New extragradient-type methods for general variational inequalities

In this paper, we consider and analyze a new class of extragradient-type methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is weaker condition than monotonicity. Our proof of convergence is very simple as compared with other methods. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems.     

Projection-proximal methods for general variational inequalities

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.     

Some algorithms for hemiequilibrium problems

In this paper, we suggest and analyze a class of iterative methods for solving hemiequilibrium problems using the auxiliary principle technique. We prove that the convergence of these new methods either requires partially relaxed strongly monotonicity or pseudomonotonicity, which is a weaker condition than monotonicity. Results obtained in this paper include several new and known results as special cases.     

Auxiliary Principle Technique for Equilibrium Problems

In this paper, we use the auxiliary principle technique to suggest and analyze a number of iterative methods for solving mixed quasiequilibrium problems. We prove that the convergence of these new methods requires either partially relaxed strongly monotonicity or peudomonotonicity, which is a weaker condition than monotonicity. Our proof of convergence is very simple as compared with others. These new results include several new and known results as special cases. Our results represent refinement and improvement of the previous known results for equilibrium and variational inequalities problems.     

Some projection methods with the BB step sizes for variational inequalities

Since the appearance of the Barzilai-Borwein (BB) step sizes strategy for unconstrained optimization problems, it received more and more attention of the researchers. It was applied in various fields of the nonlinear optimization problems and recently was also extended to optimization problems with bound constraints. In this paper, we further extend the BB step sizes to more general variational inequality (VI) problems, i.e., we adopt them in projection methods. Under the condition that the underlying mapping of the VI problem is strongly monotone and Lipschitz continuous and the modulus of strong monotonicity and the Lipschitz constant satisfy some further conditions, we establish the global convergence of the projection methods with BB step sizes. A series of numerical examples are presented, which demonstrate that the proposed methods are convergent under mild conditions, and are more efficient than some classical projection-like methods.     

Monotonicity of control volume methods

Robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications. Here, we discuss monotonicity for a simplified problem of single-phase flow, but where the simulation grids and media are allowed to be general, posing challenges to control-volume methods. We discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine-point control-volume discretizations for conforming quadrilateral grids in 2D. These criteria are less restrictive than the M-matrix property. It is shown that it is impossible to construct nine-point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields. Numerical examples are presented which show the validity of the criteria for monotonicity. Further, the impact of nonmonotonicity is studied. Different behavior for different discretization methods is illuminated, and simple ideas are presented for improvement in terms of monotonicity.     

On the Monotonicity of the Compromise Set in Multicriteria Problems

This paper discusses the extension of results on monotonicity of the compromise set valid for bicriteria problems to general multicriteria problems under a very general condition, which is assumable in compromise programming problems coming from economics. Mainly, the problem that we treat is the following: find and describe the compromise set when the feasible set is a convex set in the positive cone, limited by a level hypersurface of a differentiable production–transformation function. This scenario is usual in many economic applications, chiefly in production analysis.     

A class of new iterative methods for general mixed variational inequalities

In this paper, we use the auxiliary principle technique to suggest a class of predictor-corrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities and related problems.     

Multidimensional BSDEs with weak monotonicity and general growth generators

This paper aims at solving a multidimensional backward stochastic differential equation （BSDE） whose generator g satisfies a weak monotonicity condition and a general growth condition in y. We first establish an existence and uniqueness result of solutions for this kind of BSDEs by using systematically the technique of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Then, we overview some assumptions related closely to the monotonieity condition in the literature and compare them in an effective way, which yields that our existence and uniqueness result really and truly unifies the Mao condition in y and the monotonieity condition with the general growth condition in y, and it generalizes some known results. Finally, we prove a stability theorem and a comparison theorem for this kind of BSDEs, which also improves some known results.     

Coupling the Auxiliary Problem Principle and Epiconvergence Theory to Solve General Variational Inequalities

Many algorithms for solving variational inequality problems can be derived from the auxiliary problem principle introduced several years ago by Cohen. In recent years, the convergence of these algorithms has been established under weaker and weaker monotonicity assumptions: strong (pseudo) monotonicity has been replaced by the (pseudo) Dunn property. Moreover, well-suited assumptions have given rise to local versions of these results.In this paper, we combine the auxiliary problem principle with epiconvergence theory to present and study a basic family of perturbed methods for solving general variational inequalities. For example, this framework allows us to consider barrier functions and interior approximations of feasible domains. Our aim is to emphasize the global or local assumptions to be satisfied by the perturbed functions in order to derive convergence results similar to those without perturbations. In particular, we generalize previous results obtained by Makler-Scheimberg et al.     

