Annals of Operations Research

, Volume 58, Issue 2, pp 67–98

# A logarithmic barrier cutting plane method for convex programming

• D. den Hertog
• J. Kaliski
• C. Roos
• T. Terlaky
Nonlinear Programming

## Abstract

The paper presents a logarithmic barrier cutting plane algorithm for convex (possibly non-smooth, semi-infinite) programming. Most cutting plane methods, like that of Kelley, and Cheney and Goldstein, solve a linear approximation (localization) of the problem and then generate an additional cut to remove the linear program's optimal point. Other methods, like the “central cutting” plane methods of Elzinga-Moore and Goffin-Vial, calculate a center of the linear approximation and then adjust the level of the objective, or separate the current center from the feasible set. In contrast to these existing techniques, we develop a method which does not solve the linear relaxations to optimality, but rather stays in the interior of the feasible set. The iterates follow the central path of a linear relaxation, until the current iterate either leaves the feasible set or is too close to the boundary. When this occurs, a new cut is generated and the algorithm iterates. We use the tools developed by den Hertog, Roos and Terlaky to analyze the effect of adding and deleting constraints in long-step logarithmic barrier methods for linear programming. Finally, implementation issues and computational results are presented. The test problems come from the class of numerically difficult convex geometric and semi-infinite programming problems.

## Keywords

Column generation convex programming cutting plane methods decomposition interior point method linear programming logarithmic barrier function nonsmooth optimization semi-infinite programming

