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An adaptive radial basis algorithm (ARBF) for expensive black-box mixed-integer constrained global optimization
Department of Mathematics and Physics, Mälardalen University, Västerås, Sweden.
Department of Mathematics and Physics, Mälardalen University, Västerås, Sweden. (Division of Optimization)ORCID iD: 0000-0002-9881-4170
Tomlab Optimization Inc., Pullman, WA, USA.
2008 (English)In: Optimization and Engineering, ISSN 1389-4420, E-ISSN 1573-2924, Vol. 9, no 3, 311-339 p.Article in journal (Refereed) Published
Abstract [en]

Response surface methods based on kriging and radial basis function (RBF) interpolationhave been successfully applied to solve expensive, i.e. computationally costly,global black-box nonconvex optimization problems.In this paper we describe extensions of these methods to handlelinear, nonlinear, and integer constraints.In particular, algorithms for standard RBF and the new adaptive RBF (ARBF) aredescribed.Note, however, while the objective function may be expensive, we assumethat any nonlinear constraints are either inexpensive or are incorporatedinto the objective function via penalty terms.Test results are presented on standard test problems, both nonconvexproblems with linear and nonlinear constraints, and mixed-integernonlinear problems (MINLP). Solvers in the TOMLAB OptimizationEnvironment (http://tomopt.com/tomlab/) have been compared,specifically the three deterministic derivative-free solversrbfSolve, ARBFMIP and EGO with three derivative-based mixed-integernonlinear solvers, OQNLP, MINLPBB and MISQP, as well as the GENOsolver implementing a stochastic genetic algorithm. Results showthat the deterministic derivative-free methods compare well with thederivative-based ones, but the stochastic genetic algorithm solver isseveral orders of magnitude too slow for practical use.When the objective function for the test problems is costly to evaluate,the performance of the ARBF algorithm proves to be superior.

Place, publisher, year, edition, pages
Springer, 2008. Vol. 9, no 3, 311-339 p.
Keyword [en]
Global optimization, radial basis functions, response surface model, surrogate model, expensive function, CPU-intensive, optimization software, splines, mixed-integer nonlinear programming, nonconvex, derivative-free, black-box, linear constraints, nonlinear constraints
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-77076DOI: 10.1007/s11081-008-9037-3ISI: 000259577400002OAI: oai:DiVA.org:liu-77076DiVA: diva2:524839
Available from: 2012-05-04 Created: 2012-05-04 Last updated: 2017-12-07Bibliographically approved
In thesis
1. Models and Methods for Costly Global Optimization and Military Decision Support Systems
Open this publication in new window or tab >>Models and Methods for Costly Global Optimization and Military Decision Support Systems
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The thesis consists of five papers. The first three deal with topics within costly global optimization and the last two concern military decision support systems.

The first part of the thesis addresses so-called costly problems where the objective function is seen as a “black box” to which the input parameter values are sent and a function value is returned. This means in particular that no information about derivatives is available. The black box could, for example, solve a large system of differential equations or carry out   timeconsuming simulation, where a single function evaluation can take several hours! This is the reason for describing such problems as costly and why they require customized algorithms. The goal is to construct algorithms that find a (near)-optimal solution using as few function evaluations as possible. A good example of a real life application comes from the automotive industry, where the development of new engines utilizes advanced mathematical models that are governed by a dozen key parameters. The objective is to optimize the engine by changing these parameters in such a way that it becomes as energy efficient as possible, but still meets all sorts of demands on strength and external constraints. The first three papers describe algorithms and implementation details for these costly global optimization problems.

The second part deals with military mission planning, that is, problems that concern logistics, allocation and deployment of military resources. Given a fleet of resource, the decision problem is to allocate the resources against the enemy so that the overall mission success is optimized. We focus on the problem of the attacker and consider two separate problem classes. In the fourth paper we introduce an effect oriented planning approach to an advanced weapon-target allocation problem, where the objective is to maximize the expected outcome of a coordinated attack. We present a mathematical model together with efficient solution techniques. Finally, in the fifth paper, we introduce a military aircraft mission planning problem, where an aircraft fleet should attack a given set of targets. Aircraft routing is an essential part of the problem, and the objective is to maximize the expected mission success while minimizing the overall mission time. The problem is stated as a generalized vehicle routing model with synchronization and precedence side constraints.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2012. 39 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1450
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-77078 (URN)978-91-7519-891-0 (ISBN)
Public defence
2012-06-04, C3, C-huset, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2012-05-04 Created: 2012-05-04 Last updated: 2015-02-25Bibliographically approved
2. Algorithms for Costly Global Optimization
Open this publication in new window or tab >>Algorithms for Costly Global Optimization
2009 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

There exists many applications with so-called costly problems, which means that the objective function you want to maximize or minimize cannot be described using standard functions and expressions. Instead one considers these objective functions as \black box" where the parameter values are sent in and a function value is returned. This implies in particular that no derivative information is available.

The reason for describing these problems as expensive is that it may take a long time to calculate a single function value. The black box could, for example, solve a large system of dierential equations or carrying out a heavy simulation, which can take anywhere from several minutes to several hours!

These very special conditions therefore requires customized algorithms. Common optimization algorithms are based on calculating function values every now and then, which usually can be done instantly. But with an expensive problem, it may take several hours to compute a single function value. Our main objective is therefore to create algorithms that exploit all available information to the limit before a new function value is calculated. Or in other words, we want to nd the optimal solution using as few function evaluations as possible.

A good example of real life applications comes from the automotive industry, where the development of new engines utilize advanced models that are governed by a dozen key parameters. The goal is to optimize the model by changing the parameters in such a way that the engine becomes as energy ecient as possible, but still meets all sorts of demands on strength and external constraints.

Place, publisher, year, edition, pages
Västerås: Mälardalen University Press, 2009. 107 p.
Series
Mälardalen University Press Licentiate Thesis, ISSN 1651-9256 ; 105
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-114511 (URN)978-91-86135-29-4 (ISBN)
Presentation
2009-09-03, Gamma, Hus U, Högskoleplan 1, Mälardalens högskola, Västerås, 13:15 (English)
Opponent
Supervisors
Available from: 2015-02-26 Created: 2015-02-25 Last updated: 2015-02-26Bibliographically approved

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