Discontinuous first-order functional boundary value problems

In this paper we prove a new existence result for functional boundary value problems with first-order functional differential equations under weak conditions on the nonlinear part and monotonicity (but not continuity) with respect to the functional variable. Discontinuities with respect to the unknown are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of unviability or solubility condition.     

Splitting Algorithms for General Pseudomonotone Mixed Variational Inequalities

In this paper, we suggest and analyze a number of resolvent-splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution. The convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator. Proof of convergence is very simple. Our new methods differ from the existing splitting methods for solving variational inequalities and complementarity problems. The new results are versatile and are easy to implement.     

Some New Iterative Methods for General Mixed Variational Inequalities

In this paper, we use the auxiliary principle technique to suggest a class of predictorcorrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. From special cases, we obtain various known and new results for solving various classes of variational inequalities and related problems.AMS Subject Classification (1991): 49J40, 90C33.     

Modified resolvent splitting algorithms for general mixed variational inequalities

In this paper, we suggest and analyze a number of four-step resolvent splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution. The convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator. Proof of convergence is very simple. Our new methods differ from the existing splitting methods for solving variational inequalities and complementarity problems. The new results are versatile and are easy to implement.     

An inverse coefficient problem with nonlinear parabolic equation

The problem related to controlled potential experiments in electrochemistry is studied. Ion transport is regarded as the superposition of diffusion and migration. Modelling of the experiment leads to a problem for a nonlinear parabolic equation with additional condition. Driven by the needs of theoretical analysis, from the point of view a inverse coefficient problem, we analyze the monotonicity of input-output mappings in inverse coefficient and source problems for this parabolic equation. Additionally, we extend the nonlinear parabolic equation to a more general case. Under some proper conditions, we investigate the existence of quasisolution of the generalized nonlinear parabolic equation.     

On the coerciveness of some merit functions for complementarity problems over symmetric cones

One of the popular solution methods for the complementarity problem over symmetric cones is to reformulate it as the global minimization of a certain merit function. An important question to be answered for this class of methods is under what conditions the level sets of the merit function are bounded (the coerciveness of the merit function). In this paper, we introduce the generalized weak-coerciveness of a continuous transformation. Under this condition, we prove the coerciveness of some merit functions, such as the natural residual function, the normal map, and the Fukushima-Yamashita function for complementarity problems over symmetric cones. We note that this is a much milder condition than strong monotonicity, used in the current literature.     

On General Mixed Quasivariational Inequalities

In this paper, we suggest and analyze several iterative methods for solving general mixed quasivariational inequalities by using the technique of updating the solution and the auxiliary principle. It is shown that the convergence of these methods requires either the pseudomonotonicity or the partially relaxed strong monotonicity of the operator. Proofs of convergence is very simple. Our new methods differ from the existing methods for solving various classes of variational inequalities and related optimization problems. Various special cases are also discussed.     

A Strongly Convergent Combined Relaxation Method in Hilbert Spaces

We consider a combined relaxation method for variational inequalities in a Hilbert space setting. Methods of this class are known to solve finite-dimensional variational inequalities under mild monotonicity type assumptions, whereas in Hilbert space strong monotonicity is the standard assumption for strong convergence. Here, we relax this condition and show strong convergence of such a method, when strong monotonicity holds only on a subspace of finite co-dimension. Thus, the method applies to semi-coercive unilateral boundary value problems in mathematical physics.     

Proximal Methods for Mixed Variational Inequalities

A proximal point method for solving mixed variational inequalities is suggested and analyzed by using the auxiliary principle technique. It is shown that the convergence of the proposed method requires only the pseudomonotonicity of the operator, which is a weaker condition than monotonicity. As special cases, we obtain various known and new results for solving variational inequalities and related problems. Our proof of convergence is very simple as compared with other methods.     

Bemerkungen zum Gradientenverfahren

Summary Certain iterative methods are applied to unconstrained optimization problems in a general Hilbert space. Convergence and monotonicity theorems are proved for gradient, steepest descent, Newton and some related methods that converge for arbitrary initial values and converge quadratically in a neighbourhood of the optimum.