## Feasible descent algorithms for mixed complementarity problems (1999)

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Citations: | 33 - 15 self |

### BibTeX

@MISC{Ferris99feasibledescent,

author = {Michael C. Ferris and Christian Kanzow and Todd S. Munson},

title = {Feasible descent algorithms for mixed complementarity problems},

year = {1999}

}

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### Abstract

### Citations

1156 |
Optimization and Nonsmooth Analysis
- Clarke
- 1983
(Show Context)
Citation Context ...inear system of equations \Phi(x) = 0. The function \Phi is not differentiable everywhere. However, it is locally Lipschitzian and therefore has a nonempty generalized Jacobian in the sense of Clarke =-=[8]-=-. We next present an overestimation of this generalized Jacobian (see Billups [2, Lemma 3.2.10]). Proposition 2. We have @ \Phi(x) T ` fD a (x) +rF (x)D b (x)g; where D a (x) 2 IR n\Thetan and D b (x)... |

829 | Neuro-dynamic programming - Bertsekas, Tsitsiklis - 1996 |

804 |
Nonlinear Programming. Athena Scientific
- Bertsekas
- 1995
(Show Context)
Citation Context ...ake the full step, then the new point x k + d k stays in the feasible set [l; u]. Note that, due to the convexity of our feasible set [l; u], this guarantees that all the points x k + t k d k ; t k 2 =-=[0; 1]-=- are feasible, too. On the other hand, property (b) states that, under Robinson's strong regularity condition, Algorithm A is locally well-defined and generates a locally superlinearly convergent sear... |

350 | CAMS: A User’s Guide
- Kendrick, Meeraus
- 1988
(Show Context)
Citation Context ...ilar scheme in this code. We note that several heuristic procedures were described and tested in [12]; all of these appear not to be beneficial to PATH. MPSGE models are generated differently by GAMS =-=[5]-=-, and thus a solver can distinguish them from general MCP models. We have used this information to choose different default options for MPSGE models both in PATH 3.0 and PATH 4.0. For PATH 4.0, by def... |

234 |
Computing a trust region step
- Moré, Sorensen
- 1983
(Show Context)
Citation Context ...ted by Algorithm 1 converges to a solution x if this solution satisfies the strong regularity assumption. The critical tool to establish this result is the following proposition of Mor'e and Sorensen =-=[27]-=-. Proposition 7. Assume that x is an isolated accumulation point of a sequence fx k g (not necessarily generated by Algorithm 1) such that fkx k+1 \Gamma x k kgK ! 0 for any subsequence fx k gK conver... |

181 |
Strongly regular generalized equations
- Robinson
- 1980
(Show Context)
Citation Context ...he case x i = u i and F i (x ) ! 0 can be proven in a similar manner. Furthermore, statement (b) also follows by using an identical argument. 2 We next restate a useful characterization of Robinson's =-=[31]-=- strong regularity condition in the context of mixed complementarity problems. A proof may be found in [13]. We stress that, in the case of a nonlinear complementarity problem (i.e., l i = 0 and u i =... |

179 | The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems
- Dirkse, Ferris
- 1995
(Show Context)
Citation Context ...al convergence of linearization methods under conditions that are exact generalizations of those required in smooth systems. This procedure has been implemented and successfully used in the PATH code =-=[10,15]-=-. A key implementational difficulty remains what to do when the linearization subproblem has no solution. The theory assumes this situation does not happen. In practice, this occurs frequently, partic... |

118 |
A special Newton-type optimization method
- Fischer
- 1992
(Show Context)
Citation Context ... ! 0; x i 2 [l i ; u i ] and F i (x ) = 0: The first component of this paper, described in Section 2, is a reformulation of the mixed complementarity problem, based on the Fischer-Burmeister function =-=[17]-=-. This results in an equivalent nonsmooth system of equations \Phi(x) = 0 where the corresponding merit function \Psi (x) := 1 2 \Phi(x) T \Phi(x) = 1 2 k\Phi(x)k 2 is continuously differentiable. Thi... |

117 | Applied General Equilibrium Modeling with MPSGE as a GAMS Subsystem
- Rutherford
- 1997
(Show Context)
Citation Context ...oints in the MCPLIB collection of test problems [11]. Two tables are presented by splitting the problems into standard MCP models (Table 1) and models that were generated using the MPSGE preprocessor =-=[33,34]-=- in GAMS (Table 2). In order to condense the information in Table 1, we have grouped several similar models together whenever this grouping results in no loss of information; for example, problems col... |

88 | A semismooth equation approach to the solution of nonlinear complementarity problems - Luca, Facchinei, et al. - 1996 |

