Nonsingularity Study Of The Parametric FB System For Nonlinear Semidefinite Programming

Nonlinear semidefinite programming have extensive application in engineering de-sign§system control,finance,combinatorial optimization,robust optimization, mode de-tection and so on.We use the parametric FB functionΦτand its smoothing functionΦτand transform the KKT optimality conditions to nonsmooth system Eτ(·) = 0 or anaugmented system Fτ(·) = 0 in order to slove nonlinear SDP.When we use nonsmoothnewton method and classic newton method to slove the system respectively,we have toguarantee that the algorithm have rapid rate of convergence speed and want to know inwhat condition the Clark's Jacobian of Eτand Fτare nonsingular, besides the propertyof strong semismooth.The paper prove under Robinson's constraint that the nonsingularity of Clark's Ja-cobian of Eτat KKT point is equivalent to the strong regularity of the KKT pointand is equivalent to the strong second-order su?cient condition and constraint nonde-generacy by studying the properties of directional derivative and Clark's Jacobian ofΦτ.When nonlinear SDP reduces to linear SDP,we obtain that the primal and dual constraintnondegenercy, the nonsingularity of Clark's Jacobian of Eτat KKT point, the strongregularity of the KKT point are all equivalent.For the augmented system Fτ(·) = 0,weobtain similar result.On one hand, the results provide the guarantee of rapid conver-gence of the algorithm,on the other hand, they provide a new character of local KKTpoint.Furthermore, the parametric FB complementary function includes FB function andNR function,in another sense, we promote the results in [1, 2] and [3] to a new SDCfunction.The paper includes five chapter,they present as follow: chapter one is introduction,weintroduce algorithm of nonlinear SDP, especially for the smoothing method.Chapter twogives the preliminary knowledge and lemma, and prove the basic properties ofΦτandΦτ. Chapter three we study the properties of directional derivative and Clark's JacobianofΦτand establish the equivalence among the nonsingularity of Clark's Jacobian of Eτat KKT point, to the strong regularity of the KKT point and the strong second-ordersu?cient condition and constraint nondegeneracy.Chapter four,we obtain similar resultfor the smoothing functionΦτ.Chapter five,we use the algorithm proposed in [4] intoaugmented system Fτ(·) = 0, and obtain the local quadratic convergence of the algorithm and we use standard problems to test the algorithm.