J Glob Optim (2006) 36:565–580
DOI 10.1007/s10898-006-9027-y
ORIGINAL ARTICLE
The semismooth-related properties of a merit function
and a descent method for the nonlinear
complementarity problem
Jein-Shan Chen
Received: 13 March 2006 / Accepted: 20 March 2006 /
Published online: 14 June 2006
© Springer Science+Business Media B.V. 2006
Abstract This paper is a follow-up of the work [Chen, J.-S.: J. Optimiz. Theory Appl.,
Submitted for publication (2004)] where an NCP-function and a descent method were
proposed for the nonlinear complementarity problem. An unconstrained reformula-
tion was formulated due to a merit function based on the proposed NCP-function.
We continue to explore properties of the merit function in this paper. In particular,
we show that the gradient of the merit function is globally Lipschitz continuous which
is important from computational aspect. Moreover, we show that the merit function
is SC
1
function which means it is continuously differentiable and its gradient is semi-
smooth. On the other hand, we provide an alternative proof, which uses the new
properties of the merit function, for the convergence result of the descent method
considered in [Chen, J.-S.: J. Optimiz. Theory Appl., Submitted for publication (2004)].
Keywords Complementarity · SC
1
function · Merit function · Semismooth function ·
Descent method
1 Introduction
In the past decades, the well-known nonlinear complementarity problem (NCP) has
attracted much attention due to its various applications in operations research, eco-
nomics, and engineering [6, 11, 17]. The NCP is to find a point x ∈ IR
n
such that
x ≥ 0, F (x) ≥ 0, x, F (x)= 0, (1)
where ·, · is the Euclidean inner product and F = (F
1
, F
2
, ... , F
n
)
T
maps from IR
n
to IR
n
. We assume that F is continuously differentiable throughout this paper.
There have been many methods proposed for solving the NCP [9, 11, 17]. Among
which, one of the most popular approaches that has been studied intensively recently
J.-S. Chen (B )
Department of Mathematics
National Taiwan Normal University
Taipei 11677, Taiwan
e-mail: jschen@math.ntnu.edu.tw