Global optimization with spline constraints: a new branch-and-bound method based on B-splines

Grimstad, Bjarne; Sandnes, Anders T.

doi:10.1007/s10898-015-0358-4

Cited by 17 publications

(17 citation statements)

References 77 publications

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“…This formulation was also solved using BARON. We further include the results from solving the test problems using the special-purpose spline solver CENSO, which solves spline constrained problems as NLPs using a spatial branch-and-bound algorithm [16].…”

Section: Numerical Studymentioning

confidence: 99%

“…Comparing the mean solve times, we see that MINLP-EXP outperforms the other MIP formulations, including the MIQCP-CUT formulation, on all three test sets. CENSO [16] had the lowest solve time on all problems.…”

Section: Minimization Of Randomly Generated Pwpsmentioning

confidence: 99%

“…The availability of spline-compatible optimization algorithms and codes is limited. In a recent work, [16] develop a spatial branch-and-bound (sBB) algorithm for global optimization of spline-constrained problems. While the algorithm was shown to be highly efficient, it has only support for a limited set of algebraic functions, is only available as a specialized code and requires software for spline generation [17].…”

Section: Introductionmentioning

confidence: 99%

“…While the algorithm was shown to be highly efficient, it has only support for a limited set of algebraic functions, is only available as a specialized code and requires software for spline generation [17]. To address the comparably high modeling and implementation effort required for using the specialized sBB algorithm of [14,16] proposed an explicit constraint-formulation for continuous splines, yielding an ad-hoc mixedinteger quadratically constrained programming (MIQCP) model. In this paper, we build upon and significantly extend [14,16] to construct a general-purpose framework for mathematical programming of piecewise polynomial constraints, subsuming spline constraints.…”

Section: Introductionmentioning

confidence: 99%

“…To address the comparably high modeling and implementation effort required for using the specialized sBB algorithm of [14,16] proposed an explicit constraint-formulation for continuous splines, yielding an ad-hoc mixedinteger quadratically constrained programming (MIQCP) model. In this paper, we build upon and significantly extend [14,16] to construct a general-purpose framework for mathematical programming of piecewise polynomial constraints, subsuming spline constraints. The framework is based on an epigraph formulation and we show how it accommodates lower semi-continuous PWPs given in the monomial, Bernstein or B-spline basis.…”

Section: Introductionmentioning

confidence: 99%

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Mathematical programming formulations for piecewise polynomial functions

Grimstad¹,

Knudsen

2020

J Glob Optim

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This paper studies mathematical programming formulations for solving optimization problems with piecewise polynomial (PWP) constraints. We elaborate on suitable polynomial bases as a means of efficiently representing PWPs in mathematical programs, comparing and drawing connections between the monomial basis, the Bernstein basis, and B-splines. The theory is presented for both continuous and semi-continuous PWPs. Using a disjunctive formulation, we then exploit the characteristic of common polynomial basis functions to significantly reduce the number of nonlinearities, and to suggest a bound-tightening technique for PWP constraints. We derive several extensions using Bernstein cuts, an expanded Bernstein basis, and an expanded monomial basis, which upon a standard big-M reformulation yield a set of new MINLP models. The formulations are compared by globally solving six test sets of MINLPs and a realistic petroleum production optimization problem. The proposed framework shows promising numerical performance and facilitates the solution of PWP-constrained optimization problems using standard MINLP software.

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Section: Numerical Studymentioning

confidence: 99%

Section: Minimization Of Randomly Generated Pwpsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mathematical programming formulations for piecewise polynomial functions

Grimstad¹,

Knudsen

2020

J Glob Optim

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Machine Learning in Chemical Engineering: A Perspective

Schweidtmann

Esche

Fischer

et al. 2021

Chemie Ingenieur Technik

129

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The transformation of the chemical industry to renewable energy and feedstock supply requires new paradigms for the design of flexible plants, (bio-)catalysts, and functional materials. Recent breakthroughs in machine learning (ML) provide unique opportunities, but only joint interdisciplinary research between the ML and chemical engineering (CE) communities will unfold the full potential. We identify six challenges that will open new methods for CE and formulate new types of problems for ML: (1) optimal decision making, (2) introducing and enforcing physics in ML, (3) information and knowledge representation, (4) heterogeneity of data, (5) safety and trust in ML applications, and (6) creativity. Under the umbrella of these challenges, we discuss perspectives for future interdisciplinary research that will enable the transformation of CE.

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Hyper-arc consistency of polynomial constraints over finite domains using the modified Bernstein form

Bergenti

Monica

2017

Ann Math Artif Intell

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Global optimization with spline constraints: a new branch-and-bound method based on B-splines

Cited by 17 publications

References 77 publications

Mathematical programming formulations for piecewise polynomial functions

Mathematical programming formulations for piecewise polynomial functions

Machine Learning in Chemical Engineering: A Perspective

Hyper-arc consistency of polynomial constraints over finite domains using the modified Bernstein form

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