Research On Gradient-like Algorithms For Max-min Problems

The minimax problem is an important type of optimization problem,and is widely used in engineering design,economic management and other fields.In this thesis,the gradient methods for solving the minimax problem are studied.The minimax problems which we studied include minimax problem of general structure and minimax problem with tensor structure.The main contents of this thesis are as follows:(1)The thesis introduces the current research status of the minimax problem of general structure,proposes the minimax problem with tensor structure,and briefly introduces the transformation method of the minimax problem of general structure and the minimax problem with tensor structure.(2)The gradient methods for solving minimax problems are proposed,namely smoothing steepest descent method,smoothing three-term conjugate gradient method,smoothing conjugate gradient method and conjugate subgradient method.Under certain conditions,the global convergence analysis of proposed methods is given.(3)The gradient methods given in this thesis are used to solve minimax problem of general structure,minimax problem with tensor structure,minimax transformation problem of constrained optimization problem,minimax transformation problem of constrained optimization with tensor structure,and minimax transformation problem of generalized polynomial complementarity problem.Related numerical experiments show that the proposed gradient methods can effectively solve the above minimax problems.