The Method Of Overcoming The Maratos Effect In Nonlinear Semidefinite Programming

The study of nonlinear semidefinite programming(nonlinear SDP)is very important since it has practical values in financial?engineering and control theory fields,e.g.,feed back con-trol?truss topology optimization?structural design?robust control optimization?material op-timization?linear and bilinear matrix inequality problem?eigenvalue optimization and so on.Thus,the research of nonlinear SDP has become a hot area of reseach recently.Many researchers study the algorithms of nonlinear SDP and present many efficient algorithms,e.g.,augmented Lagrangian method?primal-dual interior point method?sequential semidefinite programming(SS-DP)method?homotopy method?feasible direction method and so on.The SSDP method can be seen as an extension of the classical sequential quadratic programming(SQP)method,so some researchers has applied the work which based on SQP-type method to nonlinear SDP and proposed trust region framework filter method and penalty-free method without filter.These algorithms are globally convergent,however,due to the speciality of semidefinite constraint,the research on the local convergence property of SSDP method is few.Similar to nonlinear programming,Maratos effect may also happen in nonlinear SDP.There is no research on this topic as far as we know.In this paper,a second-order correction(SOC)step technique corresponding to SSDP method is proposed first.This technique uses the null space to construct the subproblem when calculating the SOC step.We proved that the designed second-order correction step is well-defined via the theories of matrix analysis and strongly semismooth.Besides,we applied this idea to SSDP method based on l1 exact penalty function and proved that,when the sequence generated by the algorithm is close to the solution,then under the assumptions of non-degeneration,strict complementarity and second-order sufficient conditions,the full size step or the full size step with second-order correction step can be accepted by the l1 exact penalty function.Thus Maratos effect is ovorcome and the local superlinear convergence is proved.In nonlinear programming,calculation overflow may happen when the penalty factor is too large,so we also studied how to apply the penalty-free idea to nonlinear SDP,and proposed a line search filter method and a line search method without filter and penalty function.The global convergence of the algorithms are analyzed.Besides,combined the second-order correction step technique with these algorithms,we proved that when the sequence generated by the algorithm is close to the solution,then under the assumptions of non-degeneration,strict complementarity and second-order sufficient conditions,the full size step or the full size step with second-order correction step can be accepted by the these penalty-free criterions.Similar to the results before,the Maratos effect can be overcome and superlinear local convergence is proved.Many penalty-free algorithms need the restoration phase in order to deal with the incom-patibility of the SSDP subproblem since the algorithm may fail to generate a descent direction.However,this phase may cost a lot of calculations and there is no restoration algorithm,which is very effective corresponding to nonlinear SDP.In order to avoid this restoration phase,we pro-posed a two phase SSDP method.At first,we compute a steering step as the solution to a linear SDP programming,which provides the information of the lowest possible violation of the linearized constraints in a neighborhood of the current iterate.It can also " detect" the feasibility of the SSDP subproblem.Second,we adjust the penalty parameter and compute a search direction,which is the unique solution of a SSDP subproblem or a strictly convex minimization subproblem.We require that this direction yields an improvement in linearized feasibility and become a descent direction for the l1 exact penalty function.Compared with the traditional SSDP method,we do not need the default assumption that the subproblem is always feasible,which is the main advantage of our method.Besides,the global convergence analysis of the algorithm requires no constraint qualifi-cations.Motivated by another idea-nonmonotone technique,which can overcome Maratos effect in nonlinear programming,we propose a nonmonotone two phase algorithm.The global and local convergence of the algorithm is analyzed.We show that the Maratos effect can also be overcome by this nonmonotone technique.In this paper,we give some numerical experiments which show that the effect of the algorithms presented before is robust and practical.