On the Complexity of a Column Generation Algorithm for Convex or Quasiconvex Feasibility Problems

We analyze the convergence and the complexity of a potential reduction column generation algorithm for solving general convex or quasiconvex feasibility problems defined by a separation oracle. The oracle is called at the analytic center of the set given by the intersection of the linear inequalities which are the previous answers of the oracle. We show that the algorithm converges in finite time and is in fact a fully polynomial approximation algorithm, provided that the feasible region has an nonempty interior. This result is based on the works of Ye [22] and Nesterov [16].

[1]  I. J. Schoenberg,et al.  The Relaxation Method for Linear Inequalities , 1954, Canadian Journal of Mathematics.

[2]  S. Agmon.  The Relaxation Method for Linear Inequalities , 1954, Canadian Journal of Mathematics.

[3]  J. E. Kelley,et al.  The Cutting-Plane Method for Solving Convex Programs , 1960 .

[4]  G. Dantzig,et al.  THE DECOMPOSITION ALGORITHM FOR LINEAR PROGRAMS , 1961 .

[5]  Jack Elzinga,et al.  A central cutting plane algorithm for the convex programming problem , 1975, Math. Program..

[6]  Jean-Louis Goffin,et al.  The Relaxation Method for Solving Systems of Linear Inequalities , 1980, Math. Oper. Res..

[7]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .

[8]  John Darzentas,et al.  Problem Complexity and Method Efficiency in Optimization , 1983 .

[9]  Naum Zuselevich Shor,et al.  Minimization Methods for Non-Differentiable Functions , 1985, Springer Series in Computational Mathematics.

[10]  J. Hiriart-Urruty,et al.  Trends in Mathematical Optimization , 1987 .

[11]  G. Sonnevend.  New Algorithms in Convex Programming Based on a Notion of “Centre” (for Systems of Analytic Inequalities) and on Rational Extrapolation , 1988 .

[12]  James Renegar,et al.  A polynomial-time algorithm, based on Newton's method, for linear programming , 1988, Math. Program..

[13]  J. Mitchell.  Karmarkar's algorithm and combinatorial optimization problems , 1988 .

[14]  Pravin M. Vaidya,et al.  A new algorithm for minimizing convex functions over convex sets , 1989, 30th Annual Symposium on Foundations of Computer Science.

[15]  Yinyu Ye,et al.  A Potential Reduction Algorithm Allowing Column Generation , 1992, SIAM J. Optim..

[16]  J. Goffin,et al.  Decomposition and nondifferentiable optimization with the projective algorithm , 1992 .

[17]  Y. Ye,et al.  Translational Cuts for Convex Minimization , 1993 .

[18]  Leonid Khachiyan,et al.  On the complexity of approximating the maximal inscribed ellipsoid for a polytope , 1993, Math. Program..

[19]  P. Pardalos.  Complexity in numerical optimization , 1993 .

[20]  Yurii Nesterov,et al.  New variants of bundle methods , 1995, Math. Program..