On test sets for nonlinear integer maximization

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Abstract

A finite test set for an integer optimization problem enables us to verify whether a feasible point attains the global optimum. In this paper, we establish several general results that apply to integer optimization problems with nonlinear objective functions.

Section snippets

Introduction and related work

Given a feasible point x of an optimization problem P, one important concern is to establish a set of points T=T(x,P) with which one can verify whether x is optimal for P. We refer to such a set T as a test set. Usually, for a test set T, we compare feasible elements of {x+t:tT}, via objective value, against x. Our goal is to establish, in various settings, the existence of a finite test set T. An initial feasible solution together with a description of a finite test set for any feasible

Finite test sets for feasible points

Define a partial order on Zn that extends the coordinate-wise partial order on Z+n as follows: For a pair of vectors u,vZn, we write uv and say that u conforms to v if |ui||vi| and uivi0 for i=1,,n, that is, u and v lie in the same orthant of Zn, and each component of u is bounded by the corresponding component of v in absolute value. Points with some zero components are in multiple orthants, but it is easy to see that is well defined.

Here and throughout the paper, we make heavy use

Oriented sub/superadditive functions

In this section we introduce the notion of oriented subadditive (and superadditive) functions and show how to manipulate these functions. The next definition makes precise what we mean by this.

Definition 3.1

Let X,D1,D2Rn be given. A function f:ZnR is (X,D1,D2)-oriented superadditive if for all integral xX,yD1,zD2, we have f(x+y+z)+f(x)f(x+y)+f(x+z).

We note that the defining inequality is equivalent to f(x+y+z)f(x)[f(x+y)f(x)]+[f(x+z)f(x)], which is perhaps more intuitive–the incremental value of

Oriented sub/superadditive integer maximization

Our goal in this section is to use properties of oriented sub/superadditivity to establish the existence of finite test sets for a broad class of nonlinear integer programming problems. In doing so we need the notion of local optimality with respect to a specific subset of Rn.

Definition 4.1

Let O be a subset of Rn and f:ZnR be a function. A point xF(A,b,l,u) is O-optimal for P(f(x),A,b,l,u) if for all vectors tZnL(A)O such that lx+tu, we have f(x+t)f(x). That is, there is no tO that is an

Acknowledgments

We wish to thank Kazuo Murota for his helpful comments on an earlier version of this paper. This research was carried out at the Mathematisches Forschungsinstitut Oberwolfach during a stay within the Research in Pairs Programme from February 26 to March 10, 2007. Research of the second author was also supported by the ISF–Israel Science Foundation. Research of the third author was also supported by the European TMR Network ADONET 504438 and by the DFG–German Science Foundation through FOR 468.

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