A numerical approach to optimization problems with variational inequality constraints

Optimization problems with variational inequality constraints are converted to constrained minimization of a local Lipschitz function. To this minimization a non-differentiable optimization method is used; the required subgradients of the objective are computed by means of a special adjoint equation. Besides tests with some academic examples, the approach is applied to the computation of the Stackelberg—Cournot—Nash equilibria and to the numerical solution of a class of quasi-variational inequalities.Corresponding author.     

Multiobjective optimization problem with variational inequality constraints

 We study a general multiobjective optimization problem with variational inequality, equality, inequality and abstract constraints. Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived. They lead to Kuhn-Tucker type necessary optimality conditions under additional constraint qualifications including the calmness condition, the error bound constraint qualification, the no nonzero abnormal multiplier constraint qualification, the generalized Mangasarian-Fromovitz constraint qualification, the strong regularity constraint qualification and the linear constraint qualification. We then apply these results to the multiobjective optimization problem with complementarity constraints and the multiobjective bilevel programming problem. Received: November 2000 / Accepted: October 2001 Published online: December 19, 2002 Key Words. Multiobjective optimization – Variational inequality – Complementarity constraint – Constraint qualification – Bilevel programming problem – Preference – Utility function – Subdifferential calculus – Variational principle Research of this paper was supported by NSERC and a University of Victoria Internal Research Grant Research was supported by the National Science Foundation under grants DMS-9704203 and DMS-0102496 Mathematics Subject Classification (2000): Sub49K24, 90C29     

The variational inequality approach to the

Considering the one-phase Stefan problem, we present an account of some recent mathematical results within the framework of variational inequalities. We discuss several situations corresponding to different boundary conditions and different geometries, like the exterior problem, the continuous casting model, and the degenerate case of the quasi-steady model. We develop a few continuous-dependence results explaining their relevance to the stability properties of the solution and of the free boundary, including the asymptotic behaviour for large time, the stability for homogenization, and the perturbation of the Dirichlet boundary conditions.     

Parabolic variational inequality with parameter and gradient constraints

This paper concerns a singular control problem whose value function is governed by a time-dependent HJB equation with gradient constraints. The method is to transform a two-dimensional parabolic variational inequality with gradient constraints into a double obstacle problem with parameter involving two free boundaries that correspond to the investment and disinvestment policies. Moreover we analyze the behaviors of the free boundary surfaces. The main difficulties are to show the free boundary surfaces to be smooth with respect to time and to find the properties of free boundaries with respect to parameter.     

Approximate values for mathematical programs with variational inequality constraints

In general the infimal value of a mathematical program with variational inequality constraints (MPVI) is not stable under perturbations in the sense that the sequence of infimal values for the perturbed programs may not converge to the infimal value of the original problem even in presence of nice data. Thus, for these programs we consider different types of values which approximate the exact value from below or/and from above under or without perturbations.     

Multilevel multi-leader multiple-follower games with nonseparable objectives and shared constraints

Computational Management Science - Multi-leader multi-follower (MLMF) games are hierarchical games in which a collection of players in the upper-level, called leaders, compete in a Nash game...     

Differential games of fixed duration with state constraints

We consider differential games of fixed duration with phase coordinate restrictions on the players. Results of Ref. 1 on games with phase restrictions on only one of the players are extended. Using Berkovitz's definition of a game (Ref. 2), we prove the existence and continuity (or Lipschitz continuity) of the value under appropriate assumptions. We also note that the value can be characterized as the viscosity solution of the associated Hamilton-Jacobi-Isaacs equation.This work comprises a part of the author's PhD Thesis completed at Purdue University under the direction of Professor L. D. Berkovitz. The author wishes to thank Professor Berkovitz for suggesting the problem and many valuable discussions. During the research for this work, the author was supported by a David Ross Grant from Purdue University as well as by NSF Grant No. DMS-87-00813.     

Perturbation approach to generalized Nash equilibrium problems with shared constraints

This paper aims to consider an application of conjugate duality in convex optimization to the generalized Nash equilibrium problems with shared constraints and nonsmooth cost functions. Sufficient optimality conditions for the problems regarding to players are rewritten as inclusion problems and the maximal monotonicity of set-valued mappings generated by the subdifferentials of functions from data of GNEP is proved. Moreover, some assertions dealing with solutions to GNEP are obtained and applications of splitting algorithms to the oligopolistic market equilibrium models are presented.     

A projection subgradient method for solving optimization with variational inequality constraints

In this paper, we propose a projection subgradient method for solving some classical variational inequality problem over the set of solutions of mixed variational inequalities. Under the conditions that $T$ is a $\Theta $ -pseudomonotone mapping and $A$ is a $\rho $ -strongly pseudomonotone mapping, we prove the convergence of the algorithm constructed by projection subgradient method. Our algorithm can be applied for instance to some mathematical programs with complementarity constraints.     

