Conic Linear Programming (original) (raw)
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In Part I of this series of papers, we have introduced a transformation which converts a class of linear and nonlinear semidefinite programs (SDPs) into nonlinear optimization problems over "orthants" of the form ! n ++ Theta ! N , where n is the size of the matrices involved in the problem and N is a nonnegative integer dependent upon the specific problem. In doing so, we have reduced the number of variables and constraints. In this paper, we develop interior point methods for solving a subclass of the transformable linear SDP problems where the diagonal of a matrix variable is given. These new interior point methods have the advantage of working entirely within the space of the transformed problem while still maintaining close ties with both the primal and the dual of the original SDP problem. Under very mild and reasonable assumptions, global convergence of these methods is proved. Keywords: semidefinite program, semidefinite relaxation, nonlinear programming, interior-...
Self-Scaled Barriers and Interior-Point Methods for Convex Programming
Mathematics of Operations Research, 1997
This paper provides a theoretical foundation for efficient interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled. For such problems we devise long-step and symmetric primal-dual methods. Because of the special properties of these cones and barriers, our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.