Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-25T10:01:50.464Z Has data issue: false hasContentIssue false
  • English
  • Français

Interior methods for constrained optimization

Published online by Cambridge University Press:  07 November 2008

Margaret H. Wright
Margaret H. Wright
Affiliation:
AT&T Bell Laboratories Murray Hill, New Jersey 07974USA, E-mail: mhw@research.att.com
Get access Rights & Permissions [Opens in a new window]

Extract

Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, including their apparent inefficiency compared with the best available alternatives. In 1984, Karmarkar's announcement of a fast polynomial-time interior method for linear programming caused tremendous excitement in the field of optimization. A formal connection can be shown between his method and classical barrier methods, which have consequently undergone a renaissance in interest and popularity. Most papers published since 1984 have concentrated on issues of computational complexity in interior methods for linear programming. During the same period, implementations of interior methods have displayed great efficiency in solving many large linear programs of ever-increasing size. Interior methods have also been applied with notable success to nonlinear and combinatorial problems. This paper presents a self-contained survey of major themes in both classical material and recent developments related to the theory and practice of interior methods.

Type
Research Article
Information
Acta Numerica , Volume 1 , January 1992 , pp. 341 - 407
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Arioli, M., Duff, I.S. and de Rijk, P.P.M. (1989), ‘;On the augmented system approach to sparse least-squares problems’, Numer. Math. 55, 667–684.CrossRefGoogle Scholar
Avriel, M. (1976), Nonlinear Programming: Analysis and Methods, Prentice-Hall (Englewood Cliffs, NJ).Google Scholar
Bayer, D.A. and Lagarias, J.C. (1989), ‘The nonlinear geometry of linear programming, I: Affine and projective scaling trajectories’ Trans. Am. Math. Soc. 314, 499–526.Google Scholar
Bayer, D.A. and Lagarias, J.C. (1991), ‘Karmarkar's linear programming algorithm and Newton's method’ Math. Program. 50, 291–330.CrossRefGoogle Scholar
Bixby, R. (1990), ‘Implementing the simplex method: the initial basis’ Report 90-32, Department of Mathematical Sciences, Rice University, Houston, Texas.Google Scholar
Björck, A. (1991), ‘Least squares methods’ Institute of Technology, Linköping University, Linköping, Sweden.Google Scholar
Bunch, J.R. and Kaufman, L.C. (1977), ‘Some stable methods for calculating inertia and solving symmetric linear systems’ Math. Comput. 31, 162–179.CrossRefGoogle Scholar
Choi, I.C., Monma, C.L. and Shanno, D.F. (1990), ‘Further development of a primal-dual interior point method’, ORSA J. Comput. 2, 304–311.CrossRefGoogle Scholar
Chvátal, V. (1983), Linear Programming, W. H. Freeman (New York).Google Scholar
den Hertog, D., Roos, C. and Terlaky, T. (1990), ‘On the classical logarithmic barrier function method for a class of smooth convex programming problems’ Report 90-01, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, Holland.Google Scholar
Duff, I.S., Erisman, A.M. and Reid, J.K. (1986), Direct Methods for Sparse Matrices, Oxford University Press (London).Google Scholar
Fiacco, A.V. (1979), ‘Barrier methods for nonlinear programming’ in Operations Research Support Methodology (Holzman, A.G., ed.), Marcel Dekker (New York), 377–440.Google Scholar
Fiacco, A.V. and McCormick, G.P. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons (New York). Republished by SIAM (Philadelphia), 1990.Google Scholar
Fletcher, R. (1987), Practical Methods of Optimization (second edition), John Wiley and Sons (Chichester).Google Scholar
Forrest, J.J.H. and Tomlin, J.A. (1990), ‘Vector processing in simplex and interior methods for linear programming’ Annals of Operations Research 22, 71–100.CrossRefGoogle Scholar
