Generalized Weak Sharp Minima Of Nonlinear SDP And Its Application

Weak sharp minima is mainly used to study the sensitivity of perturbation problems in some convex programming models,and it is also an important tool to analyze the conver-gence of algorithms for solving these problems.In August 2006,Nemirovski introduced the theory and application of linear symmetric cone optimization problem at World Congress of Mathematicians.As a special symmetric cone optimization problem,the semi-definite cone constrained optimization problem(SDP)has certain research significance.In this paper,we mainly study the generalized weak sharp minima of nonlinear SDP,the generalized I(weak)sharp minima of nonlinear SDP,the global error bound of nonlinear SDP and the relationship between them.In the first Part,we introduce the concept of generalized weak sharp minima of nonlinear SDP(NLSDP).Then,on the basis of generalized weak sharp minima,we give some sufficient conditions,necessary conditions and sufficient and necessary conditions in Banach space and Hilbert space.In addition,we introduce a solution algorithm and prove that the algorithm has finite-time convergence under the condition that the NLSDP generalized weak sharp minima is satisfied.Finally,we give the concept of the linear SDP problem(LSDP)with generalized weak sharp minima,and prove some equivalent propositions with LSDP generalized weak sharp minima.In the second part,we study an algorithm for solving infeasible points in NLSDP problem.Firstly,we introduce the concept of NLSDP global error bound in n-dimensional Euclidean space.Then,we consider the NLSDP problem as a convex inequality system and prove several equivalent propositions under the metric regularity condition.Finally,by using the tools of metric regularity and convex analysis,we prove that the global error bound of NLSDP can be transformed into the generalized weak sharp minima of NLSDP under certain conditions.In the third part,in order to provide a convergence analysis tool for the infeasible interior point algorithm.Firstly,we use the variational analysis method to give the concepts of NLSDP generalized I-type sharp minima and NLSDP generalized I-type weak sharp minima,as well as characterize some properties with NLSDP generalized I-type weak sharp minima.Then,by using the penalty function,we establish the relationship between the existence of strong Lagrangian multipliers and the NLSDP generalized I-type sharp minima.Finally,under the generalized Slater constraints,the existence of strong Lagrangian multipliers is used to describe the solution set of the nonlinear SDP problem.The relation between Lagrangian multiplicative subset of optimal solution for cone constrained optimization problem and generalized weak sharp minima is generalized.