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Clash of MINLP Relaxations: Piecewise Linear vs. Global Parabolic

Adrian Göß

Abstract

Solving mixed-integer nonlinear programs (MINLPs) typically relies on constructing relaxations that are easier to tackle than the original problem. Recently, global parabolic (PARA) relaxations were introduced, featuring separable quadratic functions -- paraboloids -- as global under- or overestimators of general nonlinear constraint functions. So far, the paraboloids are all computed at once by solving a mixed-integer linear program (MIP). For small tolerances or wide function domains, the corresponding MIP grows in size and is eventually intractable, reventing a meaningful comparison with established relaxation techniques. We therefore propose a novel iterative method to compute PARA approximations that succeeds on all tolerance-domain combinations where the original one has failed. The computational study is preceded by a thorough theoretical explanation and analysis. Finally, the improved method enables a computational comparison with piecewise linear (PWL) relaxations in terms of runtime on general MINLP instances. The results show that the modern solver SCIP can solve PWL relaxations faster when the tolerance is high, shifting strongly in favor of PARA for tighter tolerances. We attribute the effect to the difference in the corresponding problem size: PWL relaxations introduce binary variables to identify the active linear piece and their number grows with decreasing tolerance. PARA, on the other hand, does not require additional variables such that the dimension is maintained. For problems with at least one (co)sine constraint, the effect significantly amplifies. Thereby, for medium tolerances, PARA relaxations outperform SCIP stand-alone. Applied problems like alternating current optimal power flow (AC-OPF) feature such constraint types, leaving PARA a viable relaxation strategy.

Clash of MINLP Relaxations: Piecewise Linear vs. Global Parabolic

Abstract

Solving mixed-integer nonlinear programs (MINLPs) typically relies on constructing relaxations that are easier to tackle than the original problem. Recently, global parabolic (PARA) relaxations were introduced, featuring separable quadratic functions -- paraboloids -- as global under- or overestimators of general nonlinear constraint functions. So far, the paraboloids are all computed at once by solving a mixed-integer linear program (MIP). For small tolerances or wide function domains, the corresponding MIP grows in size and is eventually intractable, reventing a meaningful comparison with established relaxation techniques. We therefore propose a novel iterative method to compute PARA approximations that succeeds on all tolerance-domain combinations where the original one has failed. The computational study is preceded by a thorough theoretical explanation and analysis. Finally, the improved method enables a computational comparison with piecewise linear (PWL) relaxations in terms of runtime on general MINLP instances. The results show that the modern solver SCIP can solve PWL relaxations faster when the tolerance is high, shifting strongly in favor of PARA for tighter tolerances. We attribute the effect to the difference in the corresponding problem size: PWL relaxations introduce binary variables to identify the active linear piece and their number grows with decreasing tolerance. PARA, on the other hand, does not require additional variables such that the dimension is maintained. For problems with at least one (co)sine constraint, the effect significantly amplifies. Thereby, for medium tolerances, PARA relaxations outperform SCIP stand-alone. Applied problems like alternating current optimal power flow (AC-OPF) feature such constraint types, leaving PARA a viable relaxation strategy.
Paper Structure (25 sections, 5 theorems, 57 equations, 12 figures, 4 tables, 2 algorithms)

This paper contains 25 sections, 5 theorems, 57 equations, 12 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Mathematical Problem
  5. Piecewise Linear Model
  6. Global One-sided Parabolic Model
  7. Computing Parabolas Fast
  8. Method
  9. Inner Loop
  10. Case (a)
  11. Case (b)
  12. Case (c)
  13. Outer Loop
  14. Computational Results
  15. Comparison to Existing Method
  16. ...and 10 more sections

Key Result

Lemma 1

Let $f : \mathcal{D} \to \mathbb{R}$ be as above and $\mathrm{int}(\mathcal{D}) \neq \emptyset$. Further, let $\varepsilon > 0$ and $\mathcal{D}_{\text{loc}} \mathrel{{\mathop:}{=}} [\underline{t}, \overline{t}] \subset \mathcal{D}$ such that $\underline{x} < \underline{t} < \overline{t} < \overline

Figures (12)

  • Figure 1: para approximations of $\sin$ with $\varepsilon=0.1$ on $[0, \ell\pi]$ for ${l \in \{1, 2, 3\}}$ by 1, 4, 6 parabolas, respectively.
  • Figure 2: pwl approximations of $\sin$ on with $\varepsilon=0.1$ on $[0, \ell\pi]$ for ${l \in \{1, 2, 3\}}$ with 4, 8, 12 pieces, respectively.
  • Figure 3: para approximations of $\ln$ with $\varepsilon = 0.1$ on $[e^{-4}, e^{2\ell}]$ for ${l \in \{-1, 0, 1\}}$ with 3, 7, 13 parabolas, respectively.
  • Figure 4: pwl approximations of $\ln$ with $\varepsilon = 0.1$ on $[e^{-4}, e^{2\ell}]$ for ${l \in \{-1, 0, 1\}}$ with 4, 7, 10 pieces, respectively.
  • Figure 5: Number (#) of instances solved faster by para or pwl per tolerance $\varepsilon$.
  • ...and 7 more figures

Theorems & Definitions (11)

  • Lemma 1: Bounds on $a$
  • proof
  • Remark 2
  • theorem 3: Correctness of Method \ref{['alg:subroutine']}
  • proof
  • theorem 4: Correctness of Method \ref{['alg:main']}
  • proof
  • theorem 4: Correctness of Method \ref{['alg:subroutine']}
  • proof
  • theorem 4: Correctness of Method \ref{['alg:main']}
  • ...and 1 more