2012, 2(1): 69-90. doi: 10.3934/naco.2012.2.69

Univariate geometric Lipschitz global optimization algorithms

1. 

DEIS, University of Calabria, Via P. Bucci, Cubo 42C, 87036 -- Rende (CS), Italy, Italy

Received  May 2011 Revised  August 2011 Published  March 2012

In this survey, univariate global optimization problems are considered where the objective function or its first derivative can be multiextremal black-box costly functions satisfying the Lipschitz condition over an interval. Such problems are frequently encountered in practice. A number of geometric methods based on constructing auxiliary functions with the usage of different estimates of the Lipschitz constants are described in the paper.
Citation: Dmitri E. Kvasov, Yaroslav D. Sergeyev. Univariate geometric Lipschitz global optimization algorithms. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 69-90. doi: 10.3934/naco.2012.2.69
References:
[1]

C. S. Adjiman, S. Dallwig, C. A. Floudas and A. Neumaier, A global optimization method, $\alpha$BB, for general twice-differentiable constrained NLPs - I., Theoretical advances,, Comput. Chem. Engng., 22 (1998), 1137. doi: 10.1016/S0098-1354(98)00027-1. Google Scholar

[2]

M. Yu. Andramonov, A. M. Rubinov and B. M. Glover, Cutting angle methods in global optimization,, Appl. Math. Lett., 12 (1999), 95. doi: 10.1016/S0893-9659(98)00179-7. Google Scholar

[3]

I. P. Androulakis, C. D. Maranas and C. A. Floudas, $\alpha$BB: A global optimization method for general constrained nonconvex problems,, J. Global Optim., 7 (1995), 337. doi: 10.1007/BF01099647. Google Scholar

[4]

C. Audet, P. Hansen and G. Savard, "Essays and Surveys in Global Optimization,", GERAD 25th Anniversary. Springer-Verlag, (2005). Google Scholar

[5]

A. M. Bagirov, A. M. Rubinov and J. Zhang, Local optimization method with global multidimensional search,, J. Global Optim., 32 (2005), 161. doi: 10.1007/s10898-004-2700-0. Google Scholar

[6]

W. Baritompa and A. Cutler, Accelerations for global optimization covering methods using second derivatives,, J. Global Optim., 4 (1994), 329. doi: 10.1007/BF01098365. Google Scholar

[7]

W. Baritompa, Customizing methods for global optimization - A geometric viewpoint,, J. Global Optim., 3 (1993), 193. doi: 10.1007/BF01096738. Google Scholar

[8]

W. Baritompa, Accelerations for a variety of global optimization methods,, J. Global Optim., 4 (1994), 37. doi: 10.1007/BF01096533. Google Scholar

[9]

K. A. Barkalov and R. G. Strongin, A global optimization technique with an adaptive order of checking for constraints,, Comput. Math. Math. Phys., 42 (2002), 1289. Google Scholar

[10]

M. C. Bartholomew-Biggs, Z. J. Ulanowski and S. Zakovic, Using global optimization for a microparticle identification problem with noisy data,, J. Global Optim., 32 (2005), 325. doi: 10.1007/s10898-004-1943-0. Google Scholar

[11]

P. Basso, Iterative methods for the localization of the global maximum,, SIAM J. Numer. Anal., 19 (1982), 781. doi: 10.1137/0719054. Google Scholar

[12]

G. Beliakov and A. Ferrer, Bounded lower subdifferentiability optimization techniques: Applications,, J. Global Optim., 47 (2010), 211. doi: 10.1007/s10898-009-9467-2. Google Scholar

[13]

D. P. Bertsekas, "Nonlinear Programming,", Athena Scientific, (1999). Google Scholar

[14]

B. Betrò, Bayesian methods in global optimization,, J. Global Optim., 1 (1991), 1. doi: 10.1007/BF00120661. Google Scholar

[15]

M. Björkman and K. Holmström, Global optimization of costly nonconvex functions using radial basis functions,, Optim. Eng., 1 (2000), 373. doi: 10.1023/A:1011584207202. Google Scholar

[16]

L. Breiman and A. Cutler, A deterministic algorithm for global optimization,, Math. Program., 58 (1993), 179. doi: 10.1007/BF01581266. Google Scholar

[17]

R. G. Carter, J. M. Gablonsky, A. Patrick, C. T. Kelley and O. J. Eslinger, Algorithms for noisy problems in gas transmission pipeline optimization,, Optim. Eng., 2 (2001), 139. doi: 10.1023/A:1013123110266. Google Scholar

[18]

