## Formulating and Solving Nonlinear Programs as Mixed Complementarity Problems (2000)

Venue: | Optimization. Lecture Notes in Economics and Mathematical Systems |

Citations: | 3 - 0 self |

### BibTeX

@INPROCEEDINGS{Ferris00formulatingand,

author = {Michael C. Ferris and Krung Sinapiromsaran},

title = {Formulating and Solving Nonlinear Programs as Mixed Complementarity Problems},

booktitle = {Optimization. Lecture Notes in Economics and Mathematical Systems},

year = {2000},

pages = {132--148},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

. We consider a primal-dual approach to solve nonlinear programming problems within the AMPL modeling language, via a mixed complementarity formulation. The modeling language supplies the first order and second order derivative information of the Lagrangian function of the nonlinear problem using automatic differentiation. The PATH solver finds the solution of the first order conditions which are generated automatically from this derivative information. In addition, the link incorporates the objective function into a new merit function for the PATH solver to improve the capability of the complementarity algorithm for finding optimal solutions of the nonlinear program. We test the new solver on various test suites from the literature and compare with other available nonlinear programming solvers. Keywords: Complementarity problems, nonlinear programs, automatic differentiation, modeling languages. 1 Introduction While the use of the simplex algorithm for linear programs in the 1940's h...

### Citations

3265 | Convex Analysis
- ROCKAFELLAR
- 1996
(Show Context)
Citation Context ...aints, respectively. The first order necessary conditions for the NLP (1) are 0 2 r x L(x; ; ) +NB (x) 0s? g(x)s0 h(x) = 0; (2) where NB (x) = fz 2 R n j(y \Gamma x) T zs0; 8y 2 Bg is the normal cone =-=[32]-=- to B at x. In the case that r i or s i is finite, the definition of the normal cone allows the first equation of (2), to be rewritten in the following manner. If x i = r i , then (r x L(x; ; )) is0; ... |

465 | Primal-Dual Interior-Point Methods
- Wright
- 1997
(Show Context)
Citation Context ...al time ellipsoid algorithm of Khachian [23]. The idea has been considerably developed; currently it appears that primal-dual methods are the most effective in large scale linear programming settings =-=[34]-=-. In nonlinear programming, a significant improvement has been observed for non-convex problems by using second order information. While QuasiNewton methods can be used for problems whose feasible reg... |

341 |
AMPL: a modeling language for mathematical programming Thomson/Brooks/Cole
- Fourer, Gay, et al.
- 2003
(Show Context)
Citation Context ...gorithm were much slower to develop. In fact, the MINOS code [26] released in 1976 was the first code that could deal reliably with problems of relatively large size. The advent of modeling languages =-=[3,14]-=- allowed these solvers to be used by modelers that were not operation research or numerical analysis specialists. Modeling languages allow optimization problems to be communicated to solvers in an eff... |

328 | SNOPT: An SQP algorithm for Large-Scale Constrained Optimization
- Gill, Murray, et al.
- 2001
(Show Context)
Citation Context ...racted from the AMPL web site. Specifically, we test all models in the Hock/Schittkowski test suite [20] and compare the results of the PATH solver with LANCELOT [4], MINOS [27], NPSOL [17] and SNOPT =-=[16]-=-. Other large scale examples, including problems from portfolio and structural optimization are also tested. We believe these results indicate this is already a promising approach and warrants further... |

310 |
Nonlinear programming: sequential unconstrained minimization techniques
- Fiacco, Mccormick
- 1968
(Show Context)
Citation Context ...line, the extension of the field to nonlinear programs (NLP) has been much more recent. The theory of NLP was extensively developed in the 1950's and 60's, culminating perhaps with the landmark books =-=[12,25]-=-. Leaving aside unconstrained optimization, practical algorithms for constrained nonlinear optimization rivaling the simplex algorithm were much slower to develop. In fact, the MINOS code [26] release... |

255 | A Users Guide
- Brooke, Kendrick, et al.
- 1998
(Show Context)
Citation Context ...gorithm were much slower to develop. In fact, the MINOS code [26] released in 1976 was the first code that could deal reliably with problems of relatively large size. The advent of modeling languages =-=[3,14]-=- allowed these solvers to be used by modelers that were not operation research or numerical analysis specialists. Modeling languages allow optimization problems to be communicated to solvers in an eff... |

