The interior-point revolution in optimization: History, recent developments, and lasting consequences
Author:
Margaret H. Wright
Journal:
Bull. Amer. Math. Soc. 42 (2005), 39-56
MSC (2000):
Primary 49M37, 65K05, 90C30
DOI:
https://doi.org/10.1090/S0273-0979-04-01040-7
Published electronically:
September 21, 2004
MathSciNet review:
2115066
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Interior methods are a pervasive feature of the optimization landscape today, but it was not always so. Although interior-point techniques, primarily in the form of barrier methods, were widely used during the 1960s for problems with nonlinear constraints, their use for the fundamental problem of linear programming was unthinkable because of the total dominance of the simplex method. During the 1970s, barrier methods were superseded, nearly to the point of oblivion, by newly emerging and seemingly more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost universally regarded as a closed chapter in the history of optimization. This picture changed dramatically in 1984, when Narendra Karmarkar announced a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have continued to transform both the theory and practice of constrained optimization. We present a condensed, unavoidably incomplete look at classical material and recent research about interior methods.
- Farid Alizadeh, Jean-Pierre A. Haeberly, and Michael L. Overton, Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results, SIAM J. Optim. 8 (1998), no. 3, 746–768. MR 1636549, DOI https://doi.org/10.1137/S1052623496304700
- Robert E. Bixby, Solving real-world linear programs: a decade and more of progress, Oper. Res. 50 (2002), no. 1, 3–15. 50th anniversary issue of Operations Research. MR 1885204, DOI https://doi.org/10.1287/opre.50.1.3.17780
- Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan, Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, vol. 15, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. MR 1284712
- Richard H. Byrd, Jean Charles Gilbert, and Jorge Nocedal, A trust region method based on interior point techniques for nonlinear programming, Math. Program. 89 (2000), no. 1, Ser. A, 149–185. MR 1795061, DOI https://doi.org/10.1007/PL00011391
- Vašek Chvátal, Linear programming, A Series of Books in the Mathematical Sciences, W. H. Freeman and Company, New York, 1983. MR 717219
- A. R. Conn, Nick Gould, and Ph. L. Toint, A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization, Numer. Math. 68 (1994), no. 1, 17–33. MR 1278446, DOI https://doi.org/10.1007/s002110050046
- Andrew R. Conn, Nicholas I. M. Gould, Dominique Orban, and Philippe L. Toint, A primal-dual trust-region algorithm for non-convex nonlinear programming, Math. Program. 87 (2000), no. 2, Ser. B, 215–249. Studies in algorithmic optimization. MR 1763849, DOI https://doi.org/10.1007/s101070050112
- A. S. El-Bakry, R. A. Tapia, T. Tsuchiya, and Y. Zhang, On the formulation and theory of the Newton interior-point method for nonlinear programming, J. Optim. Theory Appl. 89 (1996), no. 3, 507–541. MR 1393361, DOI https://doi.org/10.1007/BF02275347
- Anthony V. Fiacco and Garth P. McCormick, Nonlinear programming, 2nd ed., Classics in Applied Mathematics, vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. Sequential unconstrained minimization techniques. MR 1058438
- R. Fletcher, Practical methods of optimization, 2nd ed., A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. MR 955799
- Anders Forsgren and Philip E. Gill, Primal-dual interior methods for nonconvex nonlinear programming, SIAM J. Optim. 8 (1998), no. 4, 1132–1152. MR 1646122, DOI https://doi.org/10.1137/S1052623496305560
- Anders Forsgren, Philip E. Gill, and Margaret H. Wright, Interior methods for nonlinear optimization, SIAM Rev. 44 (2002), no. 4, 525–597 (2003). MR 1980444, DOI https://doi.org/10.1137/S0036144502414942
- Anders Forsgren, Philip E. Gill, and Joseph R. Shinnerl, Stability of symmetric ill-conditioned systems arising in interior methods for constrained optimization, SIAM J. Matrix Anal. Appl. 17 (1996), no. 1, 187–211. MR 1372930, DOI https://doi.org/10.1137/S0895479894270658
- David M. Gay, Michael L. Overton, and Margaret H. Wright, A primal-dual interior method for nonconvex nonlinear programming, Advances in nonlinear programming (Beijing, 1996) Appl. Optim., vol. 14, Kluwer Acad. Publ., Dordrecht, 1998, pp. 31–56. MR 1639869, DOI https://doi.org/10.1007/978-1-4613-3335-7_2
- Philip E. Gill, Walter Murray, Michael A. Saunders, J. A. Tomlin, and Margaret H. Wright, On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method, Math. Programming 36 (1986), no. 2, 183–209. MR 866988, DOI https://doi.org/10.1007/BF02592025
- Philip E. Gill, Walter Murray, and Margaret H. Wright, Practical optimization, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1981. MR 634376
- Philip E. Gill, Walter Murray, and Margaret H. Wright, Numerical linear algebra and optimization. Vol. 1, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1991. MR 1074004
- Michel X. Goemans and David P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. Assoc. Comput. Mach. 42 (1995), no. 6, 1115–1145. MR 1412228, DOI https://doi.org/10.1145/227683.227684
- Donald Goldfarb and Michael J. Todd, Linear programming, Optimization, Handbooks Oper. Res. Management Sci., vol. 1, North-Holland, Amsterdam, 1989, pp. 73–170. MR 1105101, DOI https://doi.org/10.1016/S0927-0507%2889%2901003-0
- Clovis C. Gonzaga, Path-following methods for linear programming, SIAM Rev. 34 (1992), no. 2, 167–224. MR 1166175, DOI https://doi.org/10.1137/1034048
- N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984), no. 4, 373–395. MR 779900, DOI https://doi.org/10.1007/BF02579150
- Adrian S. Lewis and Michael L. Overton, Eigenvalue optimization, Acta numerica, 1996, Acta Numer., vol. 5, Cambridge Univ. Press, Cambridge, 1996, pp. 149–190. MR 1624599, DOI https://doi.org/10.1017/S0962492900002646
- F. A. Lootsma, Hessian matrices of penalty functions for solving constrained-optimization problems, Philips Res. Rep. 24 (1969), 322–330. MR 305594 Luenberger D. G. Luenberger (1973). Introduction to Linear and Nonlinear Programming, Addison-Wesley, Menlo Park.
