Nonlinear optimization for matroid intersection and extensions

Abstract

We address optimization of nonlinear functions of the form $f(Wx)$, where $f:\R^d\to \R$ is a nonlinear function, $W$ is a $d\times n$ matrix, and feasible $x$ are in some large finite set $F$ of integer points in $\R^n$. 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.