Optimal deterministic algorithm generation

A formulation for the automated generation of algorithms via mathematical programming (optimization) is proposed. The formulation is based on the concept of optimizing within a parameterized family of algorithms, or equivalently a family of functions describing the algorithmic steps. The optimizatio...

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Published in:	Journal of global optimization Vol. 71; no. 4; pp. 891 - 913
Main Authors:	Mitsos, Alexander, Najman, Jaromił, Kevrekidis, Ioannis G.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.08.2018 Springer Springer Nature B.V
Subjects:	Algorithms Computer memory Computer Science Mathematical programming Mathematics Mathematics and Statistics Nonlinear equations Operations Research/Decision Theory Optimization Real Functions Nonlinear equations Algorithms Optimal control Optimization
ISSN:	0925-5001, 1573-2916
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Abstract	A formulation for the automated generation of algorithms via mathematical programming (optimization) is proposed. The formulation is based on the concept of optimizing within a parameterized family of algorithms, or equivalently a family of functions describing the algorithmic steps. The optimization variables are the parameters—within this family of algorithms—that encode algorithm design: the computational steps of which the selected algorithms consist. The objective function of the optimization problem encodes the merit function of the algorithm, e.g., the computational cost (possibly also including a cost component for memory requirements) of the algorithm execution. The constraints of the optimization problem ensure convergence of the algorithm, i.e., solution of the problem at hand. The formulation is described prototypically for algorithms used in solving nonlinear equations and in performing unconstrained optimization; the parametrized algorithm family considered is that of monomials in function and derivative evaluation (including negative powers). A prototype implementation in GAMS is provided along with illustrative results demonstrating cases for which well-known algorithms are shown to be optimal. The formulation is a mixed-integer nonlinear program. To overcome the multimodality arising from nonconvexity in the optimization problem, a combination of brute force and general-purpose deterministic global algorithms is employed to guarantee the optimality of the algorithm devised. We then discuss several directions towards which this methodology can be extended, their scope and limitations.
AbstractList	A formulation for the automated generation of algorithms via mathematical programming (optimization) is proposed. The formulation is based on the concept of optimizing within a parameterized family of algorithms, or equivalently a family of functions describing the algorithmic steps. The optimization variables are the parameters—within this family of algorithms—that encode algorithm design: the computational steps of which the selected algorithms consist. The objective function of the optimization problem encodes the merit function of the algorithm, e.g., the computational cost (possibly also including a cost component for memory requirements) of the algorithm execution. The constraints of the optimization problem ensure convergence of the algorithm, i.e., solution of the problem at hand. The formulation is described prototypically for algorithms used in solving nonlinear equations and in performing unconstrained optimization; the parametrized algorithm family considered is that of monomials in function and derivative evaluation (including negative powers). A prototype implementation in GAMS is provided along with illustrative results demonstrating cases for which well-known algorithms are shown to be optimal. The formulation is a mixed-integer nonlinear program. To overcome the multimodality arising from nonconvexity in the optimization problem, a combination of brute force and general-purpose deterministic global algorithms is employed to guarantee the optimality of the algorithm devised. We then discuss several directions towards which this methodology can be extended, their scope and limitations.
Audience	Academic
Author	Kevrekidis, Ioannis G. Mitsos, Alexander Najman, Jaromił
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Cites_doi	10.1063/1.463486 10.1145/358527.358537 10.1137/0708071 10.1007/s10107-007-0185-6 10.1090/S0025-5718-05-01772-2 10.1002/nav.3800240307 10.1137/080717341 10.1007/BF01096455 10.1109/4235.585893 10.1007/s10107-005-0581-8 10.1007/s10898-007-9260-z 10.1007/s10898-011-9685-2 10.1007/BF01580665 10.1137/S0036142902419284 10.1007/s10107-015-0949-3 10.1007/s10898-014-0176-0 10.1007/BF00121749 10.1090/S0025-5718-1976-0416018-3 10.1080/10556788.2014.924514 10.1007/s10107-013-0653-0 10.1080/02331934.2010.527970 10.1073/pnas.1614734113 10.1145/236869.237086 10.1145/309847.309915 10.1145/2890491 10.1007/978-3-540-28633-2_17 10.1109/IPDPS.2008.4536400 10.1137/1.9780898719383
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Snippet	A formulation for the automated generation of algorithms via mathematical programming (optimization) is proposed. The formulation is based on the concept of...
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SubjectTerms	Algorithms Computer memory Computer Science Mathematical programming Mathematics Mathematics and Statistics Nonlinear equations Operations Research/Decision Theory Optimization Real Functions
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Title	Optimal deterministic algorithm generation
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