Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris


Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_57


In the mid-1950s and the early 1960s, Frisch (1955) and Carroll (1961) proposed the use of Barrier Functions (BFs) for constrained optimization. Since then, the BFs have been extensively studied, with particularly major work in the area due to Fiacco and McCormick (1968) who developed the Sequential Unconstrained Minimization Technique (SUMT). Currently, methods based on barrier functions make up a considerable part of modern optimization theory.


Consider the constrained optimization problem
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  1. [1]
    Bertsekas, D. (1982). Constrained Optimization and Lagrange Multipliers Methods, Academic Press, New York.Google Scholar
  2. [2]
    Carroll, C. (1961). “The created response surface technique for optimizing nonlinear restrained systems,” Operations Research, 9, 169–184.Google Scholar
  3. [3]
    Eggermont, P. (1990). “Multiplicative Iterative Algorithm for Convex Programming,” Linear Algebral and its Applications, 130, 25–42.Google Scholar
  4. [4]
    Fiacco, A.V. and McCormick, G.P. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley, New York.Google Scholar
  5. [5]
    Frisch K. (1955). “The logarithmic potential method of convex programming,” Technical Memorandum, May 13, University Institute of Economics, Oslo.Google Scholar
  6. [6]
    Gonzaga, C. (1992). “Path Following Methods for Linear Programming,” SIAM Review, 34, 167–224.Google Scholar
  7. [7]
    Gill, P., Murray, W., Saunders, M., Tomlin, J., and Wright, M. (1986). “On Projected Barrier Methods for Linear Programming and an Equivalence to Karmarkar's Projective Method,” Mathematical Programming, 36, 183–209.Google Scholar
  8. [8]
    Golshtein, E.G. and Tretyakov, N.V. (1974). “Modified Lagrangean Functions,” Economics and Math. Methods, 10, 568–591 (Russian).Google Scholar
  9. [9]
    Hestenes, M. (1969). “Multiplier and gradient methods,” Jl. Optimization Theory & Applications, 4, 303–320.Google Scholar
  10. [10]
    Huard, P. (1967). “Resolution of Mathematical Programming with Nonlinear Constraints by the Method of Centers,” in Nonlinear Programming, J. Abadie, ed., North-Holland, Amsterdam.Google Scholar
  11. [11]
    Jensen, D. and Polyak, R. (1994). “The convergence of the modified barrier method for convex programming.” IBM Jl. R&D, 38, 307–321.Google Scholar
  12. [12]
    Karmarkar, N. (1984). “A New Polynomial-time Algorithm for Linear Programming,” Combinatorica, 4, 373–395.Google Scholar
  13. [13]
    Lustig, I., Marsten, R., and Shanno, D. (1992). “On implementing Mehrotra's Predictor-Corrector Interior Point Method for Linear Programming,” Siam Jl. Optimization, 2, 435–449.Google Scholar
  14. [14]
    Mangasarian, O. (1975). “Unconstrained Lagrangians in Nonlinear Programming,” SIAM Jl. Control, 13, 772–791.Google Scholar
  15. [15]
    Melman, A. and Polyak, R. (1996). “The Newton Modified Barrier Method for QP,” Annals of Operations Research, 62, 465–519.Google Scholar
  16. [16]
    Mehrotra, S. (1992). “On the Implementation of Primal-Dual Interior Point Method,” SIAM Jl. Optimization, 2, 575–601.Google Scholar
  17. [17]
    Nesterov, Yu. and Nemirovsky, A. (1994). Interior Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics, Philadelphia.Google Scholar
  18. [18]
    Polyak, B.T. and Tretyakov, N.V. (1973). “The method of penalty bounds for constrained extremum problems,” Zh. Vych. Mat. iMat. Fiz, 13, 34–46, USSR Computational Methods Math. Phys., 13, 42–58.Google Scholar
  19. [19]
    Polyak, R. (1986). Controlled processes in extremal and equilibrium problems. VINITY, Moscow (Russian).Google Scholar
  20. [20]
    Polyak, R. (1992). “Modified Barrier Functions (Theory and Methods),” Mathematical Programming, 54 177–222.Google Scholar
  21. [21]
    Polyak, R. (1996). “Modified Barrier Functions in Linear Programming,” Research Report, Dept. of Operations Research, George Mason University, 1–56. Google Scholar
  22. [22]
    Polyak, R. (1997). “Modified Interior Distance Functions,” Contemporary Mathematics, 209, 183–209.Google Scholar
  23. [23]
    Polyak, R. and Teboulle, M. (1997). “Nonlinear Rescaling and Proximal-like Methods in Convex Optimization,” Mathematical Programming, 76, 265–284.Google Scholar
  24. [24]
    Powell, M. (1969). “A Method for Nonlinear Constraints in Minimization Problems,” in Optimization, R. Fletcher, ed., Academic Press, New York.Google Scholar
  25. [25]
    Powell, M. (1995). “Some Convergence Properties of the Modified Log Barrier Method for Linear Programming,” SIAM Jl. Optimization, 5, 695–739.Google Scholar
  26. [26]
    Renegar, J. (1988). “A polynomial-time algorithm, based on Newton's method for linear programming,” Mathematical Programming, 40, 59–93.Google Scholar
  27. [27]
    Rockafellar, R.T. (1973). “The Multiplier Method of Hestenes and Powell Applied to Convex Programming,” Jl. Optimization Theory & Applications, 12, 555–562.Google Scholar
  28. [28]
    Roos, C., Terlaky, T., and Vial, J.-Ph. (1997). Theory and Algorithms for Linear Optimization: An Interior Point Approach, John Wiley, New York.Google Scholar
  29. [29]
    Smale, S. (1986). “A Newton method estimates from data at one point,” in The Merging of Disciplines in Pure, Applied and Computational Mathematics, R. Ewing, ed., Springer-Verlag, New York/Berlin.Google Scholar
  30. [30]
    Teboulle, M. (1993). “Entropic proximal mappings with application to nonlinear programming,” Mathematics of Operations Research, 17, 670–690.Google Scholar
  31. [31]
    Wright, S. (1997). Primal-Dual-Interior Point Methods, SIAM, Philadelphia.Google Scholar
  32. [32]
    Yinyu, Ye (1997). Interior Point Algorithms: Theory and Analysis, John Wiley, New York.Google Scholar

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© Kluwer Academic Publishers 2001

Authors and Affiliations

  1. 1.George Mason UniversityFairfaxUSA