Coordination Complexity of Parallel Price-Directive Decomposition

The general block-angular convex resource sharing problem in K blocks and M nonnegative block-separable coupling constraints is considered. Applications of this model are in combinatorial optimization, network flows, scheduling, communication networks, engineering design, and finance. This paper studies the coordination complexity of approximate price-directive decomposition PDD for this problem, i.e., the number of iterations required to solve the problem to a fixed relative accuracy as a function of K and M. First a simple PDD method based on the classical logarithmic potential is shown to be optimal up to a logarithmic factor in M in the class of all PDD methods that work with the original unrestricted blocks. It is then shown that logarithmic and exponential potentials generate a polylogarithmically-optimal algorithm for a wider class of PDD methods which can restrict the blocks by the coupling constraints. As an application, the fastest currently-known deterministic approximation algorithm for minimum-cost multicommodity flows is obtained.

[1]  Philip Wolfe,et al.  An algorithm for quadratic programming , 1956 .

[2]  Fillia Makedon,et al.  Fast approximation algorithms for multicommodity flow problems , 1991, STOC '91.

[3]  Farhad Shahrokhi,et al.  The maximum concurrent flow problem , 1990, JACM.

[4]  Éva Tardos,et al.  Fast Approximation Algorithms for Fractional Packing and Covering Problems , 1995, Math. Oper. Res..

[5]  Harvey J. Everett.  Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources , 1963 .

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

[7]  R. Polyak.  Smooth optimization methods for minimax problems , 1988 .

[8]  Leonid Khachiyan,et al.  Approximate minimum-cost multicommodity flows in $$\tilde O$$ (ɛ−2KNM) timetime , 1996, Math. Program..

[9]  J. Robinson.  AN ITERATIVE METHOD OF SOLVING A GAME , 1951, Classics in Game Theory.

[10]  Philip N. Klein,et al.  Leighton-Rao might be practical: faster approximation algorithms for concurrent flow with uniform capacities , 1990, STOC '90.

[11]  D. Bertsekas.  Approximation procedures based on the method of multipliers , 1977 .

[12]  H. N. Shapiro.  Note on a Computation Method in the Theory of Games , 1958 .

[13]  Luigi Fratta,et al.  The flow deviation method: An approach to store-and-forward communication network design , 1973, Networks.

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

[15]  Robert B Schnabel,et al.  Determining feasibility of a set of nonlinear inequality constraints , 1980 .

[16]  L. Khachiyan.  Convergence rate of the game processes for solving matrix games , 1977 .

[17]  James B. Orlin,et al.  A technique for speeding up the solution of the Lagrangean dual , 1994, Math. Program..

[18]  Leonid Khachiyan,et al.  An exponential-function reduction method for block-angular convex programs , 1995, Networks.

[19]  J. Goffin,et al.  Using central prices in the decomposition of linear programs , 1990 .

[20]  Robert R. Meyer,et al.  An Interior Point Method for Block Angular Optimization , 1991, SIAM J. Optim..

[21]  Leonid Khachiyan,et al.  Fast Approximation Schemes for Convex Programs with Many Blocks and Coupling Constraints , 1994, SIAM J. Optim..

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

[23]  Tomasz Radzik.  Fast deterministic approximation for the multicommodity flow problem , 1995, SODA '95.