POLYNOMIAL METHODS FOR SEPARABLE CONVEX-OPTIMIZATION IN UNIMODULAR LINEAR-SPACES WITH APPLICATIONS

We consider the problem of minimizing a separable convex objective fun ction over the linear space given by a system Mt = a with M a totally unimodular matrix. In particular, this generalizes the usual minimum l inear cost circulation and cocirculation problems in a network and the problems of determining the Euclidean distance from a point to the pe rfect bipartite matching polytope and the feasible flows polyhedron. W e first show that the idea of minimum mean cycle canceling originally worked out for linear cost circulations by Goldberg and Tarjan [J. Ass oc. Comput. Mach., 36 (1989), pp. 873-886.] and extended to some other problems [T. R. Ervolina and S. T. McCormick, Discrete Appl. Math, 46 (1993), pp. 133-165], [A. Frank and A. V. Karzanov, Technical Report RR 895-M, Laboratoire ARTEMIS IMAG, Universite Joseph Fourier, Grenobl e, France, 1992], [T. Ibaraki, A. V. Karzanov, and H. Nagamochi, priva te communication, 1993], [M. Hadjiat, Technical Report, Groupe Intelli gence Artificielle, Faculte des Sciences de Luminy, Marseille, France, 1994] can be generalized to give a combinatorial method with geometri c convergence for our problem. We also generalize the computationally more efficient cancel-and-tighten method. We then consider objective f unctions that are piecewise linear, pure and piecewise quadratic, or p iecewise mixed linear and quadratic, and we show how both methods can be implemented to find exact solutions in polynomial time (strongly po lynomial in the piecewise linear case). These implementations are then further specialized for finding circulations and cocirculations in a network. We finish by showing how to extend our methods to find optima l integer solutions, to linear spaces of larger fractionality, and to the case when the objective functions are given by approximate oracles .