A class of nonlinear Lagrangian algorithms for minimax problems

This paper studies a class of nonlinear Lagrangian algorithms for solving unconstrained minimax problems, which will provide an approach to constructing a concrete and novel Lagrangian algorithm and simplify the relevant theory analysis. A class of nonlinear Lagrangians is constructed and a set of mild conditions on them are proposed to guarantee the convergence theory of the corresponding algorithms. The unified convergence analysis framework for the class of algorithms is established. It is shown that the sequence solutions obtained by the class of algorithms are Q-linearly convergent when the controlling parameter is less than a threshold under some assumptions and the error bounds of the sequence solutions are obtained at the same time. The properties of dual problem framework based on the unified nonlinear Lagrangians are analyzed, which the classical Lagrangian lacks. Furthermore, it is presented that the proposed class of nonlinear Lagrangians contains four well-known nonlinear Lagrangians for unconstrained minimax problems appearing in the literatures. At last, numerical results for ten typical test problems are reported, based on the four nonlinear Lagrangians in the proposed class.