A Branch-and-Cut Algorithm Without Binary Variables for Nonconvex Piecewise Linear Optimization

We give a branch-and-cut algorithm for solving linear programs (LPs) with continuous separable piecewise-linear cost functions (PLFs). Models for PLFs use continuous variables in special-ordered sets of type 2 (SOS2). Traditionally, SOS2 constraints are enforced by introducing auxiliary binary varia...

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Bibliographic Details
Published inOperations research Vol. 54; no. 5; pp. 847 - 858
Main Authors Keha, Ahmet B, de Farias, Ismael R., Jr, Nemhauser, George L
Format Journal Article
LanguageEnglish
Published Linthicum, MD INFORMS 01.09.2006
Institute for Operations Research and the Management Sciences
Subjects
Algorithms
Applied sciences
Branch & bound algorithms
branch-and-cut
Coefficients
Continuous variables
Convexity
Exact sciences and technology
Heuristics
Hierarchies
Integer programming
Integers
Linear programming
Linear transformations
Mathematical analysis
Mathematical programming
Methods
Operational research and scientific management
Operational research. Management science
piecewise linear function
special-ordered set
Studies
Supply
Transportation
Transportation costs
United States
Convex hull
Non convex programming
branch-and-cut
piecewise linear function
Ordered set
Branch and cut method
Cost function
Linear programming
Mixed integer programming
Piecewise linearization
Piecewise-linear techniques
special-ordered set
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Summary:We give a branch-and-cut algorithm for solving linear programs (LPs) with continuous separable piecewise-linear cost functions (PLFs). Models for PLFs use continuous variables in special-ordered sets of type 2 (SOS2). Traditionally, SOS2 constraints are enforced by introducing auxiliary binary variables and other linear constraints on them. Alternatively, we can enforce SOS2 constraints by branching on them, thus dispensing with auxiliary binary variables. We explore this approach further by studying the inequality description of the convex hull of the feasible set of LPs with PLFs in the space of the continuous variables, and using the new cuts in a branch-and-cut scheme without auxiliary binary variables. We give two families of valid inequalities. The first family is obtained by lifting the convexity constraints. The second family consists of lifted cover inequalities. Finally, we report computational results that demonstrate the effectiveness of our cuts, and that branch-and-cut without auxiliary binary variables is significantly more practical than the traditional mixed-integer programming approach.
ISSN:0030-364X
1526-5463
DOI:10.1287/opre.1060.0277