The Study On Search Directions Of Interior-point Methods For Semidefinite Programming

In recent years,with its theory and algorithms developed greatly and its numerous applications found in control theory, design of industry and project, electrical engineering and combinatorial optimization. semidefiniet programming has become one of the most active research aeras in mathematical programming.In this paper, we firstly summarize the present research, theory, research status, application and the algorithm of semidefinite programming, and then, introduce our some work in algorithm and application.For detail,we conclude them as follows:Firstly, a new function is constructed and the existence and uniqueness of center path for semi-definite programming is reseached. Based on the assumption of the existence of strictly feasible solution of semidefinite programming, we structured new function with the inverse matrix and the trace of matrix, and using the strict convexity of this function derived that, central conditions for standard form semi-definite programming problem is equivalent to the existence conditions of a unique minimize of this new function satisfied equality constraints of the prime problem. Finally we prove the existence and uniqueness of the central path with this relation of equivalence and the existence of a unique minimize of the new function.Secendly,the search directions of the Quadratic semi-definite programming is discussed, By using Newton method, the perturbed KKT conditions associated with a special class of quadratic SDP are computed, and the unified form of equation solving search directions is obtained; Under the unified form, A sufficient conditions about the existence and uniqueness of the HKM search direction and the NT search direction are presented, then we give the expression of search direction and how to compute search direction concretely for some special classes.