We address optimization of nonlinear functions of the form , where is a

nonlinear function, W is a d × n matrix, and feasible x are in some large finite set of integer points

in . Generally, such problems are intractable, so we obtain positive algorithmic results by looking

at broad natural classes of f , W and .

One of our main motivations is multi-objective discrete optimization, where f trades off the linear

functions given by the rows of W . Another motivation is that we want to extend as much as possible

the known results about polynomial-time linear optimization over trees, assignments, matroids,

polymatroids, etc. to nonlinear optimization over such structures.

We assume that the convex hull of is well-described by linear inequalities (i.e., we have an efficient

separation oracle). For example, the set of characteristic vectors of common bases of a pair of matroids

on a common ground set satisfies this property for . In this setting, the problem is already known to

be intractable (even for a single matroid), for general f (given by a comparison oracle), for (i) d = 1

and binary-encoded W , and for (ii) d = n and W = I .

Our main results (a few technicalities suppressed):

1- When 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.

2-When is well described, f is a norm, and binary-encoded W is nonnegative, we give an efficient

deterministic constant-approximation algorithm for maximization.

3- When 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 constantapproximation

algorithm for minimization.

4- When 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.

By:* Y. Berstein; Jon Lee; S. Onn; R. Weismantel*

Published in: RC24610 in 2008

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