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Optimization techniques for modeling with piecewise-linear functions

Péter Dobrovoczki, Tamás Kis

TL;DR

This paper describes a simple heuristic to iteratively construct a triangulation with a small number of triangles, while decreasing the error of the piecewise-linear approximation.

Abstract

In this paper we aim to construct piecewise-linear (PWL) approximations for functions of multiple variables and to build compact mixed-integer linear programming (MILP) formulations to represent the resulting PWL function. On the one hand, we describe a simple heuristic to iteratively construct a triangulation with a small number of triangles, while decreasing the error of the piecewise-linear approximation. On the other hand, we extend known techniques for modeling PWLs in MILPs more efficiently than state-of-the-art methods permit. The crux of our method is that the MILP model is a result of solving some hard combinatorial optimization problems, for which we present heuristic algorithms. The effectiveness of our techniques is demonstrated by a series of computational experiments including a short-term hydropower scheduling problem

Optimization techniques for modeling with piecewise-linear functions

TL;DR

This paper describes a simple heuristic to iteratively construct a triangulation with a small number of triangles, while decreasing the error of the piecewise-linear approximation.

Abstract

In this paper we aim to construct piecewise-linear (PWL) approximations for functions of multiple variables and to build compact mixed-integer linear programming (MILP) formulations to represent the resulting PWL function. On the one hand, we describe a simple heuristic to iteratively construct a triangulation with a small number of triangles, while decreasing the error of the piecewise-linear approximation. On the other hand, we extend known techniques for modeling PWLs in MILPs more efficiently than state-of-the-art methods permit. The crux of our method is that the MILP model is a result of solving some hard combinatorial optimization problems, for which we present heuristic algorithms. The effectiveness of our techniques is demonstrated by a series of computational experiments including a short-term hydropower scheduling problem

Paper Structure

This paper contains 34 sections, 18 theorems, 24 equations, 10 figures, 10 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Review of related literature
  3. Piecewise-linear function fitting
  4. Biclique covers
  5. MILP formulations for PWL functions
  6. The procedure for computing a MILP-approximation of a nonlinear function
  7. Piecewise-linear function fitting
  8. Lipschitz-correction
  9. Error estimation with sampling
  10. Computational evaluation
  11. A MILP-formulation for
  12. Constructing the conflict hypergraph $\mathcal{H}^c_\mathcal{S}$
  13. Reducing the rank of $\mathcal{H}^c_{\mathcal{S}}$ by simplex splitting
  14. Resolving higher-rank conflicts by coloring a hypergraph
  15. Handling pairwise conflicts
  16. ...and 19 more sections

Key Result

Theorem 1

Let $f\colon\Omega\to\mathbb{R}$ be a Lipschitz-continuous function with Lipschitz-constant $L$, $\mathcal{T}$ a simplicial partitioning of $\Omega\subset\mathbb{R}^d$ and $\hat{f}$ the corresponding PWL interpolation of $f$. Then

Figures (10)

  • Figure 1: Flowchart of the PWL function fitting and the MILP formulation procedures.
  • Figure 2: PWL interpolation of $\sin(50\sqrt{x^2+(y-1/2)^2)}\cdot{\rm e}^{-10(x^2+(y-1/2)^2)}$ on the $\left[0,1\right]^2$ domain with absolute error $\leq 0.05$.
  • Figure 3: Convergence of the maximal absolute error of the PWL interpolation of the HPF in the process of the construction of triangulation a005.
  • Figure 4: Optimal schedule and unit commitment over triangulation a005 for June.
  • Figure A.1: Piecewise linear interpolation of $f_1$ on the $\left[0,1\right]^2$ domain with absolute error $\leq 0.005$.
  • ...and 5 more figures

Theorems & Definitions (37)

  • Definition 1: Definition 1 of vielma2011modeling
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Proposition 1
  • Remark 1
  • Theorem 3
  • Theorem 4
  • proof
  • ...and 27 more