A Multiparametric Programming Approach For Multilevel Optimization

Multilev I optimiza ion probl m ar math matical program which hav a sub t of th irrnvariabl on train d to be an optimal oluti n of other program parameteriz d by theirrnr maining variabl s. It i implicitly d termined by a n of optimization probl m whichrnmu t b olved in a pred termin d qu n . T b mor precis , th inn r lev I probl mrni a multiparametri programming probl m param terized b upp I' I v I optimizationrnvariable. The olution approa h for mul til v I programming problem i to repre ntrnthe inn r lev ,I problem with ufficient condi tion and a ugment it in the upper level constraints.rnAs a result , without convexity assumption for multilevel optimization at inn rrnlev 1 , tationary point may not be su ffi cient or optimal for the inner I vel problemrnand the set of all stationa ry point may not be connected. This impli s it is impossiblernto usc the approach of augmenting the conditions of the lower level problem into thernconstraints of th upper level problem. Recently, researchers have propo ed a olutionrnt rategy for multilevel optimization via multiparametric programming approach by conidernring the follow r 's problem as a multiparamet ric optimization problem. But, theirrnpropo ed algorithm work only for problem wi h convex , quadra tic or linear problem .rnIn this work , we pres nt the foundations of a gen ral global optimization algorithm forrnthe olu tion of general multilevel problem based on the r cent development in multiparametricrnprogramming th ory. Specifically, we outline the general global optimizationrnstra t gy for the olu tion of bilevel and tril vel programming problem with nonconvexitrnf rmulation at th inn l' I vel and w hav proved [-conv rgence of the algorithm

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