Semidefinite programming for discrete optimization and matrix completion problems
Discrete Applied Mathematics
Best ellipsoidal relaxation to solve a nonconvex problem
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
A relaxed cutting plane method for semi-infinite semi-definite programming
Journal of Computational and Applied Mathematics
Protein structure by semidefinite facial reduction
RECOMB'12 Proceedings of the 16th Annual international conference on Research in Computational Molecular Biology
A facial reduction algorithm for finding sparse SOS representations
Operations Research Letters
On the Slater condition for the SDP relaxations of nonconvex sets
Operations Research Letters
Strong duality and minimal representations for cone optimization
Computational Optimization and Applications
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It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interior-point methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps.Semidefinite linear programming (SDP) is a generalization of LP where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables. There are many applications, e.g., in systems and control theory and combinatorial optimization. However, the Lagrangian dual for SDP can have a duality gap. We discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming. These duals guarantee strong duality, i.e., a zero duality gap and dual attainment. This paper is motivated by the recent paper by Ramana where one of these duals is introduced.