Encyclopedia of Operations Research and Management Science

2013 Edition
| Editors: Saul I. Gass, Michael C. Fu

Interior-Point Methods for Conic-Linear Optimization

  • Tamás Terlaky
  • Paul T. Boggs
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-1153-7_475


Even with the success of the simplex method for linear programming (LP), there was from the earliest days of operations research a desire to create an algorithm for solving LP problems that proceeded on a path through the polytope rather than around its perimeter. Interior point methods (IPMs) were first developed in 1950s, analyzed and first implemented in the 1960s. At that time the conclusion was made that IPMs were not competitive with other algorithms, especially with simplex methods. The continuous effort to find a polynomial algorithm for LP problems led to the revitalization of IPMs. In 1984 Karmarkar first proved the polynomial complexity of an IPM, which led to the “Interior Point Revolution” (Wright 2004) in mathematical programming. In this article the motivation for desiring an interior path, the concept of the complexity of solving LP problems, a brief history of the developments in the area, and the research state of the art are discussed, including...

This is a preview of subscription content, log in to check access.


  1. Andersen, E. D., Roos, C., & Terlaky, T. (2003). On implementing a primal–dual interior–point method for conic quadratic optimization. Mathematical Programming, 95(2), 249–277.CrossRefGoogle Scholar
  2. Ben-Tal, A., & Nemirovski, A. (2001). Lectures on modern convex optimization: Analysis, algorithms, and engineering applications (MPS-SIAM series on optimization). Philadelphia, PA: SIAM.CrossRefGoogle Scholar
  3. Bixby, R. E. (2002). Solving real-world linear programs: A decade and more of progress. Operations Research, 50(1), 3–15.CrossRefGoogle Scholar
  4. Bomze, I. M., Duerr, M., de Klerk, E., Roos, C., Quist, A. J., & Terlaky, T. (2000). On copositive programming and standard quadratic optimization problems. Journal of Global Optimization, 18(2), 301–320.CrossRefGoogle Scholar
  5. Borgwardt, K. H. (1987). The simplex method: A probabilistic analysis, algorithms and combinatorics (Vol. 1). Berlin: Springer.Google Scholar
  6. Byrd, R., Nocedal, J., & Waltz, R. (2006). KNITRO: An integrated package for nonlinear optimization. In G. Di Pillo & M. Roma (Eds.), Large-scale nonlinear optimization (Nonconvex optimization and its applications, Vol. 83, pp. 35–59). Berlin: Springer.CrossRefGoogle Scholar
  7. Chu, M., Zinchenko, Y., Henderson, S. G., & Sharpe, M. B. (2008). Robust optimization for intensity modulated radiation therapy treatment planning under uncertainty. Physics in Medicine and Biology, 53, 3231–3250.CrossRefGoogle Scholar
  8. Craig, T., Sharpe, M. B., Terlaky, T., & Zinchenko, Y. (2008). Controlling the dose distribution with gEUD-type constraints within the convex IMRTP framework. Physics in Medicine and Biology, 53, 3231–3250.CrossRefGoogle Scholar
  9. de Klerk, E. (2002). Aspects of semidefinite programming: Interior point algorithms and selected applications. Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar
  10. den Hertog, D. (1994). Interior point approach to linear, quadratic and convex programming. Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar
  11. Deza, A., Nematollahi, E., Peyghami, R., & Terlaky, T. (2006). The central path visits all the vertices of the Klee-Minty cube. Optimization Methods and Software, 21, 851–865.CrossRefGoogle Scholar
  12. Deza, A., Nematollahi, E., & Terlaky, T. (2008). How good are interior point methods? Klee-Minty cubes tighten iteration-complexity bounds. Mathematical Programming, 113, 1–14.CrossRefGoogle Scholar
  13. Deza, A., Terlaky, T., & Zinchenko, Y. (2008). Polytopes and arrangements: Diameter and curvature. Operations Research Letters, 36, 215–222.CrossRefGoogle Scholar
  14. Deza, A., Terlaky, T., & Zinchenko, Y. (2009). The continuous d-step conjecture for polytopes. Discrete and Computational Geometry, 41, 318–327.CrossRefGoogle Scholar
  15. Dikin, I. I. (1967). Iterative solution of problems of linear and quadratic programming. Soviet Mathematics Doklady, 8, 674–675.Google Scholar
  16. Fiacco, A. V., & McCormick, G. P. (1968). Nonlinear programming: Sequential unconstrained minimization techniques. New York: John Wiley.Google Scholar
  17. Frish, K. R. (1954). Principles of linear programming – the double gradient form of the logarithmic potential method. Memorandum, Institute of Economics, University of Oslo, Oslo, Norway.Google Scholar
  18. Ghaffari Hadigheh, A. R., Romanko, O., & Terlaky, T. (2007). Sensitivity analysis in convex quadratic optimization: Simultaneous perturbation of the objective and right-hand-side vectors. Algorithmic Operations Research, 2(2), 4–111.Google Scholar
  19. Goldfarb, D., & Todd, M. J. (1989). Linear programming. In G. L. Nemhauser, A. H. G. Rinnooy Kan, & M. J. Todd (Eds.), Optimization (pp. 73–170). Amsterdam/New York: North Holland.CrossRefGoogle Scholar
  20. Gonzaga, C. C. (1991a). Large-steps path-following methods for linear programming, part I: Barrier function method. SIAM Journal on Optimization, 1, 268–279.CrossRefGoogle Scholar
