Non-convex Mapping Metric Regular And Non-convex Optimization

By techniques and methods of functional analysis, variational analysis and non-smooth analysis, many people have focused on such problems as metric regularity, metric subregularity and calmness of solutions for closed multifunctions, mathe-matic programming and vector optimization problems. They are widely used in many fields. Upon other researchers'work, in this dissertation, we mainly obtained four aspects of results below:1. We build representations of the Clarke tangent cone, the Clarke normal cone and the Bouligand tangent cone for the multifunction M(·)= g(·)+Ω(·) by the corresponding ones for the multifunctionΩ(·). We also discuss sub smoothness and L-subsmoothness for M. We build some equivalent conditions of metric regularity for a (σ,δ)-subsmooth multifunction, in which we generalize Ursescu-Robinson theorem and Zheng and Ng's corresponding results to the more general nonconvex case. Finally, by the normal dual mapping, we provide some sufficient and necessary conditions and point-based conditions for metric subregularity of a (σ,δ)-subsmooth multifunction, improving and generalizing the corresponding results of zheng, Ng and others.2. We research the relations between the sharp minimum and the strong KKT, between weak sharp minima and the quasi-strong KKT and the sub-quasi-strong KKT for the optimal problems with the constraint conditions of inequalities and equalities. Under the Lipschitzian and quasi-subsmooth conditions, We obtain equivalent conditions for the above strong KKTs. Our results generalize the results of Zheng and Ng in the convex-composite case to the more general nonconvex case.3. We obtain the Taylor formula of Ck,l vector functions in the form of the (k+1)th order Clarke subdifferential. Under the assumption of positive definiteness of the (k+1)th order subdifferential, we provide sufficient conditions of Pareto and weak Pareto solutions for this kind of problems in the term of high order Clarke subdifferential. In Banach spaces, we define the Michel-Penot high order subdifferential for a Ck.1 function, give some properties of it and a new Taylor formula. Using our new Taylor formula, we obtain some high order subdifferential sufficient optimal conditions for Ck.1 functions in Banach spaces.4. In Banach spaces, we prove that if the graph Gr(F) of the objective multifunction F is the union of finitely many generalized polyhedra and F(■)+C, the sum of the range F(■) and the order cone C, is convex or C is a polyhedron with nonempty interior, then its weak Pareto solution set and weak Pareto optimal value set are the unions of finitely many generalized polyhedra, respectively, and that if the graph of the objective multifunction F is the union of finitely many convex polyhedra, F(T)+Cis convex and the projection C2 of C, in finite dimensional subspace Y2 of Y with respect to F, is closed, then its Pareto solution set and Pareto optimal value set are the unions of finitely many convex polyhedra, respectively. Finally. dropping the assumption of the ordering cone C having a weakly base but requiring the objective multifunction being convex with respect to C, we prove that the Pareto solution set and Pareto optimal solution set of the objective multifunction are pathwise connected, respectively.