251 | Nonlinear programming
- Kuhn, Tucker
- 1951
(Show Context)
Citation Context ...r x L(x; ; )) i = 0: These conditions coupled with the regularity condition on the point x establish the necessary conditions for NLP which are normally called the Karush-Kuhn-Tucker (KKT) conditions =-=[22,24]-=-. Whenever the Hessian matrix of the Lagrangian function is positive definite at (x ;s;s), the first order conditions are also sufficient for x to be a strict local minimizer of NLP. 4 M. C. Ferris an... |

197 |
A new polynomial time algorithm for linear programming
- Karmarkar
- 1984
(Show Context)
Citation Context ...stems. The 1980's and 1990's have generated two significant algorithmic changes to the field. The first major change was the introduction of interior point methods for linear programming by Karmarkar =-=[21]-=- in 1984, as a practical alternative to the theoretically important polynomial time ellipsoid algorithm of Khachian [23]. The idea has been considerably developed; currently it appears that primal-dua... |

193 |
Nonlinear programming
- Mangasarian
- 1994
(Show Context)
Citation Context ...line, the extension of the field to nonlinear programs (NLP) has been much more recent. The theory of NLP was extensively developed in the 1950's and 60's, culminating perhaps with the landmark books =-=[12,25]-=-. Leaving aside unconstrained optimization, practical algorithms for constrained nonlinear optimization rivaling the simplex algorithm were much slower to develop. In fact, the MINOS code [26] release... |

156 |
More Test Examples For Nonlinear Programming Codes
- Schittkowski
- 1987
(Show Context)
Citation Context ...tion 4.2. Section 5 gives numerical results for our approach on a set of nonlinear test problems extracted from the AMPL web site. Specifically, we test all models in the Hock/Schittkowski test suite =-=[20]-=- and compare the results of the PATH solver with LANCELOT [4], MINOS [27], NPSOL [17] and SNOPT [16]. Other large scale examples, including problems from portfolio and structural optimization are also... |

149 | A package for the automatic differentiation of algorithms written in C/C
- Griewank, Juedes, et al.
- 1996
(Show Context)
Citation Context ... and K. Sinapiromsaran converting problems to the format required by a solver without modeler intervention. Furthermore, computational advances such as the use of automatic differentiation techniques =-=[19,18,28]-=- to generate the first order derivatives of the nonlinear functions can be used directly in a solver implementation. Currently, GAMS [3] and AMPL [14] are used in a large variety of applications. Most... |

148 | The path solver: A non-monotone stabilization scheme for mixed complementarity problems
- Dirkse, Ferris
- 1995
(Show Context)
Citation Context ...der conditions of the NLP (1). This simple observation allows us to solve the NLP problem using an MCP solver, which is the subject of Section 4. 3 The PATH solver and merit functions The PATH solver =-=[6] is a nonsmo-=-oth Newton type algorithm [31] which finds a zero of the normal map [30] F+ (x) := F (��(x)) + x \Gamma ��(x); Nonlinear programs as complementarity problems 5 where ��(x) is the closest p... |

114 |
A polynomial algorithm for linear programming
- Khachian
- 1979
(Show Context)
Citation Context ... the introduction of interior point methods for linear programming by Karmarkar [21] in 1984, as a practical alternative to the theoretically important polynomial time ellipsoid algorithm of Khachian =-=[23]-=-. The idea has been considerably developed; currently it appears that primal-dual methods are the most effective in large scale linear programming settings [34]. In nonlinear programming, a significan... |

98 |
A Special Newton-type Optimization Method
- Fischer
- 1992
(Show Context)
Citation Context ...erit function for the PATH solver The most recent version of the PATH solver [9] does not use the residual of the normal map for a merit function. Instead, it utilizes the Fischer-Burmeister function =-=[13]-=- defined as the mapping OE : R 2 ! R, OE(p; q) := p p 2 + q 2 \Gamma p \Gamma q; where p and q are scalar variables. This function exhibits the complementarity property when the function value is zero... |

96 |
LANCELOT: A Fortran Package for LargeScale Nonlinear Optimization (Release A
- Toint
- 1991
(Show Context)
Citation Context ...n a set of nonlinear test problems extracted from the AMPL web site. Specifically, we test all models in the Hock/Schittkowski test suite [20] and compare the results of the PATH solver with LANCELOT =-=[4]-=-, MINOS [27], NPSOL [17] and SNOPT [16]. Other large scale examples, including problems from portfolio and structural optimization are also tested. We believe these results indicate this is already a ... |

