Polynomial methods for separable convex optimization in unimodular spaces
We consider the problem of minimizing a separable convex objective function over the linear space given by system Mx = 0 with a totally unimodular matrix M. In particular, this generalizes the usual minimum linear cost circulation and co-circulation problems in a network, and the problem of determining the Euclidean distance to certain polyhedra (e.g. 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 and extended to some other problems 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 specialized objective functions, such as piecewise linear, pure and piecewise quadratic, and 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. Our methods also extend to finding optimal integer solutions, to linear spaces of larger fractionality, and to the case when the objective functions are given by approximate oracles.
- OSTI ID:
- 36273
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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