# Interior-point methods

**DOI:**https://doi.org/10.1007/1-4020-0611-X_475

- 32 Downloads

## INTRODUCTION

Notwithstanding the success of the simplex method for linear programming (LP), there was, even from the earliest days of operations research, a desire to create an algorithm for solving linear programming problems that proceeded on a path through the polytope rather than around its perimeter. Indeed, methods were considered, but until recently, none was as effective as the simplex method. In this article, the motivation for desiring an “interior” path, the concept of the complexity of solving a linear programming problem, a brief his-tory of the developments in the area, and the status of the subject as of this writing are discussed. More complete surveys are given in Gonzaga (1991a, 1991b, 1992), Goldfarb and Todd (1989), Roos and Terlaky (1997), Roos, Terlaky and Vial (1997), Terlaky (1996), Ye (1997), Wright (1996) and Wright (1998). Generalizations to nonlinear problems are briefly discussed as well. For a thorough treatment of interior-point algorithms on those...

## References

- [1]Borgwardt, K.H. (1987). The Simplex Method: A Probabilistic Analysis, Algorithms and Combinatorics, Vol. 1. Springer Verlag, Berlin.Google Scholar
- [2]Dikin, I.I. (1967). “Iterative solution of problems of linear and quadratic programming.” Soviet Mathematics Doklady, 8, 674–675.Google Scholar
- [3]Fiacco, A.V. and McCormick, G.P. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley, New York.Google Scholar
- [4]Frish, K.R. (1954). “Principles of linear programming — the double gradient form of the logarithmic potential method.” Memorandum, Institute of Economics, University of Oslo, Norway.Google Scholar
- [85]Gonzaga, C.C. (1991a). “Large-steps path-following methods for linear programming, part i: Barrier function method.” SIAM Jl. Optimization, 1, 268–279.Google Scholar
- [6]Gonzaga, C.C. (1991b). “Large-steps path-following methods for linear programming, part ii: Potential reduction method.” SIAMJl. Optimization, 1, 280–292.Google Scholar
- [7]Gonzaga, C.C. (1992). “Path following methods for linear programming.” SIAM Review, 34, 167–224.Google Scholar
- [8]Greenberg, H.J. (1994) “The use of the optimal partition in a linear programming solution for postoptimal analysis.” Operations Research Letters, 15, 179–186.Google Scholar
- [9]Goldfarb, D. and Todd, M.J. (1989). “Linear programming,” In G.L. Nemhauser, A.H.G. Rinnooy Kan, and Todd, M.J., eds., Optimization, 73–170, North Holland, Amsterdam.Google Scholar
- [10]den Hertog, D. (1994). Interior Point Approach to Linear, Quadratic and Convex Programming, Kluwer Academic Publishers, Dordrecht, The Nether-lands.Google Scholar
- [11]Huard, P. (1967). “Resolution of mathematical programming with nonlinear constraints by the method of centres.” In J. Abadie, ed., Nonlinear Programming, pages 209–219. North Holland, Amsterdam.Google Scholar
- [12]Illés, T., Peng, J., Roos, C., and Terlaky, T. (1998). “A strongly polynomial rounding scheme in interior point methods for
*P*_{∗}(κ) linear complementarity problems.” Technical Report, 98–15, Faculty of Mathematics and Computer Science, TU Delft, The Nether-lands (submitted to*SIAM Jl. Optimization*).Google Scholar - [13]Jansen, B. (1997). Interior Point Techniques in Optimization: Complexity, Sensitivity and Algorithms. Kluwer Academic Publishers, Dordrecht, The Nether-lands.Google Scholar
- [14]Karmarkar, N.K. (1984). “A new polynomial-time algorithm for linear programming.” Combinatorica, 4, 373–395.Google Scholar
- [15]Khachiyan, L.G. (1979). “A polynomial algorithm in linear programming.” Translated in Soviet Mathematics Doklady, 20, 191–194.Google Scholar
- [16]Klee, V. and Minty, G. (1972). “How good is the simplex algorithm?” In O. Sisha, ed., Inequalities III. Academic Press, New York.Google Scholar
- [17]Kojima, M., Megiddo, N., Noma, T., and Yoshise, A. (1991). A unified approach to interior point algorithms for linear complementarity problems, volume 538 of Lecture Notes in Computer Science. Springer Verlag, Berlin.Google Scholar
- [18]Nesterov, Y.E. and Nemirovskii, A.S. (1993). Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms. SIAM Publications. SIAM, Philadelphia.Google Scholar
- [19]Roos, C. and Terlaky, T. (1997). “Advances in linear optimization,” In M. DellAmico, F. Maffioli, and S. Martello, eds., Annotated Bibliography in Combinatorial Optimization, Chapter 7. John Wiley, New York.Google Scholar
- [20]Roos, C., Terlaky, T., and Vial, J.-Ph. (1997). Interior Point Approach to Linear Optimization: Theory and Algorithms, John Wiley, New York.Google Scholar
- [21]Sonnevend, Gy. (1985). “An ‘analytic center' for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming.” In A. Prekopa, J. Szelezsan, and B. Strazicky, editors, System Modeling and Optimization, Proceedings of the 12th IFIP-Conference held in Budapest, Hungary, September 1985, volume 84 of
*Lecture Notes in Control and In-formation Sciences*, pp. 866–876. Springer Verlag, Berlin.Google Scholar - [22]Saigal, R., Vandenberghe, L., and Wolkowicz, H., eds. (1998). Handbook of Semidefinite Programming, Kluwer Academic, Dordrecht, The Netherlands.Google Scholar
- [23]Terlaky, T., ed. (1996). Interior Point Methods in Mathematical Programming, Kluwer Academic, Dordrecht, The Netherlands.Google Scholar
- [24]Todd, M. (1997). “On search directions in interior-point methods for semidefinite programming,” Technical Report No. 1205, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York.Google Scholar
- [25]Vandenberghe, L. and Boyd, S. (1996). “Semidefinite Programming,” SIAM Review, 38, 49–95.Google Scholar
- [26]Wright, M.H. (1998). “The Interior-Point Revolution in Constrained Optimization”. Technical Report 98-4-09, Computing Sciences Research Center, Bell Laboratories, Murray Hill, New Jersey.Google Scholar
- [27]Wright, S.J. (1996). Primal-Dual Interior-Point Methods. SIAM, Philadelphia.Google Scholar
- [28]Ye, Y. (1997). Interior Point Algorithms, John Wiley, New York.Google Scholar