## References

1. [1]
O. Bahn, J.L. Goffin, J.-Ph. Vial and O. Du Merle, Implementation and behavior of an interior point cutting plane algorithm for convex programming: An application to geometric programming, Discr. Appl. Math. 49(1994)3–23.
2. [2]
E.W. Cheney and A.A. Goldstein, Newton's method for convex programming and Tchebycheff approximation, Numer. Math. 1(1959)243–268.
3. [3]
I.D. Coope and G.A. Watson, A projected Lagrangian algorithm for semi-infinite programming, Math. Progr. 32(1985)337–356.
4. [4]
J. Elzinga and T.G. Moore, A central cutting plane algorithm for the convex programming problem, Math. Progr. 8(1975)134–145.
5. [5]
A.V. Fiacco and A. Ghaemi, A sensitivity analysis of a nonlinear water pollution control problem using an upper Hudson River database, The George Washington University, Washington, DC (1982).Google Scholar
6. [6]
A.V. Fiacco and A. Ghaemi, A sensitivity and parametric bound analysis of an electric power GP model: Optimal steam turbine exhaust annulus and condenser sizes, serial T-437, The George Washington University, Washington, DC (1981).Google Scholar
7. [7]
J.L. Goffin, A. Haurie and J.-Ph. Vial, Decomposition and nondifferentiable optimization with the projective algorithm, Manag. Sci. 38(1992)284–302.
8. [8]
P.R. Gribik, A central-cutting-plane algorithm for semi-infinite programming problems, in:Semi-Infinite Programming, ed. R. Hettich, Lecture Notes in Control and Information Sciences Vol. 15 (Springer, New York, 1979) pp. 66–82.Google Scholar
9. [9]
J.L. Goffin and J.P. Vial, Cutting planes and column generation techniques with the projective algorithm, J. Optim. Theory Appl. 65(1988)409–429.
10. [10]
P.R. Gribik and D.N. Lee, A comparison of two central cutting plane algorithms for prototype geometric programming problems, in:Methods of Operations Research 31, ed. W. Oettli and S. Steffens (Verlag Anton/Hain/Mannheim, Germany, 1978) pp. 275–287.Google Scholar
11. [11]
S.-A. Gustafson, On numerical analysis in semi-infinite programming, in:Semi-Infinite Programming, ed. R. Hettich, Lecture Notes in Control and Information Sciences Vol. 15 (Springer, New York, 1979) pp. 51–65.Google Scholar
12. [12]
R. Hettich and K.O. Kortanek, Semi-infinite programming: Theory, methods, and applications, Working Paper, FB IV, Universität Trier, Germany (1991).Google Scholar
13. [13]
D. den Hertog,Interior Point Approach to Linear, Quadratic and Convex Programming — Algorithms and Complexity (Kluwer Academic, Dordrecht, The Netherlands, 1994).Google Scholar
14. [14]
D. den Hertog, C. Roos and J.-Ph. Vial, A complexity reduction for the long-step path-following algorithm for linear programming, SIAM J. Optim. 2(1992)71–87.
15. [15]
D. den Hertog, C. Roos and T. Terlaky, A Build-Up variant of the path-following method for LP, Oper. Res. Lett. 12(1992)181–186.
16. [16]
D. den Hertog, C. Roos and T. Terlaky, Adding and deleting constraints in the logarithmic barrier method for linear programming problems, in:Advances in Optimization and Approximation, ed. D. Du and J. Sun (Kluwer Academic, Dordrecht, The Netherlands, 1994) pp. 166–185.Google Scholar
17. [17]
S. Jha, K.O. Kortanek and H. No, Lotsizing and setup time reduction under stochastic demand: A geometric programming approach, Working Paper No. 88-12, College of Business Administration, The University of Iowa, Iowa City, IA (1988).Google Scholar
18. [18]
J. Kaliski and Y. Ye, A decomposition variant of the potential reduction algorithm for linear programming, Manag. Sci. 39(1993)757–776.
19. [19]
J.E. Kelley, Jr., The cutting-plane method for solving convex programs, J. Soc. Ind. Appl. Math. 8(1960)703–712.
20. [20]
K.O. Kortanek, Semi-infinite programming duality for order restricted statistical inference models, Working Paper No. 91-18, College of Business Administration, The University of Iowa, Iowa City, IA (1991).Google Scholar
21. [21]
K.O. Kortanek and H. No, A second order affine scaling algorithm for the geometric programming dual with logarithmic barrier, Optimization 23(1990)303–322. Earlier version: Working Paper No. 90-07, College of Business Administration, The University of Iowa, Iowa City, IA (1990).Google Scholar
22. [22]
K.O. Kortanek and H. No, A central cutting plane algorithm for convex semi-infinite programming problems, SIAM J. Optim. 3(1993)901–918.
23. [23]
N. Megiddo, Pathways to the optimal set in linear programming, in:Progress in Linear Programming, Interior and Related Methods, ed. N. Megiddo (Springer, New York, 1989) pp. 131–158.Google Scholar
24. [24]
J.E. Mitchell and M.J. Todd, Solving combinatorial optimization problems using Karmarkar's algorithm, Math. Progr. 56(1992)245–284.
25. [25]
J. Renegar, A polynomial time algorithm, based on Newton's method, for linear programming, Math. Progr. 40(1988)59–93.
26. [26]
C. Roos and J.-Ph. Vial, A polynomial method of approximate centers for linear programming, Math. Progr. 54(1992)295–305.
27. [27]
Gy. Sonnevend, An “analytic centre” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming, in:Lecture Notes in Control and Information Sciences Vol. 84 (Springer, New York, 1985) pp. 866–876.Google Scholar
28. [28]
J.-Ph. Vial, Computational experience with a primal-dual interior point method for smooth convex programming, Report April 1993, Département d'Economie Commerciale et Industrielle, Université Genève, Genève, Switzerland (1993), to appear in Optim. Meth. and Softwares.Google Scholar
29. [29]
Y. Ye, A potential reduction algorithm allowing column generation, SIAM J. Optim. 2(1992)7–20.

© J.C. Baltzer AG, Science Publishers 1995

## Authors and Affiliations

• D. den Hertog
• 1
• J. Kaliski
• 1
• C. Roos
• 1
• T. Terlaky
• 1
1. 1.Faculty of Technical Mathematics and InformaticsDelft University of TechnologyDelftThe Netherlands