Abstract. Most known continuation methods for P 0 complementarity problems require some restrictive assumptions, such as the strictly feasible condition and a properness condition, to guarantee the existence and the boundedness of certain homotopy continuation trajectory. To relax such restrictions, we propose in this paper 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.
|
391
|
The Linear Complementarity Problem
– Cottle, Pang, et al.
- 1992
|
|
275
|
Algebraic Graph Theory
– Biggs
- 1993
|
|
126
|
A Unified Approach to Interior Point Algorithms for Linear Complimentarity Problems
– Kojima, Meggido, et al.
- 1991
|
|
122
|
Interior Point Algorithms. Theory and Analysis
– YE
- 1997
|
|
117
|
Pathways to the optimal set in linear programming
– Megiddo
- 1989
|
|
108
|
Equilibrium Points of Bimatrix Games
– Lemke, Howson
- 1964
|
|
89
|
Numerical Continuation Methods
– Allgower, Georg
- 1990
|
|
69
|
Bimatrix equilibrium points and mathematical programming
– Lemke
- 1965
|
|
42
|
A penalized Fischer-Burmeister NCP-function: Theoretical investigation and numerical results
– Chen, Chen, et al.
- 1997
|
|
34
|
A non-interior-point continuation method for linear complementarity problems
– Chen, Harker
- 1993
|
|
34
|
Homotopy continuation methods for nonlinear complementarity problems
– Kojima, Megiddo, et al.
- 1991
|
|
30
|
The global linear convergence of a non-interior path-following algorithm for linear complementarity problem
– Burke, Xu
- 1998
|
|
27
|
A new continuation method for complementarity problems with uniform P -functions
– Kojima, Mizuno, et al.
- 1989
|
|
23
|
A global and local superlinear continuation-smoothing method for P 0 and R 0 NCP or monotone NCP
– Chen, Chen
- 1999
|
|
22
|
Sufficient matrices and the linear complementarity problem
– Cottle, Pang, et al.
- 1989
|
|
22
|
Beyond monotonicity in regularization methods for nonlinear complementarity problems
– Facchinei, Kanzow
- 1997
|
|
21
|
The complementarity problem for maximal monotone multifunctions
– McLinden
- 1980
|
|
20
|
Global methods for nonlinear complementarity problems
– Mor'e
- 1994
|
|
17
|
Yoshise: Global convergence of a class of non-interior point algorithms using Chen-Harker-Kanzow-Smale functions for nonlinear complementarity problems
– Hotta, Yoshise
- 1996
|
|
16
|
A general framework of continuation methods for complementarity problems
– Kojima, Megiddo, et al.
- 1993
|
|
16
|
Regularization of P 0 -functions in box variational inequality problems
– Ravindran, Gowda
- 1997
|
|
15
|
Limiting behavior of trajectories generated by a continuation method for monotone complementarity problems
– Kojima, Mizuno, et al.
- 1990
|
|
15
|
Theory of vector optimization
– Luc
- 1989
|
|
14
|
A non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem
– Burke, Xu
- 1997
|
|
13
|
A polynomial time interior-point path-following algorithm for LCP based on Chen-Harker-Kanzow smoothing techniques
– Xu, Burke
- 1999
|
|
13
|
Structural and stability properties of P 0 nonlinear complementarity problems
– Facchinei
- 1998
|
|
12
|
A regularized smoothing Newton method for box constrained variational inequality problems with P 0 -functions
– Qi
- 1997
|
|
12
|
P -matrices are just sufficient
– Valiaho
- 1996
|
|
10
|
On P - and S-functions and related classes of ndimensional nonlinear mappings", Linear Algebra and Its Applications 6
– Mor'e, Rheinboldt
- 1973
|
|
9
|
Exceptional families, topological degree and complementarity problems
– Isac, Bulavski, et al.
- 1997
|
|
8
|
On the limiting behavior of the trajectory of regularized solutions of P 0 complementarity problems
– Sznajder, Gowda
- 1998
|
|
7
|
Tawhid, Existence and limiting behavior of trajectories associated with P 0 -equations
– Gowda, Tawhid
- 1999
|
|
7
|
Properties of an interior-point mapping for mixed complementarity problems
– Monteiro, Pang
- 1996
|
|
7
|
Exceptional families and existence theorems for variational inequality problems
– Zhao, Han, et al.
- 1999
|
|
5
|
Stable monotone variational inequalities
– McLinden
- 1990
|
|
4
|
On smoothing methods for the P 0 matrix linear complementarity problem
– Chen, Ye
- 1998
|
|
4
|
A complexity analysis of a smoothing method using CHKS-functions for monotone linear complementarity problems
– Hotta, Inaba, et al.
- 1998
|
|
4
|
Improving the convergence of non-interior point algorithm for nonlinear complementarity problems
– Qi, Sun
- 2000
|
|
4
|
An algorithm for the linear complementarity problem with a P 0 -matrix
– Venkateswaran
- 1993
|
|
4
|
Existence of a solution to nonlinear variational inequality under generalized positive homogeneity
– Zhao
- 1999
|
|
3
|
Existence of interior points and interior-point paths in nonlinear monotone complementarity problems
– Guler
- 1993
|
|
3
|
Tikhonov's regularization and the complementarity problem in Hilbert space
– Isac
- 1993
|
|
3
|
A note on least two norm solutions of monotone complementarity problems
– Subramanian
- 1988
|
|
3
|
On constructing interior-point path-following methods for certain semimonotone linear complementarity problems
– Zhang, Zhang
- 1997
|
|
3
|
Characterization of a homotopy solution mapping for nonlinear complementarity problems
– Zhao, Li
- 1998
|
|
2
|
Total stability of variational inequalities, Dipartimento di Informatica e Sistemistica, Universit`a di Roma "La Sapienza", Via Buonarroti
– Facchinei, Pang
- 1998
|
|
2
|
Weak univalence and connected of inverse images of continuous functions
– Gowda, Sznajder
- 1999
|
|
2
|
Some nonlinear continuation methods for linear complementarity problems
– Kanzow
- 1996
|
|
2
|
A polynomial-time algorithm for linear complementarity problems
– Kojima, Mizuno, et al.
- 1989
|
|
1
|
Error bounds for regularized complementarity problems
– Tseng
- 1998
|
|