- W. Murray, Analytical expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions, J. Optim. Theory Appl. 7 (1971), 189–196. MR 284212, DOI https://doi.org/10.1007/BF00932477 MINOS B. A. Murtagh and M. A. Saunders (1987). MINOS 5.1 User’s Guide, Report SOL 83-20R, Department of Operations Research, Stanford University, Stanford, California.
- Yurii Nesterov and Arkadii Nemirovskii, Interior-point polynomial algorithms in convex programming, SIAM Studies in Applied Mathematics, vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. MR 1258086
- Jorge Nocedal and Stephen J. Wright, Numerical optimization, Springer Series in Operations Research, Springer-Verlag, New York, 1999. MR 1713114
- J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
- C. Roos, T. Terlaky, and J.-Ph. Vial, Theory and algorithms for linear optimization, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Ltd., Chichester, 1997. An interior point approach. MR 1450094
- Alexander Schrijver, Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Ltd., Chichester, 1986. A Wiley-Interscience Publication. MR 874114 SpielTang D. A. Spielman and S.-H. Teng (2001). Why the simplex method usually takes polynomial time, Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, 296–305.
- M. J. Todd, Semidefinite optimization, Acta Numer. 10 (2001), 515–560. MR 2009698, DOI https://doi.org/10.1017/S0962492901000071
- Lieven Vandenberghe and Stephen Boyd, Semidefinite programming, SIAM Rev. 38 (1996), no. 1, 49–95. MR 1379041, DOI https://doi.org/10.1137/1038003
- Robert J. Vanderbei, Linear programming, 2nd ed., International Series in Operations Research & Management Science, vol. 37, Kluwer Academic Publishers, Boston, MA, 2001. Foundations and extensions. MR 1845638
- Margaret H. Wright, Interior methods for constrained optimization, Acta numerica, 1992, Acta Numer., Cambridge Univ. Press, Cambridge, 1992, pp. 341–407. MR 1165729, DOI https://doi.org/10.1017/s0962492900002300
- Margaret H. Wright, Some properties of the Hessian of the logarithmic barrier function, Math. Programming 67 (1994), no. 2, Ser. A, 265–295. MR 1305569, DOI https://doi.org/10.1007/BF01582224
- Margaret H. Wright, Why a pure primal Newton barrier step may be infeasible?, SIAM J. Optim. 5 (1995), no. 1, 1–12. MR 1315702, DOI https://doi.org/10.1137/0805001
- Margaret H. Wright, Ill-conditioning and computational error in interior methods for nonlinear programming, SIAM J. Optim. 9 (1999), no. 1, 84–111. MR 1660094, DOI https://doi.org/10.1137/S1052623497322279
- Stephen J. Wright, Stability of linear equations solvers in interior-point methods, SIAM J. Matrix Anal. Appl. 16 (1995), no. 4, 1287–1307. MR 1351471, DOI https://doi.org/10.1137/S0895479893260498
- Stephen J. Wright, Primal-dual interior-point methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. MR 1422257
- Stephen J. Wright, Modified Cholesky factorizations in interior-point algorithms for linear programming, SIAM J. Optim. 9 (1999), no. 4, 1159–1191. Dedicated to John E. Dennis, Jr., on his 60th birthday. MR 1724782, DOI https://doi.org/10.1137/S1052623496304712
- Stephen J. Wright, Effects of finite-precision arithmetic on interior-point methods for nonlinear programming, SIAM J. Optim. 12 (2001), no. 1, 36–78. MR 1870586, DOI https://doi.org/10.1137/S1052623498347438
- Yinyu Ye, Interior point algorithms, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1997. Theory and analysis; A Wiley-Interscience Publication. MR 1481160
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Additional Information
Margaret H. Wright
Affiliation:
Computer Science Department, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
Email:
mhw@cs.nyu.edu
Received by editor(s):
July 9, 2004
Received by editor(s) in revised form:
August 17, 2004
Published electronically:
September 21, 2004
Additional Notes:
Lecture presented at the AMS Special Session on Current Events, Joint Mathematics Meetings, Phoenix, AZ, January 9, 2004
Article copyright:
© Copyright 2004
American Mathematical Society
