Research On Algorithms Based On Convex-concave Minimax Problem

Minimax problems,introduced by J.von Neumann in 1928,are an important area of nonlinear analysis and have been paid much attention.Minimax problems have many applications in various disciplines such as game theory,economics,optimization theory,variational inequalities,fixed point theory,differential equations,and so on.Recently,minimax problems have been applied to several important frontier disciplines,including,in particular,machine learning(such as generative adversarial nets,GANs).In this paper we mainly focus on the study of the weak and strong convergence of iterative algorithms for convex-concave minimax problems in the infinite-dimensional Hilbert space frame-work.Our contributions are threefold.First,we show that the choice range(0,1/L)of the stepsizes in the Korpelevich extragradient algorithm is sharp,while the choice range(0,1/(3L))of the stepsizes in both Popov’s and Malitsky-Semenov’s extragradient algorithms can be enlarged to the range(0,(√(2-1))/L),where L is the Lipschitz constant of the(saddle point)gradient of the objective function.Secondly,we prove the weak convergence of all three Korpelevich’s,Popov’s,and Malitsky-Semenov’s extragradient algorithms in an infinite-dimensional Hilbert space.Thirdly,we propose a new iterative algorithm,which is known as projected extra anchored gradient algorithm,and prove its strong convergence in an infinite-dimensional Hilbert space.