Regularization of P₀-Functions in Box Variational Inequality Problems

Abstract

In two recent papers, Facchinei 7] and Facchinei and Kanzow 8] have shown that for a continuously di erentiable P 0-function f , the nonlinear complementarity problem NCP(f ") corresponding to the regularization f " (x) := f (x) + "x has a unique solution for every " > 0, that dist (x("); SOL(f)) ! 0 as " ! 0 when the solution set SOL(f) of NCP(f) is nonempty and bounded, and NCP(f) is stable if and only if the solution set is nonempty and bounded. They prove these results via the the Fischer function and the Mountain Pass Theorem. In this paper, we generalize these NCP results to a Box Variational Inequality Problem corresponding to a continuous P 0-function where the regularization is described by an integral. We also describe an upper semicontinuity property of the inverse of a weakly univalent function and study its consequences.

A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming

“…It is not difficult to show that the functions H 0 : n+m → n+m and H : 1+n+m → 1+n+m defined by (2.4) and (2.8), respectively are weakly univalent functions defined in [11]. Since Assumption 3.1 implies that the inverse image H −1 0 (0) is nonempty and bounded, by using Theorem 2.5 in [29] we obtain that the sequence {z k } is bounded. Hence, by Lemma 3.2, any accumulation point of {z k } is a solution of H(z) = 0.…”

Calibrating Least Squares Semidefinite Programming with Equality and Inequality Constraints

“…Therefore, by using the assumption that E is strongly semismooth at (ε,ȳ) and the relations (38), (39), and (40), we have for all (ε k , y k ) sufficiently close to (ε,ȳ) that…”

“…Thus, E(ε, y) = 0 also has a nonempty and compact solution set. Since part (ii) of Proposition 2.3 implies that E is a P 0 -function, the boundedness of {(ε k , y k )} follows directly from [39,Theorem 2.5].…”

A Globally and Locally Superlinearly Convergent Non--Interior-Point Algorithm for P0LCPs

“…For P 0 LCPs, it is shown (see [42,43]) that most assumptions used for non-interior-point algorithms, for instance, the Condition 1.5 in [25], Condition 1.2 in Hotta and Yoshise [20], and the P 0 + R 0 assumption in Burke and Xu [3] and Chen and Chen [7], imply that the solution set of the problem is bounded. As showed by Ravindran and Gowda in [34] the P 0 complementarity problem with a bounded solution set must have a strictly feasible point, i.e., there exists an x 0 such that M x 0 + d > 0. (This implies that a P 0 LCP with no strictly feasible point either has no solution or has an unbounded solution set.)…”

“…Thus, the set {x(θ) : θ ∈ (0, 1]} forms a smooth path approaching to the solution set of the P 0 LCP as θ tends to zero. Notice that for a given θ, the term M x + d + θ p x is the Tikhonov regularization of M x + d, which has been used to study complementarity problems by several authors such as Isac [22], Venkateswaran [36], Facchinei and Kanzow [14], Ravindran and Gowda [34], and Zhao and Li [42]. We may refer the above smooth path to regularized central path.…”