Legendre-type optimality conditions for a variational problem with inequality state constraints

, they differ from the Legendre-Clebsch condition. They give information about the Hesse matrix of the integrand at not only inactive points but also active points. On the other hand, since the inequality state constraints can be regarded as an infinite number of inequality constraints, they sometimes form an envelope. According to a general theory [9], one has to take the envelope into consideration when one consider second-order necessary optimality conditions for an abstract optimization problem with a generalized inequality constraint. However, we show that we do not need to take it into account when we consider Legendre-type conditions. Finally, we show that any inequality state constraint forms envelopes except two trivial cases. We prove it by presenting an envelope in a visible form. Received April 18, 1995 / Revised version received January 5, 1998 Published online August 18, 1998     

Levitin–Polyak well-posedness of variational inequality problems with functional constraints

In this paper, we introduce several types of (generalized) Levitin–Polyak well-posednesses for a variational inequality problem with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given. Relations among these types of well-posednesses are also investigated.     

A probabilistic approach to second order variational inequalities with bilateral constraints

We study a class of second order variational inequalities with bilateral constraints. Under certain conditions we show the existence of aunique viscosity solution of these variational inequalities and give a stochastic representation to this solution. As an application, we study a stochastic game with stopping times and show the existence of a saddle point equilibrium.     

Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints

In this paper, we propose an easily implementable algorithm in Hilbert spaces for solving some classical monotone variational inequality problem over the set of solutions of mixed variational inequalities. The proposed method combines two strategies: projected subgradient techniques and viscosity-type approximations. The involved stepsizes are controlled and a strong convergence theorem is established under very classical assumptions. Our algorithm can be applied for instance to some mathematical programs with complementarity constraints.     

Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints

In this paper we consider heuristic algorithms for a special case of the generalized bilevel mathematical programming problem in which one of the levels is represented as a variational inequality problem. Such problems arise in network design and economic planning. We obtain derivative information needed to implement these algorithms for such bilevel problems from the theory of sensitivity analysis for variational inequalities. We provide computational results for several numerical examples.     

A continuum of solutions to variational inequality with nonlinear constraints: Existence and simplicial algorithm

In this paper, we obtain an existence theorem of a connected set of solutions to a nonlinear variational inequality with explicit nonlinear constraints. This result follows in a constructive way by designing a simplicial algorithm. The algorithm operates on a triangulation of the unbounded regions and generates a piecewise linear path of parametrized stationary points. Each point on the path is an approximate solution.     

On the convergence for kinetic variational inequality to quasi-static variational inequality with application to elastic-plastic systems with hardening

In this note, a priori estimates for the kinetic problem are obtained that imply, that the kinetic solutions converge to the quasi-static ones, when the size of initial perturbations and the rate of application of the forces tend to 0. An application to three-dimensional elastic-plastic systems with hardening is given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variational inequality formulation of equilibrium models for end-of-life products with nonlinear constraints

Variational inequality theory facilitates the formulation of equilibrium problems in economic networks. Examples of successful applications include models of supply chains, financial networks, transportation networks, and electricity networks. Previous economic network equilibrium models that were formulated as variational inequalities only included linear constraints; in this case the equivalence between equilibrium problems and variational inequality problems is achieved with a standard procedure because of the linearity of the constraints. However, in reality, often nonlinear constraints can be observed in the context of economic networks. In this paper, we first highlight with an application from the context of reverse logistics why the introduction of nonlinear constraints is beneficial. We then show mathematical conditions, including a constraint qualification and convexity of the feasible set, which allow us to characterize the economic problem by using a variational inequality formulation. Then, we provide numerical examples that highlight the applicability of the model to real-world problems. The numerical examples provide specific insights related to the role of collection targets in achieving sustainability goals.     

Levitin–Polyak well-posedness in generalized vector variational inequality problem with functional constraints

In this paper, we study Levitin–Polyak type well-posedness for generalized vector variational inequality problems with abstract and functional constraints. Various criteria and characterizations for these types of well-posednesses are given. This research is partially supported by the National Science Foundation of China and Shanghai Pujiang Program.     

A variational inequality approach to constrained control problems for parabolic equations

A distributed optimal control problem for parabolic systems with constraints in state is considered. The problem is transformed to control problem without constraints but for systems governed by parabolic variational inequalities. The new formulation presented enables the efficient use of a standard gradient method for numerically solving the problem in question. Comparison with a standard penalty method as well as numerical examples are given.