  21. Gonzaga, C. C. (1991b). Large-steps path-following methods for linear programming, part II: Potential reduction method. SIAM Journal on Optimization, 1, 280–292.CrossRefGoogle Scholar
  22. Gonzaga, C. C. (1992). Path following methods for linear programming. SIAM Review, 34, 167–224.CrossRefGoogle Scholar
  23. Huard, P. (1967). Resolution of mathematical programming with nonlinear constraints by the method of centres. In J. Abadie (Ed.), Nonlinear programming (pp. 209–219). Amsterdam: North Holland.Google Scholar
  24. Illés, T., Peng, J., Roos, C., & Terlaky, T. (2000). A strongly polynomial rounding scheme in interior point methods for P*(κ) linear complementarity problems. SIAM Journal on Optimization, 11(2), 320–340.CrossRefGoogle Scholar
  25. Illés, T., & Terlaky, T. (2002). Pivot versus interior point methods: Pros and cons. European Journal of Operational Research, 140(2), 6–26.CrossRefGoogle Scholar
  26. Jansen, B. (1997). Interior point techniques in ptimization. Complexity, sensitivity and algorithms. Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar
  27. Karmarkar, N. K. (1984). A new polynomial-time algorithm for linear programming. Combinatorica, 4, 373–395.CrossRefGoogle Scholar
  28. Khachiyan, L. G. (1979). A polynomial algorithm in linear programming. Translated in Soviet Mathematics Doklady, 20, 191–194.Google Scholar
  29. Klee, V. & Minty, G. J. (1972). How good is the simplex algorithm. In O. Shisha (Ed.), Inequalities III (pp. 159–175). Academic Press.Google Scholar
  30. Kojima, M., Megiddo, N., Noma, T., & Yoshise, A. (1991). A unified approach to interior point al − go − rithms for linear complementarity problems (Lecture notes in computer science, Vol. 538). Berlin, Germany: Springer.CrossRefGoogle Scholar
  31. Koltai, T., & Terlaky, T. (2000). The difference between managerial and mathematical interpretation of sensitivity analysis results in linear programming. International Journal of Production Economics, 65, 257–274.CrossRefGoogle Scholar
  32. Nematollahi, E., & Terlaky, T. (2008). A simpler and tighter redundant Klee-Minty construction. Optimization Letters, 2(3), 403–414.CrossRefGoogle Scholar
  33. Nesterov, Y. E., & Nemirovskii, A. S. (1994). Interior point polynomial methods in convex programming: Theory and algorithms. Philadelphia: SIAM.CrossRefGoogle Scholar
  34. Peng, J., Roos, C., & Terlaky, T. (2002). Self-regularity: A new paradigm for primal-dual interior-point algorithms. Princeton, NJ: Princeton University Press.Google Scholar
  35. Roos, C., & Terlaky, T. (1997). Advances in linear optimization. In M. DellAmico, F. Maffioli, & S. Martello (Eds.), Annotated bibliography in combinatorial optimization, Chap − ter 7. New York: John Wiley & Sons.Google Scholar
  36. Roos, C., Terlaky, G. J., & Vial J. -Ph. (1997). Interior point methods for linear optimization. (New York: Springer, 2nd ed., 2006). (Roos, C., Terlaky, T., Vial, J. -Ph. (1997). Theory and algorithms for linear optimization: An interior point approach. Chichester, UK: John Wiley & Sons).Google Scholar
  37. Santos, F. (2010). A counterexample to the Hirsch conjecture. arXiv:1006.2814.Google Scholar
  38. Sonnevend, G. y. (1985). An ‘analytic center’ for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In A. Prékopa, J. Szelezsán, & B. Strazicky (Eds.), System modeling and optimization : Proceedings of the 12th IFIP-Conference held in Budapest, Hungary, September 1985. Lecture notes in control and information sciences (Vol. 84, pp. 866–876). Berlin, West–Germany: Springer Verlag, 1986.Google Scholar
  39. Terlaky, T. (Ed.). (1996). Interior point methods in mathematical programming. Dordrecht, The Netherlands: Kluwer.Google Scholar
  40. Todd, M. (1999). A study of search directions in primal-dual interior-point methods for semidefinite programming. Optimization Methods and Software, 11, 1–46.CrossRefGoogle Scholar
  41. Vandenberghe, L., & Boyd, S. (1996). Semidefinite programming. SIAM Review, 38, 49–95.CrossRefGoogle Scholar
  42. Wächter, A. (2002). An interior point algorithm for large-scale nonlinear optimization with applications in process engineering. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, USA.Google Scholar
  43. Wächter, A., & Biegler, L. T. (2006). On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Mathematical Programming, 106(1), 25–57.CrossRefGoogle Scholar
  44. Wolkowicz, H., Saigal, R., Vandenberghe, L. (Eds.) (2000). Handbook of semidefinite programming: Theory, algorithms, and applications. Kluwer A.P.C.Google Scholar
  45. Wright, S. J. (1996). Primal-dual interior-point methods. Philadelphia: SIAM.Google Scholar
  46. Wright, M. H. (2004). The interior-point revolution in optimization: History, recent developments, and lasting consequences. Bulletin (New Series) of the American Mathematical Society, 42(1): 39–56.Google Scholar
  47. Ye, Y. (1997). Interior point algorithms. New York: John Wiley & Sons.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Lehigh UniversityBethlehemUSA
  2. 2.Sandia National LaboratoriesLivermoreUSA
  3. 3.National Institute of Standards and TechnologyGaithersburgUSA