Nonlinear optimization for matroid intersection and extensions
Abstract
We address optimization of nonlinear functions of the form f(Wx), where f:d is a nonlinear function, W is a d× n matrix, and feasible x are in some large finite set F of integer points in n. One motivation is multi-objective discrete optimization, where f trades off the linear functions given by the rows of W. Another motivation is to extend known results about polynomial-time linear optimization over discrete structures to nonlinear optimization. We assume that the convex hull of F is well-described by linear inequalities. For example, the set of characteristic vectors of common bases of a pair of matroids on a common ground set. When F is well described, f is convex (or even quasiconvex), and W has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic algorithm for maximization. When F is well described, f is a norm, and binary-encoded W is nonnegative, we give an efficient deterministic constant-approximation algorithm for maximization. When F is well described, f is ``ray concave'' and non-decreasing, and W has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic constant-approximation algorithm for minimization. When F is the set of characteristic vectors of common bases of a pair of vectorial matroids on a common ground set, f is arbitrary, and W has a fixed number of rows and is unary encoded, we give an efficient randomized algorithm for optimization.