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 matrix-valued 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 first-order interior-point 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 first-order and second-order interior-point algorithms for this type of nonlinear program and establish their global convergence. Computational results demonstrating the effectiveness of the first-order method are also presented.

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The authors of this paper recently introduced a transformation [4] that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based on the transformation, they proposed a globally convergent, first-order (i.e., gradient-based) log-barrier algorithm for solving a class of linear SDPs. In this paper, we discuss an efficient implementation of the proposed algorithm and report computational results on semidefinite relaxations of three types of combinatorial optimization problems. Our results demonstrate that the proposed algorithm is indeed capable of solving large-scale SDPs and is particularly effective for problems with a large number of constraints.

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We prove that all results determining the dimension and the ane hull of feasible solutions of any combinatorial optimization problem, and various more general nonconvex optimization problems, directly imply the existence of Slater points for a very wide class of semidefinite programming relaxations of these nonconvex problems. Our proofs are very concise, constructive and elementary.

DSDP implements the dual-scaling algorithm for semidefinite programming. The source code if this interior-point solver, written entirely in ANSI C, is freely available. The solver can be used as a subroutine library, as a function within the MATLAB environment, or as an executable that reads and writes to files. Initiated in 1997, DSDP has developed into an efficient and robust general purpose solver for semidefinite programming. Although the solver is written with semidefinite programming in mind, it can also be used for linear programming and other constraint cones. The features of DSDP include: • a robust algorithm with a convergence proof and polynomially bounded complexity under mild assumptions on the data, • primal and dual solutions, • feasible solutions when they exist or approximate certificates of infeasibity, • initial points that can be feasible or infeasible, • relatively low memory requirements for an interior-point method, • sparse and low-rank data structures,

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