Thesis Summary:

This thesis is concerned with the study of non-convex problems through two different approaches: first, by focusing on optimality conditions for problems with particular non-convex geometric constraint sets, without assuming convexity of the functions; and secondly, by considering a special family of quasiconvex functions. We develop algebraic descriptions of the limiting normal cone for distinct types of sets, including the union of two polyhedra and either a single quadric surface or the union of two quadric surfaces. These descriptions are then used to study M-stationary optimality conditions, yielding results on local uniqueness as well as stability properties of the solution set. Our findings regarding optimality conditions are new and can be viewed as extensions of known results that employ KKT conditions in problems without geometric constraints. In addition, we present an example of an optimization problem where M-stationary conditions correctly identify the minimizer, to which our results apply, whereas other approaches fail. Subsequently, we extend certain known results from convex analysis to quasiconvex functions, showing that, as with convex functions, some global behaviors can be understood by examining the behavior at individual points. Finally, we focus on a particular class of quasiconvex functions to prove existence of solutions and lower semicontinuity of the value function, thereby complementing or generalizing known results in the convex case.