Browse > Article

A B-spline based Branch & Bound Algorithm for Global Optimization  

Park, Sang-Kun (충주대학교 기계공학과)
Publication Information
Korean Journal of Computational Design and Engineering / v.15, no.1, 2010 , pp. 24-32 More about this Journal
Abstract
This paper introduces a B-spline based branch & bound algorithm for global optimization. The branch & bound is a well-known algorithm paradigm for global optimization, of which key components are the subdivision scheme and the bound calculation scheme. For this, we consider the B-spline hypervolume to approximate an objective function defined in a design space. This model enables us to subdivide the design space, and to compute the upper & lower bound of each subspace where the bound calculation is based on the LHS sampling points. We also describe a search tree to represent the searching process for optimal solution, and explain iteration steps and some conditions necessary to carry out the algorithm. Finally, the performance of the proposed algorithm is examined on some test problems which would cover most difficulties faced in global optimization area. It shows that the proposed algorithm is complete algorithm not using heuristics, provides an approximate global solution within prescribed tolerances, and has the good possibility for large scale NP-hard optimization.
Keywords
Branch & bound; B-spline hypervolume; Global optimization; LHS(Latin-hypercube sampling);
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 Neumaier, A., "Complete Search in Continuous Global Optimization and Constraint Satisfaction", pp. 1-99 in: Acta Numerica, Cambridge Univ. Press, 2004.
2 Le Thi Hoai An and Pham Dinh Tao, "A Branchand-Bound method via D.C. Optimization Algorithm and Ellipsoidal technique for Box Constrained Nonconvex Quadratic Programming Problems", Journal of Global Optimization, Vol. 13, pp. 171-206, 1998.   DOI   ScienceOn
3 Jansson, C. and Knuppel, O., "Branch and Bound Algorithm for Bound Constrained Optimization Problems without Derivatives", Journal of Global Optimization, Vol. 7, pp. 297-333, 1995.   DOI
4 Audet, C., Hansen, P., Jaumard, B. and Savard, G., "A Branch and Cut Algorithm for Nonconvex Quadratically Constrained Quadratic Programming", Math. Programming, Vol. 87, pp. 131-152, 2000.   DOI
5 Forrest, S., "Genetic Algorithms: Principles of Natural Selection Applied to Computation, Science, Vol. 261, pp. 872-878, 1993.   DOI
6 http://www.cs.sandia.gov/opt/survey/main.html
7 Park, S, "A Rational B-spline Hypervolume for Multidimensional Multivariate Modeling", Journal of Mechanical Science and Technology, Vol. 23, no. 1967-1981, 2009.   DOI
8 Hansen, E. R., Global Optimization Using Interval Analysis, Marcel Dekkar, New York, NY, 1992.
9 Hamma, B., Viitanen, S. and Torn, A., "Parallel Continuous Simulated Annealing for Global Optimization", Optimization Methods and Software, Vol. 13, No.2, pp. 93-116, 2000.
10 Hansen, P., Jaumard, B. and Lu, S., "Global Optimization of Univariate Lipschitz Functions: I. Survey and Properties", Math. Programming, Vol. 55, No. 3, pp. 251-272, 1992.   DOI
11 http://www.mat.univie.ac.at/-neum/glopt.html
12 McKay, M. D., Conover, W. J. and Beckman, R. J., "A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code", Technometrics, Vol. 21, pp.239-245, 1979.   DOI   ScienceOn
13 Floudas, C. A., "Overview of aBB-based Approaches In Deterministic Global Optimization", Workshop on Global Optimization, Imperial College London, 1517 Dec., 2007.
14 Tuy, H., "D.C. Optimization: Theory, Methods and Algorithms", pp. 149-216 in: Handbook of Global Optimization, (R. Horst and P.M. Pardalos eds.), Kluwer, Dordrecht 1995.
15 Fletcher, R. and Leyffer, S., "Solving Mixed Integer Nonlinear Programs by Outer Approximation", Math. Programming, Vol. 66, pp. 327-349, 1994.   DOI   ScienceOn
16 Floudas, C. A. and Visweswaran, V., "A Primal-Relaxed Dual Global Optimization Approach", Journal of Optimization, Theory, and its Applications, Vol. 2, No.1, pp. 73-99, 1992.
17 Floudas, C. A., Deterministic Global Optimization: Theory, Methods and Applications, Kluwer Academic Publishers, 2000.
18 Horst, R. and Pardalos, P. M., Handbook of Global Optimization, Kluwer, Dordrecht, 1995.
19 박상근, "분산형 볼륨 데이터의 VNURBS기반 다중 잔차 근사법", 한국CAD/CAM학회 논문집, 제12권, 제 1호, pp.27-38, 2007.
20 Pinter, J. D., Global Optimization in Action, Kluwer, Dordrecht, 1996.
21 Huyer, W. and Neumaier, A., "Global Optimization by Multilevel Coordinate Search", Journal of Global Optimization, Vol. 14, pp. 331-355, 1999.   DOI
22 Rinnoy-Kan, A. H. G and Timmer, G. T., "Stochastic Global Optimization Methods. Part I: Clustering Methods", Math Programming, Vol. 39, No.1, pp.27-56, 1987.   DOI   ScienceOn
23 Al-Khayyal, F. A. and Sherali, H. D., "On Finitely Terminating Branch-and-bound Algorithms for Some Global Optimization Problems", SIAM J. Optimization, Vol. 10, pp. 1049-1057, 2000.   DOI   ScienceOn
24 Stein, M., "Large Sample Properties of Simulations Using Latin Hypercube Sampling", Technometrics, Vol. 29, pp. 143-151, 1987.   DOI   ScienceOn