On a New Homotopy Continuation Trajectory for Nonlinear Complementarity Problems

Abstract

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.

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.…”

“…Recently, Zhao and Li [42] proposed a new continuation trajectory for complementarity problems, which is defined as follows:…”

“…where θ is a parameter in (0,1], p ∈ (0, ∞) and q ∈ [1, ∞) are two fixed scalars, and a = (a 1 , ..., a n ) T ∈ R n ++ is a fixed vector, for example, a = e. For a P 0 matrix M, it turns out (see [42]) that the above system has a unique solution for each given parameter θ, and this solution, denoted by x(θ), is continuously differentiable on (0,1). 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.…”

“…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.…”

“…In particular, the boundedness of the solution set and the strict feasibility condition are not needed for the existence of this path. Combining the CHKS function and Tikhonov regularization method, Zhao and Li [43] extended the results in [42] to non-interior-point methods, and studied the existence as well as the limiting behavior of a new non-interior-point smooth path.…”