In this paper, an exact dual is derived for semidefinite programming (SDP), for which strong duality properties hold without any regularity assumptions. Its main features are: (i) The new dual is an explicit semidefin...
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In this paper, an exact dual is derived for semidefinite programming (SDP), for which strong duality properties hold without any regularity assumptions. Its main features are: (i) The new dual is an explicit semidefinite program with polynomially many variables and polynomial size coefficient bitlengths. (ii) If the primal is feasible, then it is bounded if and only if the dual is feasible. (iii) When the primal is feasible and bounded, then its optimum value equals that of the dual, or in other words, there is no duality gap. Further, the dual attains this common optimum value. (iv) It yields a precise theorem of the alternative for semidefinite inequality systems, i.e. a characterization of the infeasibility of a semidefinite inequality in terms of the feasibility of another polynomial size semidefinite inequality. The standard duality for linear programming satisfies all of the above features, but no such explicit gap-free dual program of polynomial size was previously known for SDP, without Slater-like conditions being assumed. The dual is then applied to derive certain complexity results for SDP. The decision problem of semidefinite Feasibility (SDFP), which asks to determine if a given semidefinite inequality system is feasible, is the central problem of interest, he complexity of SDFP is unknown, but we show the following: (i) In the Turing machine model, the membership or nonmembership of SDFP in NP and Co-NP is simultaneous;hence SDFP is not NP-Complete unless NP = Co-NP. (ii) In the real number model of Blum, Shub and Smale, SDEP is in NP boolean AND Co-NP. (C) 1997 The Mathematical programming Society, Inc. Published by Elsevier Science B.V.
The Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro and Nesterov-Todd search directions have been used in many primal-dual interior-point methods for semidefinite programs. This paper proposes an effic...
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The Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro and Nesterov-Todd search directions have been used in many primal-dual interior-point methods for semidefinite programs. This paper proposes an efficient method for computing the two directions when the semidefinite program to be solved is large scale and sparse. (C) 1997 The Mathematical programming Society, Inc. Published by Elsevier Science B.V.
The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for large-scale linear programming. Many interior point methods for li...
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The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for large-scale linear programming. Many interior point methods for linear programming have now been extended to the more general semidefinite case, but the initialization problem remained unsolved. In this paper we show that the initialization strategy of embedding the problem in a self-dual skew-symmetric problem can also be extended to the semidefinite case. This method also provides a solution for the initialization of quadratic programs and it is applicable to more general convex problems with conic formulation. (C) 1997 Elsevier Science B.V.
This note establishes a new sufficient condition for the existence and uniqueness of the Alizadeh-Haeberly-Overton direction for semidefinite programming. (C) 1997 The Mathematical programming Society, Inc. Published ...
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This note establishes a new sufficient condition for the existence and uniqueness of the Alizadeh-Haeberly-Overton direction for semidefinite programming. (C) 1997 The Mathematical programming Society, Inc. Published by Elsevier Science B.V.
We present and motivate a new model of the truss topology design problem, where the rigidity of the resulting truss with respect both to given loading scenarios and small "occasional" loads is optimized. It ...
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We present and motivate a new model of the truss topology design problem, where the rigidity of the resulting truss with respect both to given loading scenarios and small "occasional" loads is optimized. It is shown that the resulting optimization problem is a semidefinite program. We derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples.
This paper deals with a class of primal-dual interior-point algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh, and Kara [SIAM J. Optim., 7 (1997), pp. 86-125]. These author...
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This paper deals with a class of primal-dual interior-point algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh, and Kara [SIAM J. Optim., 7 (1997), pp. 86-125]. These authors proposed a family of primal-dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix (XS-1/2)-S-1/2. They study three primal-dual algorithms based on this family of search directions: a short-step path-following method, a feasible potential-reduction method, and an infeasible potential-reduction method. However, they were not able to provide an algorithm which generalizes the long-step path-following algorithm introduced by Kojima, Mizuno, and Yoshise [Progress in Mathematical programming: Interior Point and Related Methods, N. Megiddor, ed., Springer-Verlag, Berlin, New York, 1989, pp. 29-47]. In this paper, we characterize two search directions within their family as being (unique) solutions of systems of linear equations in symmetric variables. Based on this characterization, we present a simplified polynomial convergence proof for one of their short-step path-following algorithms and, for the first time, a polynomially convergent long-step path-following algorithm for SDP which requires an extra root n factor in its iteration-complexity order as compared to its linear programming counterpart, where n is the number of rows (or columns) of the matrices involved.
In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but conve...
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In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primal-dual interior-point methods for their solution.
Primal-dual path-following algorithms are considered for determinant maximization problem (maxdet-problem). These algorithms apply Newton's method to a primal-dual central path equation similar to that in semidefi...
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Primal-dual path-following algorithms are considered for determinant maximization problem (maxdet-problem). These algorithms apply Newton's method to a primal-dual central path equation similar to that in semidefinite programming (SDP) to obtain a Newton system which is then symmetrized to avoid nonsymmetric search direction. Computational aspects of the algorithms are discussed, including Mehrotra-type predictor-corrector variants. Focusing on three different symmetrizations, which leads to what are known as the AHO, H..K..M and NT directions in SDP, numerical results for various classes of maxdet-problem are given. The computational results show that the proposed algorithms are efficient, robust and accurate.
Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for non-convex problems, where the quality ...
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Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for non-convex problems, where the quality of the lower bound depends on the duality gap. Quadratically constrained quadratic programs (QQPs) provide important examples of non-convex programs. For the simple case of one quadratic constraint (the trust-region subproblem) strong duality holds. In addition, necessary and sufficient (strengthened) second-order optimality conditions exist. However, these duality results already fail for the two trust-region subproblem. Surprisingly, there are classes of more complex, non-convex QQPs where strong duality holds. One example is the special case of orthogonality constraints, which arise naturally in relaxations for the quadratic assignment problem (QAP). In this paper we show that strong duality also holds for a relaxation of QAP where the orthogonality constraint is replaced by a semidefinite inequality constraint. Using this strong duality result, and semidefinite duality, we develop new trust-region type necessary and sufficient optimality conditions for these problems. Our proof of strong duality introduces and uses a generalization of the Hoffman-Wielandt inequality. (C) 1999 Published by Elsevier Science Inc. All rights reserved.
SeDuMi is an add-on for MATLAB, which lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large s...
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SeDuMi is an add-on for MATLAB, which lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox.
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