Models for representing piecewise linear cost functions

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Abstract

We study formulations of linear programs with piecewise linear objective functions with and without additional binary variables. We show that the two formulations without additional binary variables have the same LP bounds as those of the corresponding formulations with binary variables and therefore are preferable for efficient computation.

Introduction

Piecewise linear functions (PLFs) are widely used to approximate nonlinear functions. Any arbitrary continuous function of one variable can be approximated by a PLF, with the quality of the approximation controlled by the size of the linear segments. However, when minimizing (maximizing) a PLF, it is necessary to introduce nonlinearities in the model if the function is not convex (concave). One way to formulate these nonlinearities is by adding binary variables and new inequalities to the model, yielding a mixed-integer program (MIP), see, for example, [17]. Two well-known MIP formulations for PLFs are the incremental cost [16] and the convex combination [6] formulations. The polyhedra of both MIPs were studied in [15], [20].

Recently, Padberg [18] compared the linear programming (LP) relaxations of the two MIP models for PLFs in the simplest case when there are no constraints. He showed that the feasible set of the LP relaxation of the incremental cost formulation is integral, i.e. the binary variables are integer in every vertex of the set. He called such formulations locally ideal. On the other hand, the convex combination formulation is not locally ideal, and it strictly contains the feasible set of the LP relaxation of the incremental cost formulation. Shortly after, Sherali [19] proposed a modification of the convex combination formulation that is locally ideal.

Alternatively, Beale and Tomlin [3] suggested a formulation for PLFs similar to convex combination, except that no binary variables are included in the model and the nonlinearities are enforced algorithmically, directly in the branch-and-bound algorithm, by branching on sets of variables, which they called special ordered sets of type 2 (SOS2). It is also possible to formulate PLFs similar to incremental cost but without binary variables and enforcing the nonlinearities directly in the branch-and-bound algorithm. Two advantages of eliminating binary variables are the substantial reduction in the size of the model and the use of the problem structure, see [10]. Beale and Tomlin's approach has been applied in other contexts, see, for example, [1], [2], [4], [7], [8], [9], [11], [12], [13], [14].

In this paper, we show that the LP relaxations of both MIP formulations produce the same bound, which is a result recently obtained independently by Croxton et al. [5]. Then we show our main result that the bound remains the same when we remove the binary variables from the two formulations. We also show that both formulations correspond to the same polyhedron in the space of the continuous variables, and that all the vertices of the feasible sets of the corresponding LP relaxations are feasible. Because the continuous formulations are considerably smaller than the MIP formulations and one can take advantage of the structure of the problem by branching on sets of variables, we believe that a continuous formulation should be the model of choice here. However, neither continuous formulation seems to have any advantage over the other.

In the remainder of the paper we will use the term locally ideal in the broader sense to mean that a needed property like integrality or SOS2 is obtained for free. Thus, in this broader sense both the special ordered set formulation and the continuous formulation based on incremental cost are locally ideal.

In Section 2, we review the incremental cost and the convex combination formulations, and Padberg's result. In Section 3, we show that the bounds of both MIP formulations, as well as the bounds of the continuous formulations are the same. We also show that the two MIP formulations correspond to the same polyhedron in the space of the continuous variables, and that both continuous formulations are locally ideal.

Section snippets

MIP formulations

In this section, we review the incremental cost and the convex combination formulations for PLFs, and Padberg's result. Suppose we have a PLF f(x) specified by the points (ai,f(ai)), i∈{0,…,t}. Let ui=aiai−1 and gi=f(ai)−f(ai−1)∀i∈{1,…,t}. Then for any a0xat we havex=a0+i=1tyiandf(x)=f(a0)+i=1tgiuiyi,where 0⩽yi⩽ui∀i∈{1,…,t}, andyi+1=⋯=yt=0wheneveryi<ui,i∈{1,…,t−1}.To enforce (2), one can introduce binary variables zi, i∈{1,…,t−1}, and the constraintsu1z1⩽y1⩽u1,uizi⩽yi⩽uizi−1∀i∈{2,…,t−1}and

Two related relaxations and bound comparisons

In this section, we show that the LP bounds of both MIP formulations, as well as the bounds of the continuous formulations are the same. We also show that the two MIP formulations correspond to the same polyhedron in the space of the continuous variables, and that both continuous formulations are locally ideal.

Beale and Tomlin [3] suggested formulating PLFs with the λ variables and constraints (5) in the model, and branching on SOS2 to enforce (6). For example, if {λ0,…,λt} is SOS2, (λ̃0,…,λ̃t)

Conclusions

Since the formulations without binary variables give the same LP bound as those with binary variables, are locally ideal, and are more compact, they should be better models for PLFs than the MIP models. Moreover, when additional constraints are present, the structure of the models without binary variables can be dealt with explicitly in the space of the continuous variables through branching and cuts. Theorem 2 and Corollary 1 seem to indicate that there should be no difference between the

Acknowledgements

We are grateful to Manfred Padberg for providing comments on an earlier draft of the paper. This research was partially supported by NSF under grants DMI-0100020 and DMI-0121495.

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