We prove sufficient conditions for the monotonicity and the strong monotonicity of fixed point and normal maps associated with variational inequality problems over a general closed convex set. Sufficient conditions for the strong monotonicity of their perturbed versions are also shown. These results include some well known in the literature as particular instances. Inspired by these results, we propose a modified Solodov and Svaiter iterative algorithm for the variational inequality problem whose fixed point map or normal map is monotone.

We consider a rather general class of mathematical programming problems with data uncertainty, where the uncertainty set is represented by a system of convex inequalities. We prove that the robust counterparts of this class of problems can be equivalently reformulated as finite and explicit optimization problems. Moreover, we develop simplified reformulations for problems with uncertainty sets defined by convex homogeneous functions. Our results provide a unified treatment of many situations that have been investigated in the literature, and are applicable to a wider range of problems and more complicated uncertainty sets than those considered before. The analysis in this paper makes it possible to use existing continuous optimization algorithms to solve more complicated robust optimization problems. The analysis also shows how the structure of the resulting reformulation of the robust counterpart depends both on the structure of the original nominal optimization problem and on the structure of the uncertainty set.

Most known continuation methods for P 0 complementarity problems require some restrictive assumptions, such as the strictly feasible condition and the properness condition, to guarantee the existence and the boundedness of certain homotopy continuation trajectory. To relax such restrictions, we propose a new homotopy formulation for the complementarity problem based on which a new homotopy continuation trajectory is generated. For P 0 complementarity problems, the most promising feature of this trajectory is the assurance of the existence and the boundedness of the trajectory under a condition that is strictly weaker than the standard ones used widely in the literature of continuation methods. Particularly, the often-assumed strictly feasible condition is not required here. When applied to P * complementarity problems, the boundedness of the proposed trajectory turns out to be equivalent to the solvability of the problem, and the entire trajectory converges to the (unique) least element solution provided that it exists. Moreover, for monotone complementarity problems, the whole trajectory always converges to a least 2-norm solution provided that the solution set of the problem is nonempty. The results presented in this paper can serve as a theoretical basis for constructing a new path-following algorithm for solving complementarity problems, even for the situations where the solution set is unbounded.

No abstract

Abstract. For P 0 -complementarity problems, most existing non-interior-point path-following methods require the existence of a strictly feasible point. (For a P * -complementarity problem, the existence of a strictly feasible point is equivalent to the nonemptyness and the boundedness of the solution set). In this paper, we propose a new homotopy formulation for complementarity problems by which a new non-interior-point continuation trajectory is generated. The existence and the boundedness of this non-interior-point trajectory for P 0 -complementarity problems are proved under a very mild condition that is weaker than most used conditions in the literature. One prominent feature of this condition is that it may hold even when the often-assumed strict feasibility condition fails to hold. In particular, for a P * -problem, it turns out that the new non-interior-point trajectory exists and is bounded if and only if the problem has a solution. We also study the convergence of this trajectory and the characterization of its limiting point as the parameter approaches zero.

No abstract

No abstract

No abstract