zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. (English) Zbl 1134.90542
Summary: We present a primal-dual interior-point algorithm with a filter line-search method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix. Heuristics are also considered that allow faster performance. This method has been implemented in the IPOPT code, which we demonstrate in a detailed numerical study based on 954 problems from the CUTEr test set. An evaluation is made of several line-search options, and a comparison is provided with two state-of-the-art interior-point codes for nonlinear programming.

90C51Interior-point methods
90C06Large-scale problems (mathematical programming)
90C30Nonlinear programming
[1]Benson, H. Y., Shanno, D. F., Vanderbei, R. J.: Interior-point methods for nonconvex nonlinear programming: Filter methods and merit functions. Computational Optimization and Applications, 23 (2), 257–272 (2002) · Zbl 1022.90017 · doi:10.1023/A:1020533003783
[2]Byrd, R. H., Gilbert, J. Ch., Nocedal, J.: A trust region method based on interior point techniques for nonlinear programming. Mathematical Programming, 89, 149–185 (2000) · doi:10.1007/PL00011391
[3]Byrd, R. H., Hribar, M. E., Nocedal, J.: An interior point algorithm for large-scale nonlinear programming. SIAM Journal on Optimization, 9 (4), 877–900 (1999) · Zbl 0957.65057 · doi:10.1137/S1052623497325107
[4]Byrd, R. H., Liu, G., Nocedal, J.: On the local behavior of an interior point method for nonlinear programming. In: Griffiths, D. F., Higham, D. J. (eds), Numerical Analysis 1997, pages 37–56. Addison–Wesley Longman, Reading, MA, USA, 1997
[5]Chamberlain, R. M., Lemarechal, C., Pedersen, H. C., Powell, M. J. D.: The watchdog technique for forcing convergence in algorithms for constrained optimization. Mathematical Programming Study, 16, 1–17 (1982) · Zbl 0477.90072 · doi:10.1007/BFb0120945
[6]Conn, A. R., Gould, N. I. M., Toint, Ph. L.: LANCELOT: a Fortran package for large-scale nonlinear optimization (Release A). Number 17 in Springer Series in Computational Mathematics. Springer Verlag, Heidelberg, Berlin, New York, 1992
[7]Conn, A. R., Gould, N. I. M., Toint, Ph. L.: Trust-Region Methods. SIAM, Philadelphia, PA, USA, 2000
[8]Conn, A. R., Gould, N.I.M., Orban, D., Toint, Ph. L.: A primal-dual trust-region algorithm for non-convex nonlinear programming. Mathematical Programming, 87 (2), 215–249 (2000) · doi:10.1007/s101070050112
[9]Dolan, E. D., Moré, J. J.: Benchmarking optimization software with performance profiles. Mathematical Programming, 91 (2), 201–213 (2002) · Zbl 1049.90004 · doi:10.1007/s101070100263
[10]El-Bakry, A. S., Tapia, R. A., Tsuchiya, T., Zhang, Y.: On the formulation and theory of the Newton interior-point method for nonlinear programming. Journal of Optimization Theory and Application, 89 (3), 507–541 (1996) · Zbl 0851.90115 · doi:10.1007/BF02275347
[11]Fiacco, A. V., McCormick, G. P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley, New York, USA, 1968 Reprinted by SIAM Publications, 1990.
[12]Fletcher, R.: Practical Methods of Optimization. John Wiley and Sons, New York, USA, second edition, 1987
[13]Fletcher, R., Gould, N. I. M., Leyffer, S., Toint, Ph. L., Wächter, A.: Global convergence of a trust-region SQP-filter algorithms for general nonlinear programming. SIAM Journal on Optimization, 13 (3), 635–659 (2002) · Zbl 1038.90076 · doi:10.1137/S1052623499357258
[14]Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Mathematical Programming, 91 (2), 239–269 (2002) · Zbl 1049.90088 · doi:10.1007/s101070100244
[15]Fletcher, R., Leyffer, S., Toint, Ph. L.: On the global convergence of a filter-SQP algorithm. SIAM Journal on Optimization, 13 (1), 44–59 (2002) · Zbl 1029.65063 · doi:10.1137/S105262340038081X
[16]Forsgren, A., Gill, P. E., Wright, M. H.: Interior methods for nonlinear optimization. SIAM Review, 44 (4), 525–597 (2002) · Zbl 1028.90060 · doi:10.1137/S0036144502414942
[17]Gould, N. I. M., Orban, D., Sartenaer, A., Toint, Ph. L.: Superlinear convergence of primal-dual interior point algorithms for nonlinear programming. SIAM Journal on Optimization, 11 (4), 974–1002 (2001) · Zbl 1003.65066 · doi:10.1137/S1052623400370515
[18]Gould, N. I. M., Orban, D., Toint, Ph. L.: CUTEr (and SifDec), a constrained and unconstrained testing environment, revisited. Technical Report TR/PA/01/04, CERFACS, Toulouse, France, 2001
[19]Harwell Subroutine Library, AEA Technology, Harwell, Oxfordshire, England. A catalogue of subroutines (HSL 2000), 2002
[20]Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York, NY, USA, 1999
[21]Tits, A. L., Wächter, A., Bakhtiari, S., Urban, T. J., Lawrence, C. T.: A primal-dual interior-point method for nonlinear programming with strong global and local convergence properties. SIAM Journal on Optimization, 14 (1), 173–199 (2003) · Zbl 1075.90078 · doi:10.1137/S1052623401392123
[22]Ulbrich, M., Ulbrich, S., Vicente, L. N.: A globally convergent primal-dual interior-point filter method for nonlinear programming. Mathematical Programming, 100 (2), 379–410 (2004) · Zbl 1070.90110 · doi:10.1007/s10107-003-0477-4
[23]Vanderbei, R. J., Shanno, D. F.: An interior-point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications, 13, 231–252 (1999) · Zbl 1040.90564 · doi:10.1023/A:1008677427361
[24]Wächter, A.: An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering. PhD thesis, Carnegie Mellon University, Pittsburgh, PA, USA, January 2002
[25]Wächter, A., Biegler, L. T.: Failure of global convergence for a class of interior point methods for nonlinear programming. Mathematical Programming, 88 (2), 565–574 (2000) · doi:10.1007/PL00011386
[26]Wächter, A., Biegler, L. T.: Line search filter methods for nonlinear programming: Motivation and global convergence. Technical Report RC 23036, IBM T.J. Watson Research Center, Yorktown Heights, USA, 2001; revised 2004. To appear in SIAM Journal on Optimization.
[27]Waltz, R. A., Morales, J. L., Nocedal, J., Orban, D.: An interior algorithm for nonlinear optimization that combines line search and trust region steps. Technical Report OTC 6/2003, Optimization Technology Center, Northwestern University, Evanston, IL, USA. To appear in Mathematical Programming A
[28]Waltz, R. A., Nocedal, J.: KNITRO user’s manual. Technical Report OTC 2003/05, Optimization Technology Center, Northwestern University, Evanston, IL, USA, April 2003
[29]Yamashita, H.: A globally convergent primal-dual interior-point method for constrained optimization. Optimization Methods and Software, 10, 443–469 (1998) · Zbl 0946.90110 · doi:10.1080/10556789808805723
[30]Yamashita, H., Yabe, H., Tanabe, T.: A globally and superlinearly convergent primal-dual interior point trust region method for large scale constrained optimization. Technical report, Mathematical System Institute, Inc., Tokyo, Japan, July 1997. Revised July 1998