96 |
Minima of Functions of Several Variables with Inequalities as Side Constraints
- Karush
- 1939
(Show Context)
Citation Context ...r x L(x; ; )) i = 0: These conditions coupled with the regularity condition on the point x establish the necessary conditions for NLP which are normally called the Karush-Kuhn-Tucker (KKT) conditions =-=[22,24]-=-. Whenever the Hessian matrix of the Lagrangian function is positive definite at (x ;s;s), the first order conditions are also sufficient for x to be a strict local minimizer of NLP. 4 M. C. Ferris an... |

93 |
MINOS 5.0 user's guide
- Murtagh, Saunders
- 1983
(Show Context)
Citation Context ...nonlinear test problems extracted from the AMPL web site. Specifically, we test all models in the Hock/Schittkowski test suite [20] and compare the results of the PATH solver with LANCELOT [4], MINOS =-=[27]-=-, NPSOL [17] and SNOPT [16]. Other large scale examples, including problems from portfolio and structural optimization are also tested. We believe these results indicate this is already a promising ap... |

82 |
User’s guide for NPSOL (version 4.0): A Fortran package for nonlinear programming
- Gill, Murray, et al.
- 1986
(Show Context)
Citation Context ...st problems extracted from the AMPL web site. Specifically, we test all models in the Hock/Schittkowski test suite [20] and compare the results of the PATH solver with LANCELOT [4], MINOS [27], NPSOL =-=[17]-=- and SNOPT [16]. Other large scale examples, including problems from portfolio and structural optimization are also tested. We believe these results indicate this is already a promising approach and w... |

79 | A semismooth equation approach to the solution of nonlinear complementarity problems
- Luca, Facchinei, et al.
- 1996
(Show Context)
Citation Context ...se of the second order information is the critical aspect. Moreover, by adapting the link code we have described in this paper, we can solve the NLP problem using other MCP solvers such as semismooth =-=[5]-=-, or an interior point approach [33]. This will be subject of further research. Currently the PATH solver uses a proximal point perturbation [2] to overcome singularity problems in the Jacobian matrix... |

74 | Large-Scale Linearly Constrained Optimization
- Murtagh, Saunders
- 1978
(Show Context)
Citation Context ...ooks [12,25]. Leaving aside unconstrained optimization, practical algorithms for constrained nonlinear optimization rivaling the simplex algorithm were much slower to develop. In fact, the MINOS code =-=[26]-=- released in 1976 was the first code that could deal reliably with problems of relatively large size. The advent of modeling languages [3,14] allowed these solvers to be used by modelers that were not... |

69 |
Corliss (eds.) Automatic Differentiation of Algorithms: Theory, Implementation, and Application
- Griewank, F
- 1991
(Show Context)
Citation Context ... and K. Sinapiromsaran converting problems to the format required by a solver without modeler intervention. Furthermore, computational advances such as the use of automatic differentiation techniques =-=[19,18,28]-=- to generate the first order derivatives of the nonlinear functions can be used directly in a solver implementation. Currently, GAMS [3] and AMPL [14] are used in a large variety of applications. Most... |

62 |
Normal maps induced by linear transformation
- ROBINSON
- 1992
(Show Context)
Citation Context ...roblem using an MCP solver, which is the subject of Section 4. 3 The PATH solver and merit functions The PATH solver [6] is a nonsmooth Newton type algorithm [31] which finds a zero of the normal map =-=[30] F+ (x) := F-=- (��(x)) + x \Gamma ��(x); Nonlinear programs as complementarity problems 5 where ��(x) is the closest point in B to the variable x in the Euclidean norm. It is well known [30] that findin... |

56 |
Global convergence of damped Newton’s method for nonsmooth equations, via the path search
- Ralph
- 1994
(Show Context)
Citation Context ...map F+ (x) about the current iterate to obtain a piecewise linear map whose zero is sought using a homotopy approach [7]. To monitor progress in the nonlinear model, a nonmonotone path-search is used =-=[29]-=-. Recent extensions [9] have introduced a function \Psi to be used in conjunction with the code, both as a residual and a merit function. The following pseudo code shows the main algorithm steps of th... |

48 | Interfaces to path 3.0: Design, implementation and usage
- Ferris, Munson
- 1999
(Show Context)
Citation Context ...1000 minoriterationlimit=10000"; To decrease the convergence tolerance from 1 \Theta 10 \Gamma6 to 1 \Theta 10 \Gamma8 , a user can specify options pathnlpoptions "convergencetolerance=1E-8&=-=quot;; Consult [10,11]-=- for details on these and other options. 12 M. C. Ferris and K. Sinapiromsaran 5.1 The Hock/Schittkowski test suite We tested pathnlp with and without the new merit function using the Hock/Schittkowsk... |