Forrest, J.J.H. and Tomlin, J.A. (1991), ‘Implementing the simplex method for the Optimization Subroutine Library’ Research Report RJ 8174, IBM Almaden Research Centre, San Jose, California.Google Scholar
Fourer, R. and Mehrotra, S. (1990), ‘Performance of an augmented system approach for solving least-squares problems in an interior-point method for linear programming’ Report, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois.Google Scholar
Frisch, K.R. (1955), ‘The logarithmic potential method of convex programming’ Report, University Institute of Economics, Oslo, Norway.Google Scholar
George, J.A. and Liu, J.W.H. (1981), Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall (Englewood Cliffs, NJ).Google Scholar
Gill, P.E., Murray, W., Ponceleón, D.B., and Saunders, M.A. (1990), ‘Preconditioners for indefinite systems arising in optimization’ Report SOL 90-8, Department of Operations Research, Stanford University, Stanford, California.Google Scholar
Gill, P.E., Murray, W., Ponceleón, D.B., and Saunders, M.A. (1991), ‘Solving reduced KKT systems in barrier methods for linear and quadratic programming’ Report SOL 91-7, Department of Operations Research, Stanford University, Stanford, California.CrossRefGoogle Scholar
Gill, P.E., Murray, W., Saunders, M.A., Tomlin, J.A. and Wright, M.H. (1986), ‘On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method’ Math. Program. 36, 183–209.CrossRefGoogle Scholar
Gill, P.E., Murray, W. and Wright, M.H. (1981), Practical Optimization, Academic Press (London and New York).Google Scholar
Goldfarb, D. and Todd, M.J. (1989), ‘Linear programming’ in Optimization (Nemhauser, G.L., Rinnooy Kan, A.H.G. and Todd, M.J., eds), North Holland (Amsterdam and New York), 73–170.CrossRefGoogle Scholar
Gonzaga, C.C. (1992), ‘Path following methods for linear programming’ SIAM Review 34, to appear.CrossRefGoogle Scholar
Grünbaum, B. (1967), Convex Polytopes, John Wiley and Sons (London).Google Scholar
Higham, N.J. (1990), ‘Analysis of the Cholesky decomposition of a semi-definite matrix’ in Reliable Numerical Computation (Cox, M.G. and Hammarling, S., eds), Clarendon Press (Oxford), 161–185.CrossRefGoogle Scholar
Huard, P. (1967), ‘Resolution of mathematical programming with nonlinear constraints by the method of centres’ in Nonlinear Programming (Abadie, J., ed.), North Holland (Amsterdam and New York), 207–219.Google Scholar
Jarre, F. (1991), ‘Interior-point methods for convex programming’ Report SOL 90-16, Department of Operations Research, Stanford University, Stanford, California.Google Scholar
Jittorntrum, K. (1978), Sequential Algorithms in Nonlinear Programming, Ph.D. thesis, Australian National University.CrossRefGoogle Scholar
Karmarkar, N.K. (1984), ‘A new polynomial time algorithm for linear programming’ Combinatorica 4, 373–395.CrossRefGoogle Scholar
Karmarkar, N.K. (1990), ‘Riemannian geometry underlying interior-point methods for linear programming’ in Mathematical Developments Arising from Linear Programming (Lagarias, J.C. and Todd, M.J., eds), American Mathematical Society (Providence, RI), 51–75.CrossRefGoogle Scholar
Khachian, L.G. (1979), ‘A polynomial algorithm in linear programming’ Doklady Akademiia Nauk SSSR 244, 1093–1096 (in Russian); English translation in Sov. Math. Dokl. 20, 191–194.Google Scholar
Klee, V. and Minty, G.J. (1972), ‘How good is the simplex algorithm?’ in Inequalities III (Shisha, O., ed.), Academic Press (New York), 159–175.Google Scholar
Kranich, E. (1991), ‘Interior point methods for mathematical programming: a bibliography’ Report 171, Universität Hagen, Hagen, Germany. This bibliography can be accessed electronically by sending email to ‘’; with message ‘send intbib.tex from bib’ and/or ‘send intbib.bbl from bib’.Google Scholar
Luenberger, D.G. (1984), Introduction to Linear and Nonlinear Programming, Addison-Wesley (Menlo Park, CA).Google Scholar
Lustig, I., Marsten, R.E. and Shanno, D.F. (1990), ‘On implementing Mehrotra's predictor-corrector interior point method for linear programming’ Report SOR 90-03, Department of Civil Engineering and Operations Research, Princeton University, Princeton, New Jersey.Google Scholar
