Algorithms for General Monotone Mixed Variational Inequalities

AbstractIn this paper, we suggest and analyze some new iterative methods for solving general monotone mixed variational inequalities, which are being used to study odd-order and nonsymmetric boundary value problems arising in pure and applied sciences. These new methods can be viewed as generalizations and extensions of the methods of He, Solodov and Tseng, and Noor for solving monotone (mixed) variational inequalities

Descent method for monotone mixed variational inequalities

We consider mixed variational inequalities involving a non-strictly monotone, differentiable cost mapping and a convex nondifferentiable function. We apply the Tikhonov-Browder regularization technique to these problems. We use uniformly monotone auxiliary functions for constructing regularized problems and apply the gap function approach for the perturbed uniformly monotone variational inequalities. Then we propose a combined regularization and descent method for initial monotone problems and establish convergence of its iteration sequence. Keywords. Variational inequalities, nonsmooth functions, descent methods © 2008 Springer-Verlag

A Fast Optimistic Method for Monotone Variational Inequalities

We study monotone variational inequalities that can arise as optimality conditions for constrained convex optimisation or convex-concave minimax problems and propose a novel algorithm that uses only one gradient/operator evaluation and one projection onto the constraint set per iteration. The algorithm, which we call fOGDA-VI, achieves a $o \left( \frac{1}{k} \right)$ rate of convergence in terms of the restricted gap function as well as the natural residual for the last iterate. Moreover, we provide a convergence guarantee for the sequence of iterates to a solution of the variational inequality. These are the best theoretical convergence results for numerical methods for (only) monotone variational inequalities reported in the literature. To empirically validate our algorithm we investigate a two-player matrix game with mixed strategies of the two players. Concluding, we show promising results regarding the application of fOGDA-VI to the training of generative adversarial nets.Comment: Accepted at ICML 202

New Perturbed Proximal Point Algorithms for Set-valued Quasi Variational Inclusions

In this paper, by using some new and innovative techniques, some perturbed iterative algorithms for solving generalized set-valued variational inclusions are suggested and analyzed. Since the generalized set-valued variational inclusions include many variational inclusions , variational inequalities and set-valued operator equation studied by others in recent years, the results obtained in this paper continue to hold for them and represent a significant refinement and improvement of the previously known results in this area

Equilibrium problems on Riemannian manifolds with applications

We study the equilibrium problem on general Riemannian manifolds. The results on existence of solutions and on the convex structure of the solution set are established. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. As applications, the corresponding results for the mixed variational inequality and the Nash equilibrium are obtained. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388, 61-77, 2012 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on Hadamard manifolds.National Natural Science Foundation of ChinaNatural Science Foundation of Guizhou Province (China)Dirección General de Enseñanza SuperiorJunta de AndalucíaNational Science Council of Taiwa