41 | Algorithms for Complementarity Problems and Generalized Equations
- Billups
- 1995
(Show Context)
Citation Context ..., that is OE(p; q) = 0 if and only if ps0; qs0 and pq = 0: For the MCP problem, the residual and merit function used is \Psi : R n ! R, \Psi (x) := 1 2 /(x) T /(x); where /(x) is the Fischer operator =-=[1]-=- defined in (4) from R n to R n that maps x i and F i (x) as parameters to the Fischer-Burmeister function componentwise as follows: / i (x) := 8 ? ? ! ? ? : OE(x i \Gamma l i ; F i (x)) if \Gamma 1 !... |

40 |
Newton’s method for a class of nonsmooth functions
- Robinson
- 1994
(Show Context)
Citation Context ...e observation allows us to solve the NLP problem using an MCP solver, which is the subject of Section 4. 3 The PATH solver and merit functions The PATH solver [6] is a nonsmooth Newton type algorithm =-=[31] which finds-=- a zero of the normal map [30] F+ (x) := F (��(x)) + x \Gamma ��(x); Nonlinear programs as complementarity problems 5 where ��(x) is the closest point in B to the variable x in the Euclide... |

32 | QPCOMP: a quadratic program based solver for mived compiementarity problems
- FERRIS
(Show Context)
Citation Context ... NLP problem using other MCP solvers such as semismooth [5], or an interior point approach [33]. This will be subject of further research. Currently the PATH solver uses a proximal point perturbation =-=[2]-=- to overcome singularity problems in the Jacobian matrix. This has the tendency to remove any negative curvature and may hinder progress on non-convex problems. The improvement in performance by using... |

29 | Feasible descent algorithms for mixed complementarity problems
- Ferris, Kanzow, et al.
- 1996
(Show Context)
Citation Context ...rent iterate to obtain a piecewise linear map whose zero is sought using a homotopy approach [7]. To monitor progress in the nonlinear model, a nonmonotone path-search is used [29]. Recent extensions =-=[9]-=- have introduced a function \Psi to be used in conjunction with the code, both as a residual and a merit function. The following pseudo code shows the main algorithm steps of the PATH solver to find a... |

28 | Hooking your solver to AMPL
- Gay
- 1997
(Show Context)
Citation Context ...r that methods that exploit second order information (either using negative curvature within a trust region or line search framework) are more efficient and robust. Unfortunately, it is only recently =-=[15]-=- that second order information has become available from a modeling language, namely AMPL. This paper is an attempt to combine some of the features of these last two improvements. The idea is to use a... |

15 |
Expressing complementarity problems and communicating them to solvers
- Fourer, MC, et al.
- 1999
(Show Context)
Citation Context ... are specified in more detail. 4 The PATHNLP solver for AMPL nonlinear programs To solve the NLP problem in AMPL, a user could specify the complementarity formulation directly using the AMPL language =-=[8]-=-. This would require a modeler to write down explicitly the first order conditions as detailed in Section 2.2. This process is very cumbersome and prone to error. In this paper, we propose to use the ... |

15 |
Complementarity problems
- Ferris, Munson
- 2000
(Show Context)
Citation Context ...t. Otherwise use the merit function to find a descent direction and search along this direction for a new point. Details on the solution of linearization and the path-search mechanism can be found in =-=[6,10]-=-. In this paper, we just indicate the changes specific to solving NLP's. The Newton-type PATH solver uses the Jacobian matrix of the MCP function (3) to find its path-searching direction. In the above... |

14 |
A short course in solving equations with PL homotopies
- Eaves
- 1976
(Show Context)
Citation Context ...verview of the algorithm The essential idea of the code is to linearize the normal map F+ (x) about the current iterate to obtain a piecewise linear map whose zero is sought using a homotopy approach =-=[7]-=-. To monitor progress in the nonlinear model, a nonmonotone path-search is used [29]. Recent extensions [9] have introduced a function \Psi to be used in conjunction with the code, both as a residual ... |

10 |
Interior point methods for linear and nonlinear programming
- Shanno, Simantiraki
- 1997
(Show Context)
Citation Context ...is the critical aspect. Moreover, by adapting the link code we have described in this paper, we can solve the NLP problem using other MCP solvers such as semismooth [5], or an interior point approach =-=[33]-=-. This will be subject of further research. Currently the PATH solver uses a proximal point perturbation [2] to overcome singularity problems in the Jacobian matrix. This has the tendency to remove an... |