86 | A new merit function for nonlinear complementarity problems and a related algorithm
- Facchinei, Soares
- 1997
(Show Context)
Citation Context ...ate the properties of this mapping. These properties are extensions of some known ones for the standard nonlinear complementarity problem where l i = 0 and u i = +1 for all i 2 I (see, in particular, =-=[9,14]-=-). Our generalizations will be important in the analysis of subsequent sections. Let us first define the mapping OE : IR 2 ! IR by OE(a; b) := p a 2 + b 2 \Gamma a \Gamma b: This function was introduc... |

80 | MCPLIB: A collection of nonlinear mixed complementarity problems
- Dirkse, Ferris
- 1995
(Show Context)
Citation Context ...d in [2]. In the following two tables, we give the number of successes and failures of our new code, PATH 4.0, and the PATH 3.0 code from all starting points in the MCPLIB collection of test problems =-=[11]-=-. Two tables are presented by splitting the problems into standard MCP models (Table 1) and models that were generated using the MPSGE preprocessor [33,34] in GAMS (Table 2). In order to condense the ... |

80 |
Extension of gams for complementarity problems arising in applied economic analysis
- Rutherford
- 1995
(Show Context)
Citation Context ...oints in the MCPLIB collection of test problems [11]. Two tables are presented by splitting the problems into standard MCP models (Table 1) and models that were generated using the MPSGE preprocessor =-=[33,34]-=- in GAMS (Table 2). In order to condense the information in Table 1, we have grouped several similar models together whenever this grouping results in no loss of information; for example, problems col... |

69 |
Normal maps induced by linear transformations
- Robinson
- 1992
(Show Context)
Citation Context ...erywhere differentiable provided that the equation itself is everywhere differentiable. In complementarity, the two classical merit functions are based on the natural residual [25] and the normal map =-=[32]-=-. Both the natural residual and the normal map provide reformulations of the complementarity problem as a system of equations; unfortunately, the systems and corresponding residual merit functions are... |

62 |
NE/SQP: A robust algorithm for the nonlinear complementarity problem
- Pang, Gabriel
- 1993
(Show Context)
Citation Context ... has these two properties. Note that this method is the basis for the PATH solver by Dirkse and Ferris [10], to which we will come back in our numerical section. The NE/SQP method by Pang and Gabriel =-=[28]-=- is another possible candidate for Algorithm A as is the inexact QPbased solver by Kanzow [21] (these two methods have been used to solve the standard complementarity problem only, but it is not diffi... |

56 |
Newton’s method for generalized equations
- Josephy
- 1979
(Show Context)
Citation Context ...s say nothing about the way in which we generate the sequence fx k g. Several methods satisfy the above two conditions. For example, one may take the Josephy-Newton method as Algorithm A, see Josephy =-=[20,26]-=-. Alternatively, the method suggested by Ralph [30] has these two properties. Note that this method is the basis for the PATH solver by Dirkse and Ferris [10], to which we will come back in our numeri... |

53 | Interfaces to PATH 3.0: Design, implementation and usage
- Ferris, Munson
- 1999
(Show Context)
Citation Context ...al convergence of linearization methods under conditions that are exact generalizations of those required in smooth systems. This procedure has been implemented and successfully used in the PATH code =-=[10,15]-=-. A key implementational difficulty remains what to do when the linearization subproblem has no solution. The theory assumes this situation does not happen. In practice, this occurs frequently, partic... |

49 | A penalized Fischer-Burmeister NCP-function
- Chen, Chen, et al.
(Show Context)
Citation Context ... known properties of the merit function \Psi , see [9,14,18]. We also believe that appropriate modifications of the above theory will enable us to use other merit functions such as those described in =-=[6,29,35]-=-. 4. Algorithmic Framework In this section, we present our class of algorithms for the solution of the mixed complementarity problem and the corresponding global and local convergence theory. In our c... |

48 | Error bound and the convergence analysis of matrix splitting algorithms for the affine variational inequality problem
- Luo, Tseng
- 1992
(Show Context)
Citation Context ... above: namely, it is everywhere differentiable provided that the equation itself is everywhere differentiable. In complementarity, the two classical merit functions are based on the natural residual =-=[25]-=- and the normal map [32]. Both the natural residual and the normal map provide reformulations of the complementarity problem as a system of equations; unfortunately, the systems and corresponding resi... |

45 | Algorithms for Complementarity Problems and Generalized Equations
- Billups
- 1995
(Show Context)
Citation Context ...dices i 2 I where there are finite lower bounds only, finite upper bounds only, finite lower and upper bounds and no finite bounds on the variable x i , respectively. We now follow an idea of Billups =-=[2,3]-=- and define the operator \Phi : IR n ! IR n componentwise as follows: \Phi i (x) := 8 ? ? ! ? ? : OE(x i \Gamma l i ; F i (x)) if i 2 I l ; \GammaOE(u i \Gamma x i ; \GammaF i (x)) if i 2 I u ; OE(x i... |

