We introduce a new class of algorithms for solving linear semidefinite programming (SDP) problems. Our approach is based on classical tools from convex optimization such as quadratic regularization and augmented Lagra...
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We introduce a new class of algorithms for solving linear semidefinite programming (SDP) problems. Our approach is based on classical tools from convex optimization such as quadratic regularization and augmented Lagrangian techniques. We study the theoretical properties and we show that practical implementations behave very well on some instances of SDP having a large number of constraints. We also show that the "boundary point method" from Povh, Rendl, and Wiegele [Computing, 78 (2006), pp. 277-286] is an instance of this class.
Remarkable breakthroughs have been made recently in obtaining approximate solutions to some fundamental NP-hard problems, namely Max-Cut, Max k-Cut, Max-Sat, Max-Dicut, Max-bisection, k-vertex coloring, maximum indepe...
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Remarkable breakthroughs have been made recently in obtaining approximate solutions to some fundamental NP-hard problems, namely Max-Cut, Max k-Cut, Max-Sat, Max-Dicut, Max-bisection, k-vertex coloring, maximum independent set, etc. All these breakthroughs involve polynomial time randomized algorithms based upon semidefinite programming, a technique pioneered by Goemans and Williamson. In this paper, we give techniques to derandomize the above class of randomized algorithms, thus obtaining polynomial time deterministic algorithms with the same approximation ratios for the above problems. At the heart of our technique is the use of spherical symmetry to convert a nested sequence of n integrations, which cannot be approximated sufficiently well in polynomial time, to a nested sequence of just a constant number of integrations, which can be approximated sufficiently well in polynomial time.
In this paper we study optimization problems involving convex nonlinear semidefinite programming(CSDP).Here we convert CSDP into eigenvalue problem by exact penalty function,and apply the U-Lagrangian theory to the fu...
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In this paper we study optimization problems involving convex nonlinear semidefinite programming(CSDP).Here we convert CSDP into eigenvalue problem by exact penalty function,and apply the U-Lagrangian theory to the function of the largest eigenvalues,with matrix-convex valued *** give the first-and second-order derivatives of U-Lagrangian in the space of decision variables Rm when transversality condition ***,an algorithm frame with superlinear convergence is ***,we give one application:bilinear matrix inequality(BMI)optimization;meanwhile,list their UV decomposition results.
In this paper, we consider the problem of optimal design of experiments. A two-step inference strategy is proposed. The first step consists in minimizing the condition number of the so-called information matrix. This ...
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In this paper, we consider the problem of optimal design of experiments. A two-step inference strategy is proposed. The first step consists in minimizing the condition number of the so-called information matrix. This step can be turned into a semidefinite programming problem. The second step is more classical, and it entails the minimization of a convex integral functional under linear constraints. This step is formulated in some infinite-dimensional space and is solved by means of a dual approach. Numerical simulations will show the relevance of our approach.
A proximal augmented method for solving semidefinite programs is introduced. This method is based on the Augmented Lagrangian method. We study the theoretical properties and show the convergence of the new approach un...
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A proximal augmented method for solving semidefinite programs is introduced. This method is based on the Augmented Lagrangian method. We study the theoretical properties and show the convergence of the new approach under weaker conditions. Numerical results on some semidefinite programming problems are reported and show that the new method is potentially efficient. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.
A new method for the efficient solution of a class of convex semidefinite programming (SDP) problems is introduced. The method extends the sequential convex programming (SCP) concept to optimization problems with matr...
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A new method for the efficient solution of a class of convex semidefinite programming (SDP) problems is introduced. The method extends the sequential convex programming (SCP) concept to optimization problems with matrix variables. The basic idea of the new method is to approximate the original optimization problem by a sequence of subproblems, in which nonlinear functions (defined in matrix variables) are approximated by block separable convex functions. The subproblems are semidefinite programs with a favorable structure which can be efficiently solved by existing SDP software. The new method is shown to be globally convergent. The article is concluded by a series of numerical experiments with free material optimization problems demonstrating the effectiveness of the generalized SCP approach.
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 received energy has been becoming an efficient and attractive measure for acoustic source localization due to its cost saving in both energy and computation capability. We investigated the acoustic source localiza...
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The received energy has been becoming an efficient and attractive measure for acoustic source localization due to its cost saving in both energy and computation capability. We investigated the acoustic source localization problem based on received energy measurements in sensor networks. Focusing on the non-logarithmic energy attenuation model, we developed and compared a suite of semidefinite programming (SDP)-based source localization methods due to computational efficiency and numerical reliability. First, we proposed a general SDP-based estimator by jointly estimating the source location and the source radiation power. It yields an efficient estimate for both the scenario where the source is located inside the convex hull formed by sensors and the scenario where the source is located outside the convex hull. Next, a min-max approximation is given to cope with the applicable application of the existing energy-based source localization algorithms relying on the Gaussian energy noise assumption. Furthermore, a novel norm approximation method is proposed according to norm equivalence, which can provide a comparable performance with lower computational complexity. Simulations show that our proposed methods exhibit a superior performance than the existing energy-based source localization estimators.
The constant rank constraint qualification, introduced by Janin in 1984 for nonlinear programming, has been extensively used for sensitivity analysis, global convergence of first- and second-order algorithms, and for ...
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The constant rank constraint qualification, introduced by Janin in 1984 for nonlinear programming, has been extensively used for sensitivity analysis, global convergence of first- and second-order algorithms, and for computing the directional derivative of the value function. In this paper we discuss naive extensions of constant rank-type constraint qualifications to second-order cone programming and semidefinite programming, which are based on the Approximate-Karush-Kuhn-Tucker necessary optimality condition and on the application of the reduction approach. Our definitions are strictly weaker than Robinson's constraint qualification, and an application to the global convergence of an augmented Lagrangian algorithm is obtained.
We introduce a novel method for clustering using a semidefinite programming (SDP) re-laxation of the Max k-Cut problem. The approach is based on a new methodology for rounding the solution of an SDP relaxation using i...
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We introduce a novel method for clustering using a semidefinite programming (SDP) re-laxation of the Max k-Cut problem. The approach is based on a new methodology for rounding the solution of an SDP relaxation using iterated linear optimization. We show the vertices of the Max k-Cut relaxation correspond to partitions of the data into at most k sets. We also show the vertices are attractive fixed points of iterated linear optimization. Each step of this iterative process solves a relaxation of the closest vertex problem and leads to a new clustering problem where the underlying clusters are more clearly defined. Our experiments show that using fixed point iteration for rounding the Max k-Cut SDP relaxation leads to significantly better results when compared to randomized rounding.
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