Finite convergence of algorithms for nonlinear programs and variational inequalities (Q809897)

The authors derive a sufficient condition for nonlinear programs with convex feasible sets such that any convergent algorithm can be modified, by adding a convex subproblem with a linear objective function, to guarantee finite convergence in a generalized sense. When the feasible set is polyhedral, the subproblem is a linear program and finite convergence is obtained. Similar results are also developed for the variational inequality problem which calls for finding a point \(x'\in S\) such that \(G(x')^ T(x-x')\geq 0\), for all \(x\in S\), where G is a continuous mapping from \(R^ n\) to \(R^ n.\) Theorems 2.1 and 3.1 are two main results of the paper. In them, the assumption \(-\nabla f(x')\in int N(x')\) (resp., \(-G(x')\in int N(x'))\) is very crucial, where \(N(x')\) is the normal cone of the feasible set S at \(x'\).