34 | A semismooth Newton method for variational inequalities: The case of box constraints
- Facchinei, Fischer, et al.
- 1997
(Show Context)
Citation Context ...ws by using an identical argument. 2 We next restate a useful characterization of Robinson's [31] strong regularity condition in the context of mixed complementarity problems. A proof may be found in =-=[13]-=-. We stress that, in the case of a nonlinear complementarity problem (i.e., l i = 0 and u i = 1 for all i 2 I), this characterization reduces to a standard characterization from Robinson [31]. Proposi... |

32 | QPCOMP: A quadratic programming based solver for mixed complementarity problems
- Billups, Ferris
- 1997
(Show Context)
Citation Context ...olver by Kanzow [21] (these two methods have been used to solve the standard complementarity problem only, but it is not difficult to extend both methods to mixed complementarity problems, see, e.g., =-=[2,4]-=-). Our class of algorithms globalizes Algorithm A as follows. We use the merit function \Psi to measure any progress. If the point generated by Algorithm A has a function value of \Psi sufficiently sm... |

31 | A new class of semismooth Newton-type methods for nonlinear complementarity problems
- Kanzow, Kleinmichel
- 1998
(Show Context)
Citation Context ...at all iterates stay in the feasible set [l; u]. In effect, our class of methods is an algorithmic framework for the solution of the box constrained optimization problem min \Psi (x) s.t. x 2 [l; u]: =-=(22)-=- We now give a detailed statement of our class of methods, where the projection of an arbitrary point z 2 IR n on the feasible set [l; u] is denoted by [z] + . Algorithm 1 (General Descent Framework) ... |

28 |
An algorithm based on a sequence of linear complementarity problems applied to a Walrasian equilibrium model: an example
- Mathiesen
- 1987
(Show Context)
Citation Context ...s say nothing about the way in which we generate the sequence fx k g. Several methods satisfy the above two conditions. For example, one may take the Josephy-Newton method as Algorithm A, see Josephy =-=[20,26]-=-. Alternatively, the method suggested by Ralph [30] has these two properties. Note that this method is the basis for the PATH solver by Dirkse and Ferris [10], to which we will come back in our numeri... |

21 |
Regular pseudo-smooth NCP and BVIP functions and globally and quadratically convergent generalized Newton methods for complementarity and variational inequality problems
- Qi
- 1999
(Show Context)
Citation Context ... known properties of the merit function \Psi , see [9,14,18]. We also believe that appropriate modifications of the above theory will enable us to use other merit functions such as those described in =-=[6,29,35]-=-. 4. Algorithmic Framework In this section, we present our class of algorithms for the solution of the mixed complementarity problem and the corresponding global and local convergence theory. In our c... |

20 | A QP-free constrained Newton-type method for variational inequality problems
- Kanzow, Qi
- 1997
(Show Context)
Citation Context ...method inherits the local convergence properties of the locally superlinearly convergent Algorithm A used in Step (S.2) of Algorithm 1. To simplify the proof, we invoke the following proposition from =-=[23]-=- (see also [14] for a similar result). Proposition 8. Let G : IR n ! IR n be locally Lipschitzian, x 2 IR n with G(x ) = 0 be such that all elements in @G(x ) are nonsingular, and assume that there ar... |

19 |
Projected gradient methods for nonlinear complementarity problems via normal maps
- Ferris, Ralph
- 1995
(Show Context)
Citation Context ...Ferris, Christian Kanzow, and Todd S. Munson Other methods have attempted to use projected gradient steps in conjunction with steps that give fast local convergence. See for example, Ferris and Ralph =-=[16]-=-. Unfortunately, these hybrid algorithms are difficult to implement and numerical testing has therefore only been carried out on small test examples. A key difference in the approach outlined here is ... |

18 |
A new constrained optimization reformulation for complementarity problems
- Fischer
- 1998
(Show Context)
Citation Context ...entarity problem. 2 We note that, if we apply the main results of this section to the standard nonlinear complementarity problem, then we obtain some known properties of the merit function \Psi , see =-=[9,14,18]-=-. We also believe that appropriate modifications of the above theory will enable us to use other merit functions such as those described in [6,29,35]. 4. Algorithmic Framework In this section, we pres... |

