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.
Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inne...
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Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive semidefinite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space-classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semidefinite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm- using the labeled part of the data one can learn an embedding also for the unlabeled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method for learning the 2-norm soft margin parameter in support vector machines, solving an important open problem.
In this paper we present an extension to SDP of the well known infeasible Interior Point method for linear programming of Kojima, Megiddo and Mizuno (A primal-dual infeasible-interior-point algorithm for Linear Progra...
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In this paper we present an extension to SDP of the well known infeasible Interior Point method for linear programming of Kojima, Megiddo and Mizuno (A primal-dual infeasible-interior-point algorithm for Linear programming, Math. Progr., 1993). The extension developed here allows the use of inexact search directions;i.e., the linear systems defining the search directions can be solved with an accuracy that increases as the solution is approached. A convergence analysis is carried out and the global convergence of the method is proved.
In this paper, we present a simple algorithm to obtain mechanically SDP relaxations for any quadratic or linear program with bivalent variables, starting from an existing linear relaxation of the considered combinator...
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In this paper, we present a simple algorithm to obtain mechanically SDP relaxations for any quadratic or linear program with bivalent variables, starting from an existing linear relaxation of the considered combinatorial problem. A significant advantage of our approach is that we obtain an improvement on the linear relaxation we start from. Moreover, we can take into account all the existing theoretical and practical experience accumulated in the linear approach. After presenting the rules to treat each type of constraint, we describe our algorithm, and then apply it to obtain semidefinite relaxations for three classical combinatorial problems: the K-CLUSTER problem, the Quadratic Assignment Problem, and the Constrained-Memory Allocation Problem. We show that we obtain better SDP relaxations than the previous ones, and we report computational experiments for the three problems.
A number of recent papers on approximation algorithms have used the square roots of unity, -1 and 1, to represent binary decision variables for problems in combinatorial optimization, and have relaxed these to unit ve...
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A number of recent papers on approximation algorithms have used the square roots of unity, -1 and 1, to represent binary decision variables for problems in combinatorial optimization, and have relaxed these to unit vectors in real space using semidefinite programming in order to obtain near optimum solutions to these problems. In this paper, we consider using the cube roots of unity, 1, e(i2pi/3), and e(i4pi/3), to represent ternary decision variables for problems in combinatorial optimization. Here the natural relaxation is that of unit vectors in complex space. We use an extension of semidefinite programming to complex space to solve the natural relaxation, and use a natural extension of the random hyperplane technique introduced by the authors in Goemans and Williamson (J. ACM 42 (1995) 1115-1145) to obtain near-optimum solutions to the problems. In particular, we consider the problem of maximizing the total weight of satisfied equations x(u) - x(v) drop c (mod 3) and inequations x - x(v) not equivalent to c (mod 3), where x(u) epsilon {0, 1, 2} for all u. This problem can be used to model the MAx-3-CUT problem and a directed variant we call MAX-3-DICUT. For the general problem, we obtain a 0.793733-approximation algorithm. If the instance contains only inequations (as it does for MAX-3-CUT), we obtain a performance guarantee of (7)/(12) + (3)/(4pi2) arccos(2)(- 1/4) - epsilon>0.836008. This compares with proven performance guarantees of 0.800217 for MAX-3-CUT (by Frieze and Jerrum (Algorithmica 18 (1997) 67-81) and 1 + 10(-8) for the general problem (by Andersson et al. (J. Algorithms 3 39 (2001) 162-204)). It matches the guarantee of 0.836008 for MAX-3-CUT found independently by de Klerk et al. (On approximate graph colouring and Max-k-Cut algorithms based on the 9-function, Manuscript, October 2000). We show that all these algorithms are in fact equivalent in the case of MAX-3CUT, and that our algorithm is the same as that of Andersson et al. in the more gener
Side chain positioning is an important subproblem of the general protein-structure-prediction problem, with applications in homology modeling and protein design. The side chain positioning problem takes a fixed backbo...
