## QPCOMP: A Quadratic Programming Based Solver for Mixed Complementarity Problems (1997)

Venue: | Mathematical Programming |

Citations: | 32 - 15 self |

### BibTeX

@ARTICLE{Billups97qpcomp:a,

author = {Stephen C. Billups and Michael C. Ferris},

title = {QPCOMP: A Quadratic Programming Based Solver for Mixed Complementarity Problems},

journal = {Mathematical Programming},

year = {1997},

volume = {76},

pages = {533--562}

}

### OpenURL

### Abstract

QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel[14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudo-monotone mixed nonlinear complementarity problem. QPCOMP also extends the NE/SQP method for the nonlinear complementarity problem to the more general mixed nonlinear complementarity problem. Computational results are provided, which demonstrate the effectiveness of the algorithm. 1 Introduction This paper describes a new algorithm for solving the mixed nonlinear complementarity problem (MCP), which provides a significant improvement in robustness over previous superlinearly or quadratically convergent algorithms, while preserving these fast local convergence properties. The MCP is defined in...

### Citations

989 |
Arsenin V. Solution of Ill-Posed Problems
- Tikhonov
- 1977
(Show Context)
Citation Context ...ng point. In particular the centering point for one subproblem is the solution of the previous subproblem. This is very reminiscent of the proximal point algorithm [16] and of Tikhonov regularization =-=[18]-=-. The following lemma gives sufficient conditions for a subsequence of these iterates to converge to a solution of MCP(f; IB). Theorem 2.3 Lets? 0 and let fx k g; k = 0; 1; ::: be a sequence of points... |

610 | Rheinboldt, Iterative Solution of Nonlinear Equations - Ortega, C - 1970 |

225 |
Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications
- Harker, Pang
- 1990
(Show Context)
Citation Context ...udo-monotone at a point x 2 IB if 8y 2 IB, f(x ) ? (y \Gamma x )s0 implies f(y) ? (y \Gamma x )s0: (1) f is said to be pseudo-monotone on IB if it is pseudo-monotone at every point in IB. It is known =-=[9]-=- that if a function g : IR n ! IR is pseudo-convex [11, Definition 9.3.1], then rg is a pseudo-monotone function. However, if g is only pseudo-convex at a point x , it does not necessarily follow that... |

188 | The PATH Solver: A Non-Monotone Stabilization Scheme for Mixed Complementarity Problems
- Dirkse, Ferris
- 1995
(Show Context)
Citation Context ...ersity of Wisconsin, Madison, Wisconsin 53706 1 chosen so that zeros of the merit function correspond to solutions of the complementarity problem. Among the algorithms included in this class are PATH =-=[5, 15]-=-, MILES [17], SMOOTH [3], NE/SQP [14], and BDIFF [10]. Within this basic framework, there are substantial differences between the algorithms; the algorithms differ in the choice of merit function, the... |

164 |
GAMS: A User’s Guide, The Scientific
- Brooke, Kendrick, et al.
- 1992
(Show Context)
Citation Context ...fied NE/SQP algorithm is called from Step 2 of QPCOMP, and is increased by 4 whenever the Modified NE/SQP algorithm fails, up to a maximum of 30. QPCOMP was interfaced with the GAMS modeling language =-=[2, 6]-=-, allowing problems to be easily specified in GAMS, and the algorithm to be tested using MCPLIB [4] and GAMSLIB [2]. Specifically, we tested QPCOMP on every problem with fewer than 110 variables in MC... |

159 | Engineering and economic applications of complementarity problems”, Society for industrial and applied mathematics
- Ferris, Pang
- 1997
(Show Context)
Citation Context ...the MCP reduces to the standard nonlinear complementarity problem (NCP), which is to find xs0 such that f(x)s0 and x ? f(x) = 0: Complementarity problems (both MCP and NCP) arise in many applications =-=[4, 7]-=- and are the subject of much recent computational work. Indeed in recent years, a significant number of algorithms have been developed to solve complementarity problems. Most of these algorithms can b... |

114 |
A class of smoothing functions for nonlinear and mixed complementarity problems
- Chen, Mangasarian
- 1996
(Show Context)
Citation Context ..., Wisconsin 53706 1 chosen so that zeros of the merit function correspond to solutions of the complementarity problem. Among the algorithms included in this class are PATH [5, 15], MILES [17], SMOOTH =-=[3]-=-, NE/SQP [14], and BDIFF [10]. Within this basic framework, there are substantial differences between the algorithms; the algorithms differ in the choice of merit function, the techniques used for det... |

106 |
MINOS 5.0 users guide
- BA, MA
- 1983
(Show Context)
Citation Context ...1 ) ! `(x k ) =) `(x k ) !sk `(x 0 ); so `(x k ) converges to zero. 5 Implementation and Testing The QPCOMP algorithm was coded in ANSI C, using double precision arithmetic. The Fortran package MINOS =-=[12]-=- was used to solve the quadratic subproblems. The algorithm allows for a great deal of flexibility in the choice of parameters, which can be specified in an options file. For testing purposes, we used... |

81 | MCPLIB – A Collection of Nonlinear Mixed Complementarity Problems
- Dirkse, Ferris
- 1994
(Show Context)
Citation Context ...the MCP reduces to the standard nonlinear complementarity problem (NCP), which is to find xs0 such that f(x)s0 and x ? f(x) = 0: Complementarity problems (both MCP and NCP) arise in many applications =-=[4, 7]-=- and are the subject of much recent computational work. Indeed in recent years, a significant number of algorithms have been developed to solve complementarity problems. Most of these algorithms can b... |

64 |
NE/SQP: a robust algorithm for the nonlinear complementarity problem
- Pang, Gabriel
- 1993
(Show Context)
Citation Context ...6 Abstract QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel=-=[14]-=-, this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable,... |

60 | Global Convergence of Damped Newton’s Method for Nonsmooth Equations, via the Path Search - RALPH |

44 | Algorithms for complementarity problems and generalized equations
- Billups
- 1995
(Show Context)
Citation Context ...ction will be used to guarantee sufficient decrease in the merit function at each iteration. The following proposition summarizes some essential properties of the functions OE and z: Proposition 3.1 (=-=[1]-=-, Lemmas 2.2.5 and 2.2.6) The following properties hold: 1. If x k 2 IB, then (QP k ) has at least one optimal solution. 2. OE(x; d) \Gamma OE(x; 0) \Gamma z(x; d)s` 0 (x; d) for all (x; d) 2 IB \Thet... |

36 |
Continuation method for nonlinear complementarity problems via normal maps
- Chen, Harker, et al.
- 1999
(Show Context)
Citation Context ...o that zeros of the merit function correspond to solutions of the complementarity problem. Among the algorithms included in this class are PATH [5, 15], MILES [17], SMOOTH [3], NE/SQP [14], and BDIFF =-=[10]-=-. Within this basic framework, there are substantial differences between the algorithms; the algorithms differ in the choice of merit function, the techniques used for determining search directions, a... |

22 | The GAMS callable program library for variational and complementarity solvers
- Dirkse, Ferris, et al.
(Show Context)
Citation Context ...fied NE/SQP algorithm is called from Step 2 of QPCOMP, and is increased by 4 whenever the Modified NE/SQP algorithm fails, up to a maximum of 30. QPCOMP was interfaced with the GAMS modeling language =-=[2, 6]-=-, allowing problems to be easily specified in GAMS, and the algorithm to be tested using MCPLIB [4] and GAMSLIB [2]. Specifically, we tested QPCOMP on every problem with fewer than 110 variables in MC... |

19 | MILES: A mixed inequality and nonlinear equation solver. Working Paper
- Rutherford
- 1993
(Show Context)
Citation Context ...nsin, Madison, Wisconsin 53706 1 chosen so that zeros of the merit function correspond to solutions of the complementarity problem. Among the algorithms included in this class are PATH [5, 15], MILES =-=[17]-=-, SMOOTH [3], NE/SQP [14], and BDIFF [10]. Within this basic framework, there are substantial differences between the algorithms; the algorithms differ in the choice of merit function, the techniques ... |

16 | Nonlinear Programming (McGraw-Hill - Mangasarian - 1969 |

12 |
Monotone operators and augmented Lagrangian methods in nonlinear programming
- Rockafellar
- 1978
(Show Context)
Citation Context ... , but a different choice of centering point. In particular the centering point for one subproblem is the solution of the previous subproblem. This is very reminiscent of the proximal point algorithm =-=[16]-=- and of Tikhonov regularization [18]. The following lemma gives sufficient conditions for a subsequence of these iterates to converge to a solution of MCP(f; IB). Theorem 2.3 Lets? 0 and let fx k g; k... |

3 |
Algorithms for the Nonlinear Complementarity Problem: The NE/SQP Method and Extensions
- Gabriel
- 1992
(Show Context)
Citation Context ... 3.5. However, detailed proofs for these results are given in [1, Chapter 2]. Once the extended NE/SQP algorithm is presented we will then modify it to ensure finite termination. We note that Gabriel =-=[8]-=- also extended NE/SQP to address the upper bound nonlinear complementarity problem, a special case of MCP where l = 0 and u ? 0 is finite. 3.1 Extension of NE/SQP to the MCP Framework Recall that a ve... |