L. G. Casado, I. García and Ya. D. Sergeyev, Interval algorithms for finding the minimal root in a set of multiextremal non-differentiable one-dimensional functions,, SIAM J. Sci. Comput., 24 (2002), 359. doi: 10.1137/S1064827599357590. Google Scholar

[19]

M. H. Chang, Y. C. Park, and T. Y. Lee, A new global optimization method for univariate constrained twice-differentiable NLP problems,, J. Global Optim., 39 (2007), 79. doi: 10.1007/s10898-006-9121-1. Google Scholar

[20]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley & Sons, (1983). Google Scholar

[21]

J. J. Cochran, "Wiley Encyclopedia of Operations Research and Management Science (8 Volumes),", Wiley, (2011). doi: 10.1002/9780470400531. Google Scholar

[22]

A. R. Conn, K. Scheinberg and L. N. Vicente, "Introduction to Derivative-Free Optimization,", SIAM, (2009). doi: 10.1137/1.9780898718768. Google Scholar

[23]

S. E. Cox, R. T. Haftka, C. A. Baker, B. Grossman, W. H. Mason and L. T. Watson, A comparison of global optimization methods for the design of a high-speed civil transport,, J. Global Optim., 21 (2001), 415. doi: 10.1023/A:1012782825166. Google Scholar

[24]

A. E. Csallner, T. Csendes and M. Cs. Markót, Multisection in interval branch-and-bound methods for global optimization - I. Theoretical results,, J. Global Optim., 16 (2000), 371. doi: 10.1023/A:1008354711345. Google Scholar

[25]

Yu. M. Danilin, Estimation of the efficiency of an absolute-minimum-finding algorithm,, USSR Comput. Math. Math. Phys., 11 (1971), 261. doi: 10.1016/0041-5553(71)90020-6. Google Scholar

[26]

V. F. Demyanov and V. N. Malozemov, "Introduction to Minimax,", John Wiley & Sons, (1974). Google Scholar

[27]

V. F. Demyanov and A. M. Rubinov, "Quasidifferential Calculus,", Optimization Software Inc., (1986). Google Scholar

[28]

S. M. Elsakov and V. I. Shiryaev, Homogeneous algorithms for multiextremal optimization,, Comput. Math. Math. Phys., 50 (2010), 1642. doi: 10.1134/S0965542510100027. Google Scholar

[29]

Yu. G. Evtushenko, V. U. Malkova and A. A. Stanevichyus, Parallel global optimization of functions of several variables,, Comput. Math. Math. Phys., 49 (2009), 246. doi: 10.1134/S0965542509020055. Google Scholar

[30]

Yu. G. Evtushenko, M. A. Posypkin and I. Kh. Sigal, A framework for parallel large-scale global optimization,, Comp. Sci. - Res. Dev., 23 (2009), 211. doi: 10.1007/s00450-009-0083-7. Google Scholar

[31]

Yu. G. Evtushenko and M. A. Posypkin, Coverings for global optimization of partial-integer nonlinear problems,, Doklady Mathematics, 83 (2011), 268. doi: 10.1134/S1064562411020074. Google Scholar

[32]

Yu. G. Evtushenko, Numerical methods for finding global extrema (Case of a non-uniform mesh),, USSR Comput. Math. Math. Phys., 11 (1971), 38. doi: 10.1016/0041-5553(71)90065-6. Google Scholar

[33]

Yu. G. Evtushenko, "Numerical Optimization Techniques,", Translations Series in Mathematics and Engineering. Springer-Verlag, (1985). doi: 10.1007/978-1-4612-5022-7. Google Scholar

[34]

D. E. Finkel and C. T. Kelley, Additive scaling and the DIRECT algorithm,, J. Global Optim., 36 (2006), 597. doi: 10.1007/s10898-006-9029-9. Google Scholar

[35]

R. Fletcher, "Practical Methods of Optimization,", John Wiley & Sons, (2000). Google Scholar

[36]

C. A. Floudas and C. E. Gounaris, A review of recent advances in global optimization,, J. Global Optim., 45 (2009), 3. doi: 10.1007/s10898-008-9332-8. Google Scholar

[37]

C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. Esposito, Z. Gümüs, S. Harding, J. Klepeis, C. Meyer and C. Schweiger, "Handbook of Test Problems in Local and Global Optimization,", Kluwer Academic Publishers, (1999). Google Scholar

[38]

C. A. Floudas and P. M. Pardalos, "Encyclopedia of Optimization (6 Volumes),", Kluwer Academic Publishers, (2001). Google Scholar

[39]