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Side chain positioning is an important subproblem of the general protein-structure-prediction problem, with applications in homology modeling and protein design. The side chain positioning problem takes a fixed backbone and a protein sequence and predicts the lowest energy conformation of the protein's side chains on this backbone. We study a widely used version of the problem where the side chain positioning procedure uses a rotamer library and an energy function that can be expressed as a sum of pairwise terms. The problem is NP-complete;we show that it cannot even be approximated. In practice, it is tackled by a variety of general search techniques and specialized heuristics. Here, we propose formulating the side chain positioning problem as an instance of semidefinite programming (SDP). We introduce two novel rounding schemes and provide theoretical justification for their effectiveness under various conditions. We apply our method on simulated data, as well as on the computational redesign of two naturally occurring protein cores, and show that our SDP approach generally finds good solutions. Beyond the context of side chain positioning, our very general rounding schemes should be applicable elsewhere.
The trust region (TR) subproblem (the minimization of a quadratic objective subject to one quadratic constraint and denoted TRS) has many applications in diverse areas, e.g., function minimization, sequential quadrati...
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The trust region (TR) subproblem (the minimization of a quadratic objective subject to one quadratic constraint and denoted TRS) has many applications in diverse areas, e.g., function minimization, sequential quadratic programming, regularization, ridge regression, and discrete optimization. In particular, it determines the step in TR algorithms for function minimization. Trust region algorithms are popular for their strong convergence properties. However, a drawback has been the inability to exploit sparsity as well as the difficulty in dealing with the so-called hard case. These concerns have been addressed by recent advances in the theory and algorithmic development. In particular, this has allowed Lanczos techniques to replace Cholesky factorizations. This article provides an in depth study of TRS and its properties as well as a survey of recent advances. We emphasize large scale problems and robustness. This is done using semidefinite programming (SDP) and the modern primal-dual approaches as a unifying framework. The SDP framework arises naturally and solves TRS efficiently. In addition, it shows that TRS is always a well-posed problem, i.e., the optimal value and an optimum can be calculated to a given tolerance. We provide both theoretical and empirical evidence to illustrate the strength of the SDP and duality approach. In particular, this includes new insights and techniques for handling the hard case, as well as numerical results on large test problems.
In this paper we present a primal-dual inexact infeasible interior-point algorithm for semidefinite programming problems (SDP). This algorithm allows the use of search directions that are calculated from the defining ...
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In this paper we present a primal-dual inexact infeasible interior-point algorithm for semidefinite programming problems (SDP). This algorithm allows the use of search directions that are calculated from the defining linear system with only moderate accuracy, and does not require feasibility to be maintained even if the initial iterate happened to be a feasible solution of the problem. Under a mild assumption on the inexactness, we show that the algorithm can find an epsilon-approximate solution of an SDP in O(n(2)ln(1/epsilon)) iterations. This bound of our algorithm is the same as that of the exact infeasible interior point algorithms proposed by Y. Zhang.
Based on the semidefinite programming relaxation of the CDMA maximum likelihood multiuser detection problem, a detection strategy by the successive quadratic programming algorithm is presented. Coupled with the random...
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Based on the semidefinite programming relaxation of the CDMA maximum likelihood multiuser detection problem, a detection strategy by the successive quadratic programming algorithm is presented. Coupled with the randomized cut generation scheme, the suboptimal solution of the multiuser detection problem in obtained. Compared to the interior point methods previously reported based on semidefmite programming, simulations demonstrate that the successive quadratic programming algorithm often yields the similar BER performances of the multiuser detection problem. But the average CPU time of this approach is significantly reduced.
We consider the attitude control problem augmented with an arbitrary number of nonconvex quadratic constraints. By utilizing the implicit magnitude constraint of the state vector (i.e., quaternion), we are able to red...
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We consider the attitude control problem augmented with an arbitrary number of nonconvex quadratic constraints. By utilizing the implicit magnitude constraint of the state vector (i.e., quaternion), we are able to reduce this problem to a convex quadratically constrained quadratic program or a semidefinite program. This result leads to an efficient solution strategy for general quadratically constrained attitude control problems. An example demonstrates the proposed methodology.
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