The zero duality gap that underpins the duality theory is one of the central ingredients in optimisation. In convex programming, it means that the optimal values of a given convex program and its associated dual progr...
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The zero duality gap that underpins the duality theory is one of the central ingredients in optimisation. In convex programming, it means that the optimal values of a given convex program and its associated dual program are equal. It allows, in particular, the development of efficient numerical schemes. However, the zero duality gap property does not always hold even for finite-dimensional problems and it frequently fails for problems with non-polyhedral constraints such as the ones in semidefinite programming problems. Over the years, various criteria have been developed ensuring zero duality gaps for convex programming problems. In the present work, we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem. Globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions. For convex semidefinite programs, we establish necessary and sufficient dual conditions for stable zero duality gaps, as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right-hand side and convex objective function. Zero duality gap results for second-order cone programming problems are also given. Our approach makes use of elegant conjugate analysis and Fenchel's duality. (C) 2009 Elsevier Inc. All rights reserved.
In this paper we present penalty and barrier methods for solving general convex semidefinite programming problems. More precisely, the constraint set is described by a convex operator that takes its values in the cone...
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In this paper we present penalty and barrier methods for solving general convex semidefinite programming problems. More precisely, the constraint set is described by a convex operator that takes its values in the cone of negative semidefinite symmetric matrices. This class of methods is an extension of penalty and barrier methods for convex optimization to this setting. We provide implementable stopping rules and prove the convergence of the primal and dual paths obtained by these methods under minimal assumptions. The two parameters approach for penalty methods is also extended. As for usual convex programming, we prove that after a finite number of steps all iterates will be feasible.
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 discuss computational enhancements for the low-rank semidefinite programming algorithm, including the extension to block semidefinite programs (SDPs), an exact linesearch procedure, and a dynamic rank reduction sch...
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We discuss computational enhancements for the low-rank semidefinite programming algorithm, including the extension to block semidefinite programs (SDPs), an exact linesearch procedure, and a dynamic rank reduction scheme. A truncated-Newton method is also introduced, and several preconditioning strategies are proposed. Numerical experiments illustrating these enhancements are provided on a wide class of test problems. In particular, the truncated-Newton variant is able to achieve high accuracy in modest amounts of time on maximum-cut-type SDPs.
Can we detect low dimensional structure in high dimensional data sets of images? In this paper, we propose an algorithm for unsupervised learning of image manifolds by semidefinite programming. Given a data set of ima...
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Can we detect low dimensional structure in high dimensional data sets of images? In this paper, we propose an algorithm for unsupervised learning of image manifolds by semidefinite programming. Given a data set of images, our algorithm computes a low dimensional representation of each image with the property that distances between nearby images are preserved. More generally, it can be used to analyze high dimensional data that lies on or near a low dimensional manifold. We illustrate the algorithm on easily visualized examples of curves and surfaces, as well as on actual images of faces, handwritten digits, and solid objects.
We consider the problem of constructing an optimal set of orthogonal vectors from a given set of vectors in a real Hilbert space. The vectors are chosen to minimize the sum of the squared norms of the errors between t...
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We consider the problem of constructing an optimal set of orthogonal vectors from a given set of vectors in a real Hilbert space. The vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors. We show that the design of the optimal vectors, referred to as the least-squares (LS) orthogonal vectors, can be formulated as a semidefinite programming (SDP) problem. Using the many well-known algorithms for solving SDPs, which are guaranteed to converge to the global optimum, the LS vectors can be computed very efficiently in polynomial time within any desired accuracy. By exploiting the connection between our problem and a quantum detection problem we derive a closed form analytical expression for the LS orthogonal vectors, for vector sets with a broad class of symmetry properties. Specifically, we consider geometrically uniform (GU) sets with a possibly non-abelian generating group, and compound GU sets which consist of subsets that are GU. (c) 2005 Elsevier Inc. All rights reserved.
We give a proximal bundle method for minimizing a convex function f over a convex set C. It requires evaluating f and its subgradients with a fixed but possibly unknown accuracy epsilon > 0. Each iteration involves...
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We give a proximal bundle method for minimizing a convex function f over a convex set C. It requires evaluating f and its subgradients with a fixed but possibly unknown accuracy epsilon > 0. Each iteration involves solving an unconstrained proximal subproblem and projecting a certain point onto C. The method asymptotically finds points that are epsilon-optimal. In Lagrangian relaxation of convex programs, it allows for epsilon-accurate solutions of Lagrangian subproblems and finds epsilon-optimal primal solutions. For semidefinite programming problems, it extends the highly successful spectral bundle method to the case of inexact eigenvalue computations.
Cutting plane methods provide the means to solve large scale semidefinite programs ( SDP) cheaply and quickly. They can also conceivably be employed for the purposes of re-optimization after branching or the addition ...
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Cutting plane methods provide the means to solve large scale semidefinite programs ( SDP) cheaply and quickly. They can also conceivably be employed for the purposes of re-optimization after branching or the addition of cutting planes. We give a survey of various cutting plane approaches for SDP in this paper. These cutting plane approaches arise from various perspectives, and include techniques based on interior point cutting plane approaches, non-differentiable optimization, and finally an approach which mimics the simplex method for linear programming ( LP). We present an accessible introduction to various cutting plane approaches that have appeared in the literature. We place these methods in a unifying framework which illustrates how each approach arises as a natural enhancement of a primordial LP cutting plane scheme based on a semi-infinite formulation of the SDP.
We give a new upper bound on the maximum size A(q) (n, d) of a code of word length n and minimum Hamming distance at least d over the alphabet of q >= 3 letters. By block-diagonalizing the Terwilliger algebra of th...
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We give a new upper bound on the maximum size A(q) (n, d) of a code of word length n and minimum Hamming distance at least d over the alphabet of q >= 3 letters. By block-diagonalizing the Terwilliger algebra of the nonbinary Hamming scheme, the bound can be calculated in time polynomial in n using semidefinite programming. For q = 3, 4, 5 this gives several improved upper bounds for concrete values of n and d. This work builds upon previous results of Schrijver [A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory 51 (2005) 2859-2866] on the Terwilliger algebra of the binary Hamming scheme. (c) 2006 Elsevier Inc. All rights reserved.
We investigate solution of the maximum cut problem using a polyhedral cut and price approach. The dual of the well-known SDP relaxation of maxcut is formulated as a semi-infinite linear programming problem, which is s...
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We investigate solution of the maximum cut problem using a polyhedral cut and price approach. The dual of the well-known SDP relaxation of maxcut is formulated as a semi-infinite linear programming problem, which is solved within an interior point cutting plane algorithm in a dual setting;this constitutes the pricing (column generation) phase of the algorithm. Cutting planes based on the polyhedral theory of the maxcut problem are then added to the primal problem in order to improve the SDP relaxation;this is the cutting phase of the algorithm. We provide computational results, and compare these results with a standard SDP cutting plane scheme.
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