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

Abstract

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.

Cutting-Planes for Optimization of Convex Functions over Nonconvex Sets

“…The extension of lifting to the nonlinear or continuous setting is not new: see [17], [27], [34], and [7]. Also see [16], which lifts "tangent" inequalities to approximate multilinear functions.…”

“…The construction that culminates in Definition 2.5 is a generalization of the classical lifting construction in mixed-integer programming (see [44], [27], [34], [7]). The LFO inequality at y uses the local structure of P to strengthen the linearization inequality (2); the strengthening is only local, however the LFO inequality is (globally) valid.…”

Mixed Integer Linear Programming Formulation Techniques

“…Some recent work concerning piecewise linear functions that are not referenced in [155] include [151,51,57,124,62,123,126,127,43,156,153]. Finally, we note that there is also a vast literature on incorporating piecewise linear functions into optimization models without using MIP [17,54,55,56,38,99,158,47,169,46] Functions with bounded MIP representable graphs and epigraphs include most piecewise linear functions. In particular, for continuous multivariate functions with bounded domain we can give the following precise characterization.…”

Global optimization of non-convex piecewise linear regression splines

“…They can be solved with algorithms such as the one proposed by Keha et al [89], a branch-and-cut algorithm without auxiliary binary variables for solving non-convex separable piecewise linear optimization problems that uses cuts and applies SOS2 branching. They can also be modeled as mixed integer programming (MIP) problems, following the work shown by Croxton et al [90], where it was demonstrated that the linear programming relaxation of three textbook mixedinteger programming models for non-convex piecewise linear minimization problems are equivalent, each approximating the cost function with its lower convex envelope.…”