**Abstract**

A well-studied nonlinear extension of the minimum-cost flow problemis to minimize the objective \sum_{ij\in E} C_{ij}(f_{ij}) over feasible flows f, where on each arc ij of the network, C_{ij} is a convex function. We give a strongly polynomial algorithm for finding an exact optimal solution for a broad class of such problems, The class includes convex quadratic objectives; thereby we give the first strongly polynomial algorithms for separable convex quadratic

minimum-cost flows, settling a long-standing open question. Further applications include market equilibrium problems, in particular, we give the first strongly polynomial algorithm for Fisher's market with spending constraint utilities.