Abstract
The most common approach for solving constrained optimization problems is based on penalty functions, where the constrained problem is transformed into an unconstrained problem by penalizing the objective function when constraints are violated. In this paper, we analyze the implementation of penalty functions, within the DIRECT algorithm. In order to assess the applicability and performance of the proposed approaches, some benchmark problems from engineering design optimization are considered.
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References
Barbosa, H.J.C., Lemonge, A.C.C.: An adaptive penalty method for genetic algorithms in constrained optimization problems. In: Iba, H. (ed.) Frontiers in Evolutionary Robotics, pp. 9–34. I-Tech Education Publ., Austria (2008)
Bertsekas, D.P.: Constrained Optimization and Lagrange Multipliers Methods. Academic Press, New York (1982)
Carroll, C.W.: The created response surface technique for optimizing nonlinear restrained systems. Operations research 184, 9–169 (1961)
Carter, R.G., Gablonsky, J.M., Patrick, A., Kelley, C.T., Eslinger, O.J.: Algorithms for noisy problems in gas transmission pipeline optimization. Optimization and Engineering 2, 139–157 (2002)
Coello Coello, C.A.: Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Computer Methods in Applied Mechanics and Engineering 191, 1245–1287 (2002)
Costa, M.F.P., Fernandes, E.M.G.P.: Efficient solving of engineering design problems by an interior point 3-D filter line search method. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) AIP Conference Proceedings, vol. 1048, pp. 197–200 (2008)
Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations. Bulletin of the American Mathematical Society 49, 1–23 (1943)
Fiacco, A.V., McCormick, G.P.: Extensions of SUMT for nonlinear programming: equality constraints and extrapolation. Management Science 12(11), 816–828 (1966)
Finkel, D.E.: Global optimization with the DIRECT algorithm. PhD thesis, North Carolina state University (2005)
Finkel, D.E., Kelley, C.T.: Convergence Analysis of the DIRECT Algorithm. North Carolina State University: Center for Research in Scientific Computation, Raleigh (2004)
Gablonsky, J.M.: Modifications of the DIRECT algorithm. PhD Thesis, North Carolina State University, Raleigh, North Carolina (2001)
Henderson, S.G., Biller, B., Hsieh, M.-H., Shortle, J., Tew, J.D., Barton, R.R.: Extension of the direct optimization algorithm for noisy functions. In: Proceedings of the 2007 Winter Simulation Conference (2007)
Joines, J., Houck, C.: On the use of nonstationary penalty functions to solve nonlinear constrained optimization problems with GAs. In: Proceedings of the First IEEE Congress on Evolutionary Computation, Orlando, FL, pp. 579–584 (1994)
Jones, D.R.: The DIRECT Global Optimization Algorithm. The Encyclopedia of Optimization. Kluwer Academic (1999)
Lee, K.S., Geem, Z.W.: A new meta-heuristic algorithm for continuous engineering optimization: Harmony search theory and practice. Computer Methods in Applied Mechanics and Engineering 194, 3902–3933 (2005)
Lewis, R., Torczon, V.: A Globally Convergent Augmented Lagrangian Pattern Search Algorithm for Optimization with General Constraints and Simple Bounds. SIAM Journal on Optimization 4(4), 1075–1089 (2012)
Liu, J.-L., Lin, J.-H.: Evolutionary computation of unconstrained and constrained problems using a novel momentum-type particle swarm optimization. Engineering Optimization 39, 287–305 (2007)
Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operations Research (1999)
Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the lipschitz constant. Journal of Optimization Theory and Application 79(1), 157–181 (1993)
Petalas, Y.G., Parsopoulos, K.E., Vrahatis, M.N.: Memetic particle swarm optimization. Annals of Operations Research 156, 99–127 (2007)
Ray, T., Liew, K.M.: A swarm metaphor for multiobjective design optimization. Engineering Optimization 34(2), 141–153 (2002)
Rocha, A.M.A.C., Fernandes, E.M.G.P.: Hybridizing the electromagnetism-like algorithm with descent search for solving engineering design problems. International Journal of Computer Mathematics 86, 1932–1946 (2009)
Tessema, B., Yen, G.G.: A Self Adaptive Penalty Function Based Algorithm for Constrained Optimization. In: IEEE Congress on Evolutionary Computation, pp. 246–253 (2006)
Vilaça, R., Rocha, A.M.A.C.: An Adaptive Penalty Method for DIRECT Algorithm in Engineering Optimization. In: Simos, T.E., Psihoyios, G., Tsitouras, C., Anastassi, Z. (eds.) AIP Conference Proceedings, vol. 1479, pp. 826–829 (2012)
Vilaça, R.: Sofia Pinto: Extending the DIRECT algorithm to solve constrained nonlinear optimization problems: a case study. MSc Thesis, University of Minho (2012)
Xavier, A.E.: Hyperbolic penalty: a new method for nonlinear programming with inequalities. International Transactions in Operational Research 8, 659–671 (2001)
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Rocha, A.M.A.C., Vilaça, R. (2013). A Computational Study on Different Penalty Functions with DIRECT Algorithm. In: Murgante, B., et al. Computational Science and Its Applications – ICCSA 2013. ICCSA 2013. Lecture Notes in Computer Science, vol 7971. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39637-3_26
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DOI: https://doi.org/10.1007/978-3-642-39637-3_26
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