Minimax methods for finding multiple saddle critical points in Banach spaces and their applications

This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study

Minimax methods for finding multiple saddle critical points in Banach spaces and their applications

This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study

AdaBiM: An adaptive proximal gradient method for structured convex bilevel optimization

Bilevel optimization is a comprehensive framework that bridges single- and multi-objective optimization. It encompasses many general formulations, including, but not limited to, standard nonlinear programs. This work demonstrates how elementary proximal gradient iterations can be used to solve a wide class of convex bilevel optimization problems without involving subroutines. Compared to and improving upon existing methods, ours (1) can handle a wider class of problems, including nonsmooth terms in the upper and lower level problems, (2) does not require strong convexity or global Lipschitz gradient continuity assumptions, and (3) provides a systematic adaptive stepsize selection strategy, allowing for the use of large stepsizes while being insensitive to the choice of parameters

Dirichlet problems for several nonlocal operators via variational and topological methods

The main topic of the thesis is the study of elliptic differential equations with fractional order driven by nonlocal operators, as the fractional p-Laplacian, the fractional Laplacian for p=2, the general nonlocal operator and its anisotropic version. Recently, great attention has been focused on the study of fractional and nonlocal operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. This type of operators arises in a quite natural way in many different contexts, such as, among others, game theory, image processing, optimization, phase transition, anomalous diffusion, crystal dislocation, water waves, population dynamics and geophysical fluid dynamics. The main reason is that nonlocal operators are the infinitesimal generators of Lévy-type stochastic processes. Such processes extend the concept of Brownian motion, where the infinitesimal generator is the Laplace operator, and may contain jump discontinuities. Our aim is to show existence and multiplicity results for nonlinear elliptic Dirichlet problems, driven by a nonlocal operator, by applying variational and topological methods. Such methods usually exploit the special form of the nonlinearities entering the problem, for instance its symmetries, and offer complementary information. They are powerful tools to show the existence of multiple solutions and establish qualitative results on these solutions, for instance information regarding their location. The topological and variational approach provides not just existence of a solution, usually several solutions, but allow to achieve relevant knowledge about the behavior and properties of the solutions, which is extremely useful because generally the problems cannot be effectively solved, so the precise expression of the solutions is unknown. As a specific example of property of a solution that we look for is the sign of the solution, for example to be able to determine whether it is positive, or negative, or nodal (i.e., sign changing)