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In semidefinite programming (SDP) we minimize a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so positive definite programs are convex optimization problems. Semidefinite programming unifies several standard problems (eg, linear and quadratic programming) and finds many applications in engineering. Although semidefinite programs are much more general than linear programs, they are just as easy to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity, and perform very well in practice. Interior-point methods [1] are one of the efficient methods developed to solve linear, semidefinite and nonlinear programming problems. In this paper, we particularly propose a logarithmic barrier interior-point method for solving semidefinite programming problem (SDP). In fact, the main difficulty to be anticipated in establishing an iteration in such a method will come from the determination and computation of the displacement step. Various approaches are developed to overcome this difficulty. It is known [4, 5] that the computation of the displacement step is expensive specifically while using line search methods. The purpose of this paper is to propose alternative ways to determine the displacement step which is more efficient than classical line-searches.

Study of a Logarithmic Barrier Approach for Linear Semidefinite Programming