Polynomial Methods for Separable Convex Optimization in Unimodular Linear Spaces with Applications

We consider the problem of minimizing a separable convex objective function over the linear space given by system Mx = 0 with M a totally unimodular matrix. In particular, this generalizes the usual minimum linear cost circulation and co-circulation problems in a network, and the problems of determining the Euclidean distance from a point to the perfect bipartite matching polytope and the feasible flows polyhedron. We first show that the idea of minimum mean cycle canceling originally worked out for linear cost circulations by Goldberg and Tarjan [5] and extended to some other problems [2, 4, 12, 7] can be generalized to give a combinatorial method with geometric convergence for our problem. We also generalize the computationally more efficient Cancel and Tighten method. We then consider objective functions that are piecewise linear, or pure and piecewise quadratic, or piecewise mixed linear and quadratic, and show how both methods can be implemented to find exact solutions in polynomial time (strongly polynomial in the piecewise linear case). These implementations are then further specialized for finding circulations and co-circulations in a network. We finish by showing how to extend our methods to find optimal integer solutions, to linear spaces of larger fractionality, and to the case when the objective functions are given by approximate