compute DC decompositions of some nonlinear semidefinite constraints appearing in applications. We also compute a DC decomposition of the maximal eigenvalue of a DC matrix-valued function. This DC decomposition can be used to reformulate DC semidefinite constraints as DC inequality constrains. Finally, we study local optimality conditions for general cone constrained DC optimization problems.
In this paper, we consider an alternating direction algorithm for the solution of semidefinite programming problems (SDP). The main idea of our algorithm is that we reformulate the complementary conditions in the prim...
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In this paper, we consider an alternating direction algorithm for the solution of semidefinite programming problems (SDP). The main idea of our algorithm is that we reformulate the complementary conditions in the primal-dual optimality conditions as a projection equation. By using this reformulation, we only need to make one projection and solve a linear system of equation with reduced dimension in each iterate. We prove that the generated sequence converges to the solution of the SDP under weak conditions. (c) 2005 Elsevier B.V. All rights reserved.
We study computability and applicability of error bounds for a given semidefinite pro-gramming problem under the assumption that the recession function associated with the constraint system satisfies the Slater condit...
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We study computability and applicability of error bounds for a given semidefinite pro-gramming problem under the assumption that the recession function associated with the constraint system satisfies the Slater condition. Specifically, we give computable error bounds for the distances between feasible sets, optimal objective values, and optimal solution sets in terms of an upper bound for the condition number of a constraint system, a Lipschitz constant of the objective function, and the size of perturbation. Moreover, we are able to obtain an exact penalty function for semidefinite programming along with a lower bound for penalty parameters. We also apply the results to a class of statistical problems.
This paper deals with the computation of exact solutions of a classical NP-hard problem in combinatorial optimization, the -cluster problem. This problem consists in finding a heaviest subgraph with nodes in an edge w...
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This paper deals with the computation of exact solutions of a classical NP-hard problem in combinatorial optimization, the -cluster problem. This problem consists in finding a heaviest subgraph with 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.
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