Column Generation Algorithms for Nonlinear Optimization, I: Convergence Analysis

Column generation is an increasingly popular basic tool for the solution of large-scale mathematical programming problems. As problems being solved grow bigger, column generation may however become less efficient in its present form, where columns typically are not optimizing, and finding an optimal solution instead entails finding an optimal convex combination of a huge number of them. We present a class of column generation algorithms in which the columns defining the restricted master problem may be chosen to be optimizing in the limit, thereby reducing the total number of columns needed. This first article is devoted to the convergence properties of the algorithm class, and includes global (asymptotic) convergence results for differentiable minimization, finite convergence results with respect to the optimal face and the optimal solution, and extensions of these results to variational inequality problems. An illustration of its possibilities is made on a nonlinear network flow model, contrasting its convergence characteristics to that of the restricted simplicial decomposition (RSD) algorithm.

[1]  Balder Von Hohenbalken,et al.  Simplicial decomposition in nonlinear programming algorithms , 1977, Math. Program..

[2]  Ron S. Dembo,et al.  Computing equilibria on large multicommodity networks: An application of truncated quadratic programming algorithms , 1988, Networks.

[3]  James V. Burke,et al.  Characterization of solution sets of convex programs , 1991, Oper. Res. Lett..

[4]  W. Rudin.  Principles of mathematical analysis , 1964 .

[5]  A. Goldstein.  Convex programming in Hilbert space , 1964 .

[6]  E. M. L. Beale,et al.  Nonlinear Programming: A Unified Approach. , 1970 .

[7]  Warren B. Powell,et al.  Multicommodity network flows: The impact of formulation on decomposition , 1993, Math. Program..

[8]  P. Marcotte,et al.  Coupling the auxiliary problem principle with descent methods of pseudoconvex programming , 1995 .

[9]  R. Jayakrishnan,et al.  A FASTER PATH-BASED ALGORITHM FOR TRAFFIC ASSIGNMENT , 1994 .

[10]  Michael Patriksson,et al.  Cost Approximation: A Unified Framework of Descent Algorithms for Nonlinear Programs , 1998, SIAM J. Optim..

[11]  Patrice Marcotte,et al.  Conditions de régularité géométrique pour les inéquations variationnelles , 1988 .

[12]  Boris Polyak,et al.  Constrained minimization methods , 1966 .

[13]  M. Guignard.  Generalized Kuhn–Tucker Conditions for Mathematical Programming Problems in a Banach Space , 1969 .

[14]  George B. Dantzig,et al.  Decomposition Principle for Linear Programs , 1960 .

[15]  D. Bertsekas.  Projected Newton methods for optimization problems with simple constraints , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[16]  J. Dunn.  On the convergence of projected gradient processes to singular critical points , 1987 .

[17]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[18]  Elijah Polak,et al.  Minimizing pseudoconvex functions on convex compact sets , 1990 .

[19]  P. Tseng.  Decomposition algorithm for convex differentiable minimization , 1991 .

[20]  G. Nemhauser,et al.  Integer Programming , 2020 .

[21]  John L. Nazareth,et al.  The decomposition principle and algorithms for linear programming , 1991 .

[22]  Patrice Marcotte,et al.  Weak Sharp Solutions of Variational Inequalities , 1998, SIAM J. Optim..

[23]  Leon S. Lasdon,et al.  Optimization Theory of Large Systems , 1970 .

[24]  J. Burkey,et al.  WEAK SHARP MINIMA IN MATHEMATICAL PROGRAMMING , 1993 .

[25]  Masao Fukushima,et al.  Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems , 1992, Math. Program..

[26]  J. J. Moré,et al.  On the identification of active constraints , 1988 .

[27]  James V. Burke,et al.  Exposing Constraints , 1994, SIAM J. Optim..

[28]  Hector A. Rosales-Macedo.  Nonlinear Programming: Theory and Algorithms (2nd Edition) , 1993 .

[29]  Athanasios Migdalas,et al.  A regularization of the Frank—Wolfe method and unification of certain nonlinear programming methods , 1994, Math. Program..

[30]  Torbjörn Larsson,et al.  Simplicial Decomposition with Disaggregated Representation for the Traffic Assignment Problem , 1992, Transp. Sci..

[31]  Zdzisław Denkowski,et al.  Set-Valued Analysis , 2021 .

[32]  Robert Hooke,et al.  `` Direct Search'' Solution of Numerical and Statistical Problems , 1961, JACM.

[33]  Michael Patriksson,et al.  An algorithm for the combined distribution and assignment model , 1998 .

[34]  J. A. Ventura,et al.  Finiteness in restricted simplicial decomposition , 1985 .

[35]  M. Patriksson.  Partial linearization methods in nonlinear programming , 1993 .

[36]  M. Patriksson.  Nonlinear Programming and Variational Inequality Problems: A Unified Approach , 1998 .

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

[38]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[39]  M G Bell.  Transportation networks: recent methodological advances: selected proceedings of the 4th EURO Transportation Meeting , 1998 .

[40]  Jose A. Ventura,et al.  Restricted simplicial decomposition for convex constrained problems , 1993, Math. Program..

[41]  M. Patriksson.  A unified description of iterative algorithms for traffic equilibria , 1993 .

[42]  P. Marcotte,et al.  A sequential linear programming algorithm for solving monotone variational inequalities , 1989 .

[43]  J. A. Ventura,et al.  Restricted simplicial decomposition: computation and extensions , 1987 .

[44]  Torbjörn Larsson,et al.  Applications of Simplicial Decomposition with Nonlinear Column Generation to Nonlinear Network Flows , 1997 .

[45]  Charles A. Holloway.  An extension of the frank and Wolfe method of feasible directions , 1974, Math. Program..

[46]  J. Hammond.  Solving asymmetric variational inequality problems and systems of equations with generalized nonlinear programming algorithms , 1984 .

[47]  R. Wets,et al.  On the convergence of sequences of convex sets in finite dimensions , 1979 .

[48]  Suzanne P. Evans,et al.  DERIVATION AND ANALYSIS OF SOME MODELS FOR COMBINING TRIP DISTRIBUTION AND ASSIGNMENT , 1976 .

[49]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[50]  Martin W. P. Savelsbergh,et al.  Branch-and-Price: Column Generation for Solving Huge Integer Programs , 1998, Oper. Res..