Solving variational inequalities defined on a domain with infinitely many linear constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed method

Relaxed Cutting Plane Method for Solving Linear Semi-Infinite Programming Problems

One of the major computational tasks of using the traditional cutting plane approach to solve linear semi-infinite programming problems lies in finding a global optimizer of a nonlinear and nonconvex program. This paper generalizes the Gustafson and Kortanek scheme to relax this requirement. In each iteration, the proposed method chooses a point at which the infinite constraints are violated to a degree, rather than a point at which the violations are maximized. A convergence proof of the proposed scheme is provided. Some computational results are included. An explicit algorithm which allows the unnecessary constraints to be dropped in each iteration is also introduced to reduce the size of computed programs.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45251/1/10957_2004_Article_411711.pd

Proximal-based regularization methods and successive approximation of variational inequalities in Hilbert spaces

For variational inequalities with multi-valued, maximal monotone operators in Hilbert spaces we study proximal-based methods with an improvement of the data approximation after each (approximately performed) proximal iteration. The standard conditions on a distance functional of Bregman's type are weakened, depending on a "reserve of monotonicity" of the operator in the variational inequality, and the enlargement concept is used for approximating the operator. Weak convergence of the proxinnal iterates to a solution of tire original problem is proved. The construction of the [epsilon]-enlargement of monotone operators is analyzed for some particular cases

Regularized penalty method for non-coercive parabolic optimal control problems

The application of a proximal point approach to illposed convex control problems governed by linear parabolic equations is studied. A stable penalty method is constructed by means of multi-step proximal regularization (only w.r.t. the control functions) in the penalized problems. For distributed control problems with state constraints convergence of the approximately determined solutions of the regularized problems to an optimal process is proved

Case of Non-paramonotone Operators ∗

For variational inequalities characterizing saddle points of Lagragians associated with convex programming problems in Hilbert spaces, the convergence of an interior proximal method based on Bregman distance functionals is studied. The convergence results admit a successive approximation of the variational inequality and an inexact treatment of the proximal iterations. Key words Variational inequalities, monotone operators, proximal point methods, regularization, Bregman function. AMS subject classification: 65J20, 65K10, 90C25, 90C48