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

Finding Multiple Saddle Points for G-differential Functionals and Defocused Nonlinear Problems

We study computational theory and numerical methods for finding multiple unstable solutions (saddle points) for two types of nonlinear variational functionals. The first type consists of Gateaux differentiable (G-differentiable) M-type (focused) problems. Motivated by quasilinear elliptic problems from physical applications, where energy functionals are at most lower semi-continuous with blow-up singularities in the whole space and G-differntiable in a subspace, and mathematical results and numerical methods for C1 or nonsmooth/Lipschitz saddle points existing in the literature are not applicable, we establish a new mathematical frame-work for a local minimax method and its numerical implementation for finding multiple G-saddle points with a new strong-weak topology approach. Numerical implementation in a weak form of the algorithm is presented. Numerical examples are carried out to illustrate the method. The second type consists of C^1 W-type (defocused) problems. In many applications, finding saddles for W-type functionals is desirable, but no mathematically validated numerical method for finding multiple solutions exists in literature so far. In this dissertation, a new mathematical numerical method called a local minmaxmin method (LMMM) is proposed and numerical examples are carried out to illustrate the efficiency of this method. We also establish computational theory and present the convergence results of LMMM under much weaker conditions. Furthermore, we study this algorithm in depth for a typical W-type problem and analyze the instability performances of saddles by LMMM as well

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

Finding Multiple Saddle Points for G-differential Functionals and Defocused Nonlinear Problems

We study computational theory and numerical methods for finding multiple unstable solutions (saddle points) for two types of nonlinear variational functionals. The first type consists of Gateaux differentiable (G-differentiable) M-type (focused) problems. Motivated by quasilinear elliptic problems from physical applications, where energy functionals are at most lower semi-continuous with blow-up singularities in the whole space and G-differntiable in a subspace, and mathematical results and numerical methods for C1 or nonsmooth/Lipschitz saddle points existing in the literature are not applicable, we establish a new mathematical frame-work for a local minimax method and its numerical implementation for finding multiple G-saddle points with a new strong-weak topology approach. Numerical implementation in a weak form of the algorithm is presented. Numerical examples are carried out to illustrate the method. The second type consists of C^1 W-type (defocused) problems. In many applications, finding saddles for W-type functionals is desirable, but no mathematically validated numerical method for finding multiple solutions exists in literature so far. In this dissertation, a new mathematical numerical method called a local minmaxmin method (LMMM) is proposed and numerical examples are carried out to illustrate the efficiency of this method. We also establish computational theory and present the convergence results of LMMM under much weaker conditions. Furthermore, we study this algorithm in depth for a typical W-type problem and analyze the instability performances of saddles by LMMM as well

Distributed convergence to Nash equilibria in two-network zero-sum games

This paper considers a class of strategic scenarios in which two networks of agents have opposing objectives with regards to the optimization of a common objective function. In the resulting zero-sum game, individual agents collaborate with neighbors in their respective network and have only partial knowledge of the state of the agents in the other network. For the case when the interaction topology of each network is undirected, we synthesize a distributed saddle-point strategy and establish its convergence to the Nash equilibrium for the class of strictly concave-convex and locally Lipschitz objective functions. We also show that this dynamics does not converge in general if the topologies are directed. This justifies the introduction, in the directed case, of a generalization of this distributed dynamics which we show converges to the Nash equilibrium for the class of strictly concave-convex differentiable functions with locally Lipschitz gradients. The technical approach combines tools from algebraic graph theory, nonsmooth analysis, set-valued dynamical systems, and game theory

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more