Marxen, A. (1989), ‘Primal barrier methods for linear programming’ Ph.D. thesis, Stanford University, Stanford, California.CrossRefGoogle Scholar
McCormick, G.P. (1991), ‘The superlinear convergence of a nonlinear primal-dual algorithm’ Report T-550/91, Department of Operations Research, George Washington University, Washington, DC.Google Scholar
Megiddo, N. (1987), ‘Pathways to the optimal set in linear programming’ in Progress in Mathematical Programming (Megiddo, N., ed.), Springer-Verlag (New York), 131–158.Google Scholar
Mehrotra, S. (1990), ‘On the implementation of a (primal-dual) interior method’ Report 90-03, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois.Google Scholar
Mehrotra, S. and Sun, J. (1990), ‘An interior point algorithm for solving smooth convex programs based on Newton's method’ in Mathematical Developments Arising from Linear Programming (Lagarias, J.C. and Todd, M.J., eds), American Mathematical Society (Providence, RI), 265–284.CrossRefGoogle Scholar
Monteiro, R.D.C. and Adler, I. (1989a), ‘Interior path following primal-dual algorithms, Part I: Linear programming’ Math. Program. 44, 27–41.CrossRefGoogle Scholar
Monteiro, R.D.C. and Adler, I. (1989b), ‘Interior path following primal-dual algorithms, Part II: Convex quadratic programming’ Math. Program. 44, 43–66.CrossRefGoogle Scholar
Moré, J.J. and Wright, S.J. (1990), private communication.Google Scholar
Murray, W. (1971), Analytical expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions, J. Optim. Theory Appl. 7, 189–196.CrossRefGoogle Scholar
Murray, W. and Wright, M.H. (1991), ‘Line search procedures for the logarithmic barrier function’ Manuscript, AT&T Bell Laboratories, Murray Hill, New Jersey.Google Scholar
Nesterov, Y. and Nemirovsky, A. (1989), Self-Concordant Functions and Polynomial-Time Methods in Convex Programming, USSR Academy of Science (Moscow).Google Scholar
Ortega, J.M. and Rheinboldt, W.C. (1970), Iterative Solution of Nonlinear Equations in Several Variables, Academic Press (London and New York).Google Scholar
Papadimitriou, C.R. and Steiglitz, K. (1982), Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall (Englewood Cliffs, NJ).Google Scholar
Ponceleón, D.B. (1990), Barrier Methods for Large-scale Quadratic Programming, Ph.D. thesis, Stanford University, Stanford, California.CrossRefGoogle Scholar
Powell, M.J.D. (1972), ‘Problems related to unconstrained optimization’ in Numerical Methods for Unconstrained Optimization (Murray, W., ed.), Academic Press (London and New York), 29–55.Google Scholar
Powell, M.J.D. (1990), ‘Karmarkar's algorithm: a view from nonlinear programming’ IMA Bulletin 26, 165–181.Google Scholar
Renegar, J. (1988), ‘A polynomial-time algorithm based on Newton's method for linear programming’ Math. Program. 40, 59–94.CrossRefGoogle Scholar
Rockafellar, R. T. (1970), Convex Analysis, Princeton University Press (Princeton, NJ).CrossRefGoogle Scholar
Roos, C. and Vial, J.-Ph. (1988), ‘A polynomial method of approximate centres for linear programming’ Report 88-68, Faculty of Mathematics and Informatics, Delft University of Technology, Delft, Holland. To appear in Math. Program.Google Scholar
Sonnevend, G. (1986), ‘An analytic centre for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming’ in Lecture Notes in Control and Information Science, Vol. 84, Springer-Verlag (New York), 866–876.Google Scholar
Vanderbei, R.J. (1991), ‘Symmetric quasi-definite matrices’ Report 91-10, Department of Civil Engineering and Operations Research, Princeton University, Princeton, New Jersey.Google Scholar
Wright, M.H. (1976), Numerical Methods for Nonlinearly Constrained Optimization, Ph.D. thesis, Stanford University, California.CrossRefGoogle Scholar
Wright, M.H. (1991), ‘Determining subspace information from the Hessian of a barrier function’ Manuscript, AT&T Bell Laboratories, Murray Hill, New Jersey.Google Scholar