Reformulations for utilizing separability when solving convex MINLP problems

Several deterministic methods for convex mixed integer nonlinear programming generate a polyhedral approximation of the feasible region, and utilize this approximation to obtain trial solutions. Such methods are, e.g., outer approximation, the extended cutting plane method and the extended supportin...

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Vydané v:Journal of global optimization Ročník 71; číslo 3; s. 571 - 592
Hlavní autori: Kronqvist, Jan, Lundell, Andreas, Westerlund, Tapio
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York Springer US 01.07.2018
Springer
Springer Nature B.V
Predmet:
Approximation
Computer Science
Mathematics
Mathematics and Statistics
Mixed integer
Nonlinear programming
Norms
Operations Research/Decision Theory
Optimization
Real Functions
Solvers
Transformations (mathematics)
Lifted polyhedral approximation
Extended cutting plane algorithm
Convex MINLP
Outer approximation
Extended supporting hyperplane algorithm
Separable MINLP
ISSN:0925-5001, 1573-2916
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Shrnutí:Several deterministic methods for convex mixed integer nonlinear programming generate a polyhedral approximation of the feasible region, and utilize this approximation to obtain trial solutions. Such methods are, e.g., outer approximation, the extended cutting plane method and the extended supporting hyperplane method. In order to obtain the optimal solution and verify global optimality, these methods often require a quite accurate polyhedral approximation. In case the nonlinear functions are convex and separable to some extent, it is possible to obtain a tighter approximation by using a lifted polyhedral approximation, which can be achieved by reformulating the problem. We prove that under mild assumptions, it is possible to obtain tighter linear approximations for a type of functions referred to as almost additively separable. Here it is also shown that solvers, by a simple reformulation, can benefit from the tighter approximation, and a numerical comparison demonstrates the potential of the reformulation. The reformulation technique can also be combined with other known transformations to make it applicable to some nonseparable convex functions. By using a power transform and a logarithmic transform the reformulation technique can for example be applied to p -norms and some convex signomial functions, and the benefits of combining these transforms with the reformulation technique are illustrated with some numerical examples.
Bibliografia:ObjectType-Article-1
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-018-0616-3