We study the modeling of nonconvex piecewise-linear functions as mixed-integer programming (MIP) problems. We review several new and existing MIP formulations for continuous piecewise-linear functions with special attention paid to multivariate nonseparable functions. We compare these formulations with respect to their theoretical properties and their relative computational performance. In addition, we study the extension of these formulations to lower semicontinuous piecewise-linear functions.

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Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions

Many combinatorial constraints over continuous variables such as SOS1 and SOS2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of a finite number of specially structured polyhedra. Known mixed integer binary formulations for these constraints have a number of binary variables and extra constraints linear in the number of polyhedra. We give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints logarithmic in the number of polyhedra. Using these conditions we introduce mixed integer binary formulations for SOS1 and SOS2 constraints that have a number of binary variables and extra constraints logarithmic in the number of continuous variables. We also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints logarithmic in the number of linear pieces of the functions. We prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations.

Modeling disjunctive constraints with a logarithmic number of binary variables and constraints

Many combinatorial constraints over continuous variables such as SOS1 and SOS2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of m specially structured polyhedra. Known mixed integer binary formulations for these constraints have a number of binary variables and extra constraints that is linear in m. We give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints that is logarithmic in m. Using these conditions we introduce the first mixed integer binary formulations for SOS1 and SOS2 constraints that use a number of binary variables and extra constraints that is logarithmic in the number of continuous variables. We also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints that is logarithmic in the number of linear pieces of the functions. We prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations.

Inventory placement in acyclic supply chain networks

We study mixed-integer programming formulations, based upon variable disaggregation, for generic multicommodity network flow problems with nonconvex piecewise linear costs, a problem class that arises frequently in many application domains in telecommunications, transportation, and logistics. We present several structural results for these formulations, and we analyze the results of extensive experiments on a large set of instances with various characteristics. In particular, we show that the linear programming relaxation of an extended disaggregated model approximates the objective function by its lower convex envelope in the space of commodity flows. Together, the theoretical and computational results allow us to suggest which formulation might be the most appropriate, depending on the characteristics of the problem instances.

Variable Disaggregation in Network Flow Problems with Piecewise Linear Costs

We show how to approximate a separable concave minimization problem over a general closed ground set by a single piecewise linear minimization problem. The approximation is to arbitrary 1+e precision in optimal cost. For polyhedral ground sets in \(\mathbb{R}^n_+\) and nondecreasing cost functions, the number of pieces is polynomial in the input size and proportional to 1/log(1+e). For general polyhedra, the number of pieces is polynomial in the input size and the size of the zeroes of the concave objective components.

http://www.optimization-online.org/DB_FILE/2012/01/3316.pdf

Separable Concave Optimization Approximately Equals Piecewise Linear Optimization

We show how to approximate a separable concave minimization problem over a general closed ground set by a single piecewise linear minimization problem. The approximation is to arbitrary 1 + e precision in optimal cost. For polyhedral ground sets in R n + and nondecreasing cost functions, the number of pieces is polynomial in the input size and proportional to 1/ / log(1 + e). For general polyhedra, the number of pieces is polynomial in the input size and the size of the zeroes of the concave objective components. We illustrate our approach on the concave-cost uncapacitated multi-commodity flow problem. By formulating the resulting piecewise linear approximation problem as a fixed charge, mixed integer model and using a dual ascent solution procedure, we solve randomly generated instances to within five to twenty percent of guaranteed optimality. The problem instances contain up to 50 nodes, 500 edges, 1,225 commodities, and 1,250,000 flow variables.

Separable concave optimization approximately equals piecewise linear optimization

In the first part, we study the problem of minimizing a separable concave function over a polyhedron. We assume the concave functions are nonnegative nondecreasing on R+ , and the polyhedron is in Rn+ (these assumptions can be relaxed further under suitable technical conditions). We show how to approximate this problem to 1+e precision in optimal value by a piecewise linear minimization problem so that the number of resulting pieces is polynomial in the input size of the original problem and linear in 1/e. For several concave cost problems, the resulting piecewise linear problem can be reformulated as a classical combinatorial optimization problem. As a result of our bound, a variety of polynomial-time heuristics, approximation algorithms, and exact algorithms for classical combinatorial optimization problems immediately yield polynomial-time heuristics, approximation algorithms, and fully polynomial-time approximation schemes for the corresponding concave cost problems. For example, we obtain a new approximation algorithm for concave cost facility location, and a new heuristic for concave cost multicommodity flow. 
In the second part, we study several concave cost problems and the corresponding combinatorial optimization problems. We develop an algorithm design technique that yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the corresponding combinatorial optimization problem. Our technique preserves constant-factor approximation ratios as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms. For example, we obtain new approximation algorithms for concave cost facility location and concave cost joint replenishment, and a new exact algorithm for concave cost lot-sizing. 
In the third part, we study a real-time optimization problem arising in the operations of a leading internet retailer. The problem involves the assignment of orders that arrive via the retailer's website to the retailer's warehouses. We model it as a concave cost facility location problem, and employ existing primal-dual algorithms and approximations of concave cost functions to solve it. On past data, we obtain solutions on average within 1.5% of optimality, with running times of less than 100ms per problem. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)

Combinatorial optimization problems with concave costs

We study the problem of minimizing a nonnegative separable concave function over a compact feasible set. We approximate this problem to within a factor of 1+epsilon by a piecewise-linear minimization problem over the same feasible set. Our main result is that when the feasible set is a polyhedron, the number of resulting pieces is polynomial in the input size of the polyhedron and linear in 1/epsilon. For many practical concave cost problems, the resulting piecewise-linear cost problem can be formulated as a well-studied discrete optimization problem. As a result, a variety of polynomial-time exact algorithms, approximation algorithms, and polynomial-time heuristics for discrete optimization problems immediately yield fully polynomial-time approximation schemes, approximation algorithms, and polynomial-time heuristics for the corresponding concave cost problems. 
We illustrate our approach on two problems. For the concave cost multicommodity flow problem, we devise a new heuristic and study its performance using computational experiments. We are able to approximately solve significantly larger test instances than previously possible, and obtain solutions on average within 4.27% of optimality. For the concave cost facility location problem, we obtain a new 1.4991+epsilon approximation algorithm.

Dan Stratila

Papers

Separable Concave Optimization Approximately Equals Piecewise Linear Optimization

Separable concave optimization approximately equals piecewise linear optimization

Combinatorial optimization problems with concave costs

Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization