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P-algorithm based on a simplicial statistical model of multimodal functions. (English) Zbl 1206.90131
Summary: A well-recognized one-dimensional global optimization method is generalized to the multidimensional case. The generalization is based on a multidimensional statistical model of multimodal functions constructed by generalizing computationally favorable properties of a popular one-dimensional model-the Wiener process. A simplicial partition of a feasible region is essential for the construction of the model. The basic idea of the proposed method is to search where improvements of the objective function are most probable; a probability of improvement is evaluated with respect to the statistical model. Some results of computational experiments are presented.

MSC:
 90C26 Nonconvex programming, global optimization
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References:
 [1] Bundfuss S, Dür M (2008) Algorithmic copositivity detection by simplicial partition. Linear Algebra Appl 428(7):1511–1523 · Zbl 1138.15007 [2] Calvin J, Žilinskas A (1999) On the convergence of the P-algorithm for one-dimensional global optimization of smooth functions. J Optim Theory Appl 102:479–495 · Zbl 0985.90075 [3] Calvin J, Žilinskas A (2000) On one-dimensional P-algorithm with convergence rate O(n(+delta)) for smooth functions. J Optim Theory Appl 106:297–307 · Zbl 0992.90053 [4] Dür M, Stix V (2005) Probabilistic subproblem selection in branch-and-bound algorithms. J Comput Appl Math 182(1):67–80 · Zbl 1078.65050 [5] Fishburn P (1964) Decision and value theory. Wiley, New York · Zbl 0149.16203 [6] Gonçalves E, Palhares R, Takahashi R, Mesquita R (2006) Algorithm 860: SimpleS–an extension of Freudenthal’s simplex subdivision. ACM Trans Math Softw 32:609–621 · Zbl 1230.65066 [7] Gutmann H (2001) A radial basis function method for global optimization. J Glob Optim 19:201–227 · Zbl 0972.90055 [8] Hansen P, Jaumard B (1995) Lipschitz optimization. In: Horst R, Pardalos PM (eds) Handbook of global optimization. Springer, Berlin, pp 404–493 · Zbl 0833.90105 [9] Holmström K (2008) An adaptive radial basis algorithm (ARBF) for expensive black-box global optimization. J Glob Optim 41:447–464 · Zbl 1152.90609 [10] Hooker J (1995) Testing heuristics: we have it all wrong. J Heuristics 1:33–45 · Zbl 0853.68155 [11] Horst R (1997) On generalised bisection of n-simplices. Math Comput 66:691–698 · Zbl 0863.51018 [12] Horst R, Tuy H (1996) Global optimization–deterministic approaches, 3rd edn. Springer, Berlin · Zbl 0867.90105 [13] Jones D (2001) A taxonomy of global optimization methods based on response surfaces. J Glob Optim 21:345–383 · Zbl 1172.90492 [14] Kushner H (1962) A versatile stochastic model of a function of unknown and time-varying form. J Math Anal Appl 5:150–167 · Zbl 0111.33001 [15] Mockus J (1988) Bayesian approach to global optimization. Kluwer, Dordrecht · Zbl 0513.62084 [16] Mockus J, Eddy W, Reklaitis G (1996) Bayesian heuristic approach to discrete and global optimization. Kluwer, Dordrecht · Zbl 0864.65036 [17] Paulavičius R, Žilinskas J (2007) Analysis of different norms and corresponding Lipschitz constants for global optimization in multidimensional case. Inf Technol Control 36(4):383–387 [18] Paulavičius R, Žilinskas J (2008) Improved Lipschitz bounds over multidimensional simplex. Math Model Anal 13(4):553–563 · Zbl 1182.90073 [19] Paulavičius R, Žilinskas J, Grothey A (2010) Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds. Optim Lett 4(2):173–183 · Zbl 1189.90203 [20] Pinter J (1996) Global optimization in action: Continuous and Lipschitz optimization: algorithms, implementations and applications. Kluwer, Dordrecht [21] Santler T, Wiliams B, Notz W (2003) The design and analysis of computer experiments. Springer, Berlin [22] Sergeyev Y, Kvasov D (2006) Global search based on efficient diagonal partitions and a set of Lipschitz constants. SIAM J Optim 16(3):910–937 · Zbl 1097.65068 [23] Stein M (1999) Interpolation of spatial data: some theory of kriging. Springer, Berlin · Zbl 0924.62100 [24] Strongin R, Sergeyev Y (2000) Global optimization with non-convex constraints. Kluwer, Dordrecht · Zbl 0987.90068 [25] Törn A, Žilinskas A (1989) Global optimization. Lect Notes Comput Sci 350:1–252 [26] Zhigljavsky A, Žilinskas A (2008) Stochastic global optimization. Springer, Berlin · Zbl 1136.90003 [27] Žilinskas A (1985) Axiomatic characterization of a global optimization algorithm and investigation of its search strategies. Oper Res Lett 4:35–39 · Zbl 0568.90082 [28] Žilinskas A (1992) A review of statistical models for global optimization. J Glob Optim 2:145–153 · Zbl 0768.90071 [29] Žilinskas J (2007) Reducing of search space of multidimensional scaling problems with data exposing symmetries. Inf Technol Control 36(4):377–382 [30] Žilinskas J (2008) Branch and bound with simplicial partitions for global optimization. Math Model Anal 13(1):145–159 · Zbl 1146.49029 [31] Žilinskas A, Žilinskas J (2002) Global optimization based on a statistical model and simplicial partitioning. Comput Math Appl 44(7):957–967 · Zbl 1047.90036
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