We extend the concept of epsilon-sensitivity analysis developed for linear programming to that for semidefinite programming. First, the notion of epsilon-optimality for a given semidefinite programming problem is defi...
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We extend the concept of epsilon-sensitivity analysis developed for linear programming to that for semidefinite programming. First, the notion of epsilon-optimality for a given semidefinite programming problem is defined, and then a generic c-sensitivity analysis for semidefinite programming is introduced. Based on the definitions, we develop an implementation of the generic epsilon-sensitivity analysis under perturbations of either the cost parameters or the right-hand side. (C) 2004 Elsevier B.V. All rights reserved.
The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in ...
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The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. It is well known that SAT is in general NP-complete, although several important special cases can be solved in polynomial time. semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a matrix variable X is maximized (or minimized) subject to linear constraints on the elements of X and the additional constraint that X be positive semidefinite. We are interested in the application of SDP to satisfiability problems, and in particular in how SDP can be used to detect unsatisfiability. In this paper we introduce a new SDP relaxation for the satisfiability problem. This SDP relaxation arises from the recently introduced paradigm of "higher liftings" for constructing semidefinite programming relaxations of discrete optimization problems. To derive the SDP relaxation, we first formulate SAT as an optimization problem involving matrices. Relaxing this formulation yields an SDP which significantly improves on the previous relaxations in the literature. The important characteristics of the SDP relaxation are its ability to prove that a given SAT formula is unsatisfiable independently of the lengths of the clauses in the formula, its potential to yield truth assignments satisfying the SAT instance if a feasible matrix of sufficiently low rank is computed, and the fact that it is more amenable to practical computation than previous SDPs arising from higher liftings. We present theoretical and computational results that support these claims.
In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lovasz and Schrijver to the generalized stable set problem. We define a convex set which s...
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In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lovasz and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a linear function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semidefinite programming relaxation of Lovasz and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof.
Primal-dual interior-point (p-d i-p) methods for semidefinite programming (SDP) are generally based on solving a system of matrix equations for a Newton type search direction for a symmetrization of the optimality con...
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Primal-dual interior-point (p-d i-p) methods for semidefinite programming (SDP) are generally based on solving a system of matrix equations for a Newton type search direction for a symmetrization of the optimality conditions. These search directions followed the derivation of similar p-d i-p methods for linear programming (LP). Among these, a computationally interesting search direction is the AHO direction. However, in contrast to the LP case, existence and uniqueness of the AHO search direction is not guaranteed Linder the standard nondegeneracy assumptions. Two different sufficient conditions are known that guarantee the existence and uniqueness independent of the specific linear constraints. The first is given by Shida-Shindoh-Kojima and is based on the semidefiniteness of the symmetrization of the product SX at the current iterate. The second is a centrality condition given first by Monteiro-Zanjacomo and then improved by Monteiro-Todd. In this paper, we revisit and strengthen both of the above mentioned sufficient conditions. We include characterizations for existence and uniqueness in the matrix equations arising from the linearization of the optimality conditions. As well, we present new results on the relationship between the Kronecker product and the symmetric Kronecker product that arise from these matrix equations. We conclude with a proof that the existence and uniqueness of the AHO direction is a generic property for every SDP problem and extend the results to the general Monteiro-Zhang family of search directions. (c) 2005 Elsevier Inc. All rights reserved.
The presence of complementarity constraints brings a combinatorial flavour to an optimization problem. A quadratic programming problem with complementarity constraints can be relaxed to give a semidefinite programming...
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The presence of complementarity constraints brings a combinatorial flavour to an optimization problem. A quadratic programming problem with complementarity constraints can be relaxed to give a semidefinite programming problem. The solution to this relaxation can be used to generate feasible solutions to the complementarity constraints. A quadratic programming problem is solved for each of these feasible solutions and the best resulting solution provides an estimate for the optimal solution to the quadratic program with complementarity constraints. Computational testing of such an approach is described for a problem arising in portfolio optimization.
We give a new upper bound on the maximum size A (n, d) of a binary code of word length n and minimum distance at least d. It is based on block-diagonalizing the Terwilliger algebra of the Hamming cube. The bound stren...
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We give a new upper bound on the maximum size A (n, d) of a binary code of word length n and minimum distance at least d. It is based on block-diagonalizing the Terwilliger algebra of the Hamming cube. The bound strengthens the Delsarte bound, and can be calculated with semidefinite programming in time bounded by a polynomial in n. We show that it improves a number of known upper bounds for concrete values of n and d. From this we also derive a new upper bound on the maximum size A(n, d, w) of a binary code of word length n, minimum distance at least d, and constant weight w, again strengthening the Delsarte bound and yielding several improved upper bounds for concrete values of n, d, and w.
The low-rank semidefinite programming problem LRSDPr is a restriction of the semidefinite programming problem SDP in which a bound r is imposed on the rank of X, and it is well known that LRSDPr is equivalent to SDP i...
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The low-rank semidefinite programming problem LRSDPr is a restriction of the semidefinite programming problem SDP in which a bound r is imposed on the rank of X, and it is well known that LRSDPr is equivalent to SDP if r is not too small. In this paper, we classify the local minima of LRSDPr and prove the optimal convergence of a slight variant of the successful, yet experimental, algorithm of Burer and Monteiro [5], which handles LRSDPr via the nonconvex change of variables X = RRT. In addition, for particular problem classes, we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm. Computational results are presented on a set of combinatorial optimization relaxations, including some of the largest quadratic assignment SDPs solved to date.
In many theoretical proposals appeared recently, semidefinite programming was considered as a way to express quantum entanglement. Using semidefinite optimization method, we prove the Lewenstein-Sanpera lemma in a sim...
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In many theoretical proposals appeared recently, semidefinite programming was considered as a way to express quantum entanglement. Using semidefinite optimization method, we prove the Lewenstein-Sanpera lemma in a simple elegant manner. Particularly, using this method we obtain Lewenstein-Sanpera decomposition for some examples such as: generic two qubit state in Wootters's basis, Iso-concurrence state, Bell decomposable state and 2 x 3 Bell decomposable state.
作者：
Popescu, IINSEAD
Decis Sci Area F-77300 Fontainebleau France
We provide an optimization framework for computing optimal upper and lower bounds on functional expectations of distributions with special properties, given moment constraints. Bertsimas and Popescu (Optimal inequalit...
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We provide an optimization framework for computing optimal upper and lower bounds on functional expectations of distributions with special properties, given moment constraints. Bertsimas and Popescu (Optimal inequalities in probability theory: a convex optimization approach. SIAM J Optim. 2004. Forthcoming) have already shown how to obtain optimal moment inequalities for arbitrary distributions via semidefinite programming. These bounds are not sharp if the underlying distributions possess additional structural properties, including symmetry, unimodality, convexity, or smoothness. For convex distribution classes that are in some sense generated by an appropriate parametric family, we use conic duality to show how optimal moment bounds can be efficiently computed as semidefinite programs. In particular, we obtain generalizations of Chebyshev's inequality for symmetric and unimodal distributions and provide numerical calculations to compare these bounds, given higher-order moments. We also extend these results for multivariate distributions.
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