K. R. Fowler, J. P. Reese, C. E. Kees, J. E. Dennis Jr., C. T. Kelley, C. T. Miller, C. Audet, A. J. Booker, G. Couture, R. W. Darwin, M. W. Farthing, D. E. Finkel, J. M. Gablonsky, G. Gray and T. G. Kolda, Comparison of derivative-free optimization methods for groundwater supply and hydraulic capture community problems,, Adv. Water Res., 31 (2008), 743. doi: 10.1016/j.advwatres.2008.01.010. Google Scholar

[40]

J. M. Gablonsky and C. T. Kelley, A locally-biased form of the DIRECT algorithm, J. Global Optim., 21 (2001), 27. doi: 10.1023/A:1017930332101. Google Scholar

[41]

D. Y. Gao and H. D. Sherali, Canonical duality theory: Connection between nonconvex mechanics and global optimization,, In, (2009), 257. doi: 10.1007/978-0-387-75714-8_8. Google Scholar

[42]

D. Y. Gao, "Duality Principles in Nonconvex Systems: Theory, Methods, and Applications,", Kluwer Academic Publishers, (2000). Google Scholar

[43]

V. P. Gergel, A global search algorithm using derivatives,, In, (1992), 161. Google Scholar

[44]

V. A. Grishagin, Operating characteristics of some global search algorithms,, In, 7 (1978), 198. Google Scholar

[45]

I. E. Grossmann, "Global Optimization in Engineering Design,", Kluwer Academic Publishers, (1996). Google Scholar

[46]

H.-M. Gutmann, A radial basis function method for global optimization,, J. Global Optim., 19 (2001), 201. doi: 10.1023/A:1011255519438. Google Scholar

[47]

P. Hansen and B. Jaumard, Lipschitz optimization,, In, 1 (1995), 407. Google Scholar

[48]

E. M. T. Hendrix and B. G.-Toth, "Introduction to Nonlinear and Global Optimization,", Springer, (2010). Google Scholar

[49]

J. He, L. T. Watson, N. Ramakrishnan, C. A. Shaffer, A. Verstak, J. Jiang, K. Bae and W. H. Tranter, Dynamic data structures for a direct search algorithm,, Comput. Optim. Appl., 23 (2002), 5. doi: 10.1023/A:1019992822938. Google Scholar

[50]

J. B. Hiriart-Urruty and C. Lemaréchal, "Convex Analysis and Minimization Algorithms (Parts I and II),", Springer-Verlag, (1996). Google Scholar

[51]

R. Horst, P. M. Pardalos and N. V. Thoai, "Introduction to Global Optimization,", Kluwer Academic Publishers, (1995). Google Scholar

[52]

R. Horst and P. M. Pardalos, "Handbook of Global Optimization,", volume 1. Kluwer Academic Publishers, (1995). Google Scholar

[53]

R. Horst and H. Tuy, "Global Optimization - Deterministic Approaches,", Springer-Verlag, (1996). Google Scholar

[54]

R. Horst, Deterministic global optimization with partition sets whose feasibility is not known: Application to concave minimization, reverse convex constraints, DC-programming, and Lipschitzian optimization,, J. Optim. Theory Appl., 58 (1988), 11. doi: 10.1007/BF00939768. Google Scholar

[55]

V. V. Ivanov, On optimal algorithms for the minimization of functions of certain classes,, Cybernetics, 4 (1972), 81. Google Scholar

[56]

D. R. Jones, C. D. Perttunen and B. E. Stuckman, Lipschitzian optimization without the Lipschitz constant,, J. Optim. Theory Appl., 79 (1993), 157. doi: 10.1007/BF00941892. Google Scholar

[57]

D. R. Jones, M. Schonlau and W. J. Welch, Efficient global optimization of expensive black-box functions,, J. Global Optim., 13 (1998), 455. doi: 10.1023/A:1008306431147. Google Scholar

[58]

O. V. Khamisov, Global optimization of functions with a concave support minorant,, Comput. Math. Math. Phys., 44 (2004), 1473. Google Scholar

[59]

A. G. Korotchenko, An algorithm for seeking the maximum value of univariate functions,, USSR Comput. Math. Math. Phys., 18 (1978), 34. doi: 10.1016/0041-5553(78)90162-3. Google Scholar

[60]

D. E. Kvasov, C. Pizzuti and Ya. D. Sergeyev, Local tuning and partition strategies for diagonal GO methods,, Numer. Math., 94 (2003), 93. doi: 10.1007/s00211-002-0419-8. Google Scholar

[61]

D. E. Kvasov and Ya. D. Sergeyev, Multidimensional global optimization algorithm based on adaptive diagonal curves,, Comput. Math. