An affine scaling interior trust region method via optimal path for solving monotone variational inequality problem with linear constraints

Abstract

Based on a differentiable merit function proposed by Taji et al. in "Math. Prog. Stud., 58, 1993, 369-383", the authors propose an affine scaling interior trust region strategy via optimal path to modify Newton method for the strictly monotone variational inequality problem subject to linear equality and inequality constraints. By using the eigensystem decomposition and affine scaling mapping, the authors form an affine scaling optimal curvilinear path very easily in order to approximately solve the trust region subproblem. Theoretical analysis is given which shows that the proposed algorithm is globally convergent and has a local quadratic convergence rate under some reasonable conditions.

A new non-monotone filter trust region algorithm for solving nonlinear systems of equalities and inequalities

“…Suppose Assumptions H hold. Then whenever condition (14) holds, the condition (28) is satisfied for sufficiently large k.…”

“…Systems of nonlinear equalities and inequalities appear in a wide variety of problems. These systems play a central role in the model formulation design and analysis of numerical techniques employed in solving problems arising in optimization, complementarity, and variational inequalities [12,15,18,19,20,28,30]. For example, nonlinear complementarity problem is a special case of (1).…”

A penalty approach for a box constrained variational inequality problem

“…Extensive studies of BVIP have been done in [11], [7] and the references therein. Numerical methods for solving BVIP have been extensively investigated in the literature such as smoothing Newton methods [5], [14], [17], interior point method [2] and nonsmooth equation methods [9], [8]. However, it seems that there are few studies of penalty methods for BVIP.…”