On test sets for nonlinear integer maximization
Section snippets
Introduction and related work
Given a feasible point of an optimization problem , one important concern is to establish a set of points with which one can verify whether is optimal for . We refer to such a set as a test set. Usually, for a test set , we compare feasible elements of , via objective value, against . Our goal is to establish, in various settings, the existence of a finite test set . 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 that extends the coordinate-wise partial order on as follows: For a pair of vectors , we write and say that conforms to if and for , that is, and lie in the same orthant of , and each component of is bounded by the corresponding component of 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 be given. A function is -oriented superadditive if for all integral , we have
We note that the defining inequality is equivalent to 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 .
Definition 4.1 Let be a subset of and be a function. A point is -optimal for if for all vectors such that , we have . That is, there is no 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.
References (5)
- et al.
Optimality criterion for a class of nonlinear integer programs
Oper. Res. Lett.
(2004) Über die Auflösung linearer Gleichungen mit reellen Coefficienten
Math. Ann.
(1873)