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Semidefinite optimization
 Acta Numerica
, 2001
"... Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the ..."
Abstract

Cited by 107 (2 self)
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Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps “or ” would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems.
RankTwo Relaxation Heuristics for MaxCut and Other Binary Quadratic Programs
 SIAM Journal on Optimization
, 2000
"... The GoemansWilliamson randomized algorithm guarantees a highquality approximation to the MaxCut problem, but the cost associated with such an approximation can be excessively high for largescale problems due to the need for solving an expensive semidefinite relaxation. In order to achieve better ..."
Abstract

Cited by 35 (3 self)
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The GoemansWilliamson randomized algorithm guarantees a highquality approximation to the MaxCut problem, but the cost associated with such an approximation can be excessively high for largescale problems due to the need for solving an expensive semidefinite relaxation. In order to achieve better practical performance, we propose an alternative, ranktwo relaxation and develop a specialized version of the GoemansWilliamson technique. The proposed approach leads to continuous optimization heuristics applicable to MaxCut as well as other binary quadratic programs, for example the MaxBisection problem. A computer code based on the ranktwo relaxation heuristics is compared with two stateoftheart semidefinite programming codes that implement the GoemansWilliamson randomized algorithm, as well as with a purely heuristic code for effectively solving a particular MaxCut problem arising in physics. Computational results show that the proposed approach is fast and scalable and, more importantly, attains a higher approximation quality in practice than that of the GoemansWilliamson randomized algorithm. An extension to MaxBisection is also discussed as well as an important difference between the proposed approach and the GoemansWilliamson algorithm, namely that the new approach does not guarantee an upper bound on the MaxCut optimal value. Key words. Binary quadratic programs, MaxCut and MaxBisection, semidefinite relaxation, ranktwo relaxation, continuous optimization heuristics. AMS subject classifications. 90C06, 90C27, 90C30 1.
Interiorpoint algorithms for semidefinite programming based on a nonlinear formulation
 COMP. OPT. AND APPL
, 2002
"... Recently in Burer et al. (Mathematical Programming A, submitted), the authors of this paper introduced a nonlinear transformation to convert the positive definiteness constraint on an n × n matrixvalued function of a certain form into the positivity constraint on n scalar variables while keeping t ..."
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Cited by 19 (9 self)
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Recently in Burer et al. (Mathematical Programming A, submitted), the authors of this paper introduced a nonlinear transformation to convert the positive definiteness constraint on an n × n matrixvalued function of a certain form into the positivity constraint on n scalar variables while keeping the number of variables unchanged. Based on this transformation, they proposed a firstorder interiorpoint algorithm for solving a special class of linear semidefinite programs. In this paper, we extend this approach and apply the transformation to general linear semidefinite programs, producing nonlinear programs that have not only the n positivity constraints, but also n additional nonlinear inequality constraints. Despite this complication, the transformed problems still retain most of the desirable properties. We propose firstorder and secondorder interiorpoint algorithms for this type of nonlinear program and establish their global convergence. Computational results demonstrating the effectiveness of the firstorder method are also presented.
1. Electronic structure computation
"... – approximations – linear scaling methods – convex program for density matrix 2. Pathfollowing interior point method [Boyd and Vandenberghe (2001)] – formulation – number of iterations – cost of each iteration 3. Automatic differentiation [Griewank (1989), Vavasis (1999)] 4. Automatic CG preconditi ..."
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– approximations – linear scaling methods – convex program for density matrix 2. Pathfollowing interior point method [Boyd and Vandenberghe (2001)] – formulation – number of iterations – cost of each iteration 3. Automatic differentiation [Griewank (1989), Vavasis (1999)] 4. Automatic CG preconditioning [Morales and Nocedal (2000,2002)]