Constructing composite search directions with parameters in quadratic interpolation models. (English) Zbl 1230.90148
Summary: In this paper, we introduce the weighted composite search directions to develop the quadratic approximation methods. The purpose is to make fully use of the information disclosed by the former steps to construct possibly more promising directions. Firstly, we obtain these composite directions based on the properties of simplex methods and use them to construct trust region subproblems. Then, these subproblems are solved in the algorithm to find solutions of some benchmark optimization problems. The computation results show that for most tested problems, the improved quadratic approximation methods can obviously reduce the number of function evaluations compared with the existing ones. Finally, we conclude that the algorithm will perform better if the composite directions approach the previous steepest descent direction of the sub-simplex so far. We also point out the potential applications of this improved quadratic interpolation method in business intelligence systems.
MSC:
| 90C20 | Quadratic programming |
Keywords:
unconstrained optimization; trust region method; quadratic model; simplex methods; business intelligence systemsReferences:
| [1] | Garcia-Palomares U.M., Gonzalez-Castaño F.J., Burguillo-Rial J.C.: A combined global & local search approach to global optimization. J. Glob. Optim. 34, 409–426 (2006) · Zbl 1149.90426 · doi:10.1007/s10898-005-3249-2 |
| [2] | Vaz A.I.F., Vicente L.N.: A particle swarm pattern search method for bound constrained global optimization. J. Glob. Optim. 39, 197–219 (2007) · Zbl 1180.90252 · doi:10.1007/s10898-007-9133-5 |
| [3] | Pardalos, P.M., Resende, M. (eds): Handbook of Applied Optimization. Oxford University Press, Oxford (2002) · Zbl 0996.90001 |
| [4] | Winfield, D.: Function and functional optimization by interpolation in data tables. Ph.D Thesis, Hardvard University, Cambridge, USA (1969) |
| [5] | Winfield D.: Function minimization by interpolation in a data table. J. Inst. Math. Appl. 12, 339–347 (1973) · Zbl 0274.90060 · doi:10.1093/imamat/12.3.339 |
| [6] | Powell M.J.D.: A new algorithm for unconstrained optimization. In: Rosen, J.B., Mangasarian, O.L., Ritter, K. (eds) Nonlinear Programming, Academic Press, New York (1970) · Zbl 0228.90043 |
| [7] | Yuan Y.X.: Conditions for convergence of trust region algorithms for nonsmooth optimization. Math. Program. 31, 220–228 (1985) · Zbl 0576.90080 · doi:10.1007/BF02591750 |
| [8] | Yuan Y.X.: On the superlinear convergence of a trust region algorithm fornonsmooth optimization. Math. Program. 31, 269–285 (1985) · Zbl 0577.90066 · doi:10.1007/BF02591949 |
| [9] | Du J.Z., Pardalos P.M., Wu W.L.: Mathematical Theory of Optimization. Kluwer, Dordrecht (2001) |
| [10] | Lucia A., Dimaggio P.A., Depa P.: A geometric terrain methodology for global optimization. J. Glob. Optim. 29, 297–314 (2004) · Zbl 1133.90380 · doi:10.1023/B:JOGO.0000044771.25100.2d |
| [11] | Powell M.J.D.: UOBYQA: unconstrained optimization by quadratic approximation. Math. Program. 92, 555–582 (2002) · Zbl 1014.65050 · doi:10.1007/s101070100290 |
| [12] | Powell M.J.D.: On the use of quadratic models in unconstrained minimization without derivatives. Optim. Methods Softw. 19, 399–411 (2004) · Zbl 1140.90513 · doi:10.1080/10556780410001661450 |
| [13] | Han L.X., Liu G.H.: On the convergence of the UOBYQA method. J. Appl. Math. Comput. 16(1–2), 125–142 (2004) · Zbl 1129.90380 · doi:10.1007/BF02936156 |
| [14] | Berghen F.V., Bersini H.: CONDOR, a new parallel, constrained extension of Powell’s UOBYQA algorithm: experimental results and comparison with the DFO algorithm. J. Comput. Appl. Math. 181(1), 157–175 (2005) · Zbl 1072.65088 · doi:10.1016/j.cam.2004.11.029 |
| [15] | Zhou Q.H.: On the use of simplex methods in constructing quadratic models. Sci. China Ser. A Math. 50(7), 913–924 (2007) · Zbl 1125.90064 · doi:10.1007/s11425-007-0054-z |
| [16] | Torczon, V.: Multi-directional search: a direct search algorithm for parallel machines. Ph.D Thesis, Rice University, Houston, TX, USA (1989) |
| [17] | Moré J.J., Garbow B.S., Hillstorm K.E.: Testing unconstrained optimization software. ACM Trans. Math. Softw. 7, 17–41 (1981) · Zbl 0454.65049 · doi:10.1145/355934.355936 |
| [18] | Moré J.J., Sorensen D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4, 553–572 (1983) · Zbl 0551.65042 · doi:10.1137/0904038 |
| [19] | Parlett B.N.: The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs (1980) · Zbl 0431.65017 |
| [20] | Vercellis C.: Business Intelligence. Wiley, New York (2009) · Zbl 1169.90382 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
