Ellipsoidal approximation methods for variational inequalities over polyhedral set

Abstract

The application of variational inequalities covers a great number of areas. We concern with numerical solution method of the variational inequality over a polyhedral set. Among well known numerical methods are the projection method, Newton method and etc. The subproblems of them, however, require another iterative methods which are themselves often computationally challenging since they are also optimization problems. The purpose of this research is to propose effective methods to solve these subproblems by approximating the given set K via an inscribed ellipsoid. We propose an ellipsoidal projection method and an ellipsoidal Newton method. The subproblems of them are shown to be solved in a closed form. A custom tailored version of the proposed ellipsoidal projection method for fixed demand traffic equilibrium problem is also proposed. A practical version of the ellipsoidal projection method with additional line search step is proposed to give safety against possible risk of small step size. Convergence properties of the above methods are investigated. Limited computational experiments with small sized traffic equilibrium problems show that the proposed ellipsoidal projection method is promising.