18 |
Strong stability in variational inequalities
- Liu
- 1995
(Show Context)
Citation Context ...+ 1, and go to Step (S.1). (S.4) (Take Projected Gradient Step) Compute t k = maxfsfi ` j ` = 0; 1; 2; : : : g such that \Psi (x k (t k ))s\Psi (x k ) \Gamma oer\Psi (x k ) T (x k \Gamma x k (t k )); =-=(24)-=- where x k (t) := [x k \Gamma tr\Psi (x k )] + . Set x k+1 := x k (t k ); k / k + 1, and go to Step (S.1). 16 Michael C. Ferris, Christian Kanzow, and Todd S. Munson Other methods have attempted to us... |

16 |
Facchinei and C. Kanzow: A semismooth equation approach to the solution of nonlinear complementarity problems
- Luca, F
- 1996
(Show Context)
Citation Context ...ate the properties of this mapping. These properties are extensions of some known ones for the standard nonlinear complementarity problem where l i = 0 and u i = +1 for all i 2 I (see, in particular, =-=[9,14]-=-). Our generalizations will be important in the analysis of subsequent sections. Let us first define the mapping OE : IR 2 ! IR by OE(a; b) := p a 2 + b 2 \Gamma a \Gamma b: This function was introduc... |

15 | A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Gauss-Newton method
- Sun, Womersley
- 1999
(Show Context)
Citation Context ... known properties of the merit function \Psi , see [9,14,18]. We also believe that appropriate modifications of the above theory will enable us to use other merit functions such as those described in =-=[6,29,35]-=-. 4. Algorithmic Framework In this section, we present our class of algorithms for the solution of the mixed complementarity problem and the corresponding global and local convergence theory. In our c... |

12 | Crash techniques for large-scale complementairty problems
- Dirkse, Ferris
- 1997
(Show Context)
Citation Context ...t the solutions of the subproblem will eventually provide descent for the merit function and that local superlinear or quadratic convergence will occur under appropriate conditions. A crash procedure =-=[12]-=- is used to quickly identify an approximation to the active set at the solution; this is based on a projected Newton step for the normal map, but the direction produced is not known to be a descent di... |

10 |
Harker: A noninterior continuation method for quadratic and linear programming
- Chen, T
- 1993
(Show Context)
Citation Context ... a stationary point result for the unconstrained reformulation min \Psi (x); x 2 IR n ; of the mixed complementarity problem. To this end, we need the following characterization of P 0 -matrices, see =-=[7]-=- as well as [19] for some generalizations (note the difference between this result and the related statement in Proposition 4). Proposition 6. A matrix of the form D a +D b M is nonsingular for all ne... |

7 |
An inexact QP-based method for nonlinear complementarity problems
- Kanzow
- 1998
(Show Context)
Citation Context ...d Ferris [10], to which we will come back in our numerical section. The NE/SQP method by Pang and Gabriel [28] is another possible candidate for Algorithm A as is the inexact QPbased solver by Kanzow =-=[21]-=- (these two methods have been used to solve the standard complementarity problem only, but it is not difficult to extend both methods to mixed complementarity problems, see, e.g., [2,4]). Our class of... |

6 |
Global convergence of Newton’s method for nonsmooth equations, via the path search
- Ralph
- 1994
(Show Context)
Citation Context ...map provide reformulations of the complementarity problem as a system of equations; unfortunately, the systems and corresponding residual merit functions are nonsmooth. Even with this drawback, Ralph =-=[30]-=- showed how to construct an extension of the line search procedure for smooth nonlinear equations that enables fast local convergence of linearization methods under conditions that are exact generaliz... |

5 |
Generalizations of P 0 - and P -properties; extended vertical and horizontal linear complementarity problems
- Sznajder, Gowda
- 1995
(Show Context)
Citation Context ...t for the generalized Jacobian @ \Phi(x ) at a strongly regular solution of the mixed complementarity problem, we also need the following result whose proof can be found in [22, Proposition 2.7]; see =-=[19]-=- for several extensions. Proposition 4. A matrix of the form D a +D b M is nonsingular for all negative semidefinite diagonal matrices D a ; D b 2 IR m\Thetam such that D a +D b is negative definite i... |

4 | A homotopy based algorithm for mixed complementarity problems
- Billups
- 1998
(Show Context)
Citation Context ...dices i 2 I where there are finite lower bounds only, finite upper bounds only, finite lower and upper bounds and no finite bounds on the variable x i , respectively. We now follow an idea of Billups =-=[2,3]-=- and define the operator \Phi : IR n ! IR n componentwise as follows: \Phi i (x) := 8 ? ? ! ? ? : OE(x i \Gamma l i ; F i (x)) if i 2 I l ; \GammaOE(u i \Gamma x i ; \GammaF i (x)) if i 2 I u ; OE(x i... |

1 | Generalizations of P0- andP-properties; extended vertical and horizontal - Gowda, Snajder - 1995 |