Variational Analysis And Stability In Optimization

The dissertation is devoted to the study of the so-called full Lipschitzian stability of local solutions to finite-dimensional parameterized problems of constrained optimization, which has been well recognized as a very important property from both viewpoints of optimization theory and its applications. Employing second-order subdifferentials of variational analysis, we obtain necessary and sufficient conditions for fully stable local minimizers in general classes of constrained optimization problems including problems of composite optimization as well as problems of nonlinear programming with twice continuously differentiable data. Based on our recent explicit calculations of the second-order subdifferential for convex piecewise linear functions, we establish relationships between nondegeneracy and second-order qualification for fully amenable compositions involving piecewise linear functions and obtain new applications of the developed second-order theory to full stability in composite optimization and constrained minimax problems, strong regularity of associate generalized equations and strong stability of stationary points for composite optimization. Finally, we discuss the important concept of critical multipliers for composite optimization problems and characterize it via second-order subdifferentials. Then we demonstrate that full stability can rule out the existence of critical multipliers in the mentioned framework