Metric Subregularity For Piecewise Linear Multifunctions And Applications

The metric subregularity property is one of the core concepts in variational analysis and has important applications in optimization.Essentially,as early as 1952,Hoffman proved that a linear multifunction has the uniformly global metric subregularity property on its range.Afterwards,Robinson[Math.Program.Stud.1981]proved that a piecewise linear multifunction has the local metric subregulaxity property at each point in its range.Zheng and Ng[SIAM J.Optim.2014]further studied the bounded metric subregularity property and global metric subregularity property of a piecewise multifunction at a given point in its range.This thesis is devoted to undertaking research on the uniformly bounded metric subregularity property and uniformly global metric subregularity property of a piecewise linear multifunction on its range.The main results are stated as follows:In Part 1,we give an counterexample which shows that,even in a finite dimensional space,a piecewise linear multifunction is not necessary to have the uniformly bounded metric subregularity property on its range and prove that every piecewise linear multifunction F:X(?)Y(X and Y are normed spaces)always has the approximatively uniformly bounded metric subregularity property,namely,for any ε>0,there exits a ε-small set Oε of F(X)such that F has the uniformly bounded metric subregularity property on F(X)\Oε.In addition,we establish characterizations for a piecewise linear multifunction to have the uniformly bounded metric subregularity property on F(X).As applications,we study weak sharp minima for piecewise linear multiobjective optimization problems.In Part 2,we provide a characterization for a piecewise linear multifunction F to have the global metric subregularity property at a given point in its range F(X)and prove that the set E,which is composed of the points with the global metric subregularity property,is the union of finitely many convex polyhedra.Then we prove that F has the approximatively uniformly global metric subregularity property on the above mentioned set E.Moreover,we give characterizations for a piecewise linear multifunction to have the uniformly global metric subregularity property on F(X).