Rigidity Of Submanifolds And Algorithms For Variational Inequality And Equilibrium Problems

Manifold optimization has been widely used in the fields of applied mathematics,statistics,engineering,machine learning and so on.By using the topological structure and geomet-ric properties of Riemannian manifolds,constrained optimization problems in linear space can be seen as unconstrained optimization ones on Riemannian manifolds.Optimization problems of nonconvex objective functions in linear space become convex on Riemannian manifold by introducing appropriate Riemannian metric.In many practical applications,the natural structure of data is often modeled as constrained optimization problems with constraints being Riemannian manifolds.In order to overcome this difficulty,on the one hand,scholars use the pinching property of submanifolds to sim-plify the data modeling structure and make it homeomorphic with known manifolds.On the other hand,they actively explore the optimization theory and algorithms on Riemannian manifolds.In this doctoral thesis,pinching results of a class of spacelike submanifolds in semi Riemannian space are given by using the moving frames method and Simons' method.Based on the algorithm models in linear space,some algorithms for solving variational in-equality problems and equilibrium problems on Hadamard manifolds are proposed.The main work includes:1.By making use of moving frames method and Simons' method,an n-dimensional space-like submanifolds with parallel mean curvature vector in(n+p)-dimensional de Sitter space with index q are studied.Firstly,the rigidity structure and geometric characteristics of space-like submanifolds are described,the orthogonal frame field of spacelike submanifolds is con-structed,and the tensor expressions such as structure equations,Gauss equations,Codazzi equations and Ricci formulas are derived.Secondly,the Simons type integral inequalities under the pinching conditions on scalar curvature,Ricci curvature and sectional curvature of the submanifolds are obtained.Finally,pinching results of spacelike submanifolds with some simple submanifolds or hypersurfaces,such as totally umbilical submanifolds,totally geodesic submanifolds,Clifford torus,and Veronese surfaces are discussed.2.A class of variational inequality problems on Hadamard manifolds is studied,and six pro-jection algorithms are proposed.The convergence of each algorithm is analyzed under the condition that the vector field satisfies Lipschitz continuity and pseudomonotonicity.First-ly,two algorithms based on extragradient model are proposed.The stepsizes of algorithms adopt Armijo step rule and a new step rule without the prior knowledge of the Lipschitz constant of vector field,respectively.The latter stepsize is variable,and has no correlation with the Lipschitz constant in each iteration,which is especially useful when the Lipschitz constant cannot be solved or is difficult to solve.Secondly,two algorithms based on sub-gradient extragradient model are presented.The stepsizes of the algorithms adopt the norm form and the inner product form of step rule without the prior knowledge of the Lipschitz constant of the vector field,respectively.And the performance of the two algorithms is com-pared in numerical experiment.Finally,an inertial subgradient extragradient algorithm is proposed.The stepsizes of the algorithm adopt the step rule without the prior knowledge of the Lipschitz constant of the vector field.When the Lipschitz constant is known,the stepsize of the algorithm in each iteration is reduced to a constant step under mild conditions,and the efficiency of the two algorithms is compared by numerical experiment.3.A class of equilibrium problems on Hadamard manifolds is studied,and three new algo-rithms for solving the equilibrium problem involving pseudomonotone and Lipschitz-type bifunction are proposed.The stepsizes of the algorithms adopt step rule without the prior knowledge of the Lipschitz-type constants of the bifunction.The monotone boundedness of the sequence generated by the stepsizes is proved,the convergence of each algorithm is analyzed,and the performance of the algorithms is tested by numerical experiment.Firstly,two algorithms based on extragradient model and golden ratio model are presented.Com-pared with the former algorithm,the latter algorithm only needs to solve one quadratic pro-gramming problem per iteration.Secondly,the framework of the extragradient algorithm is improved,and a new extragradient algorithm called extragradient-like algorithm for solving equilibrium problem is proposed.On this basis,the extragradient-like algorithm is applied to solve variational inequality problems,and a new algorithm for solving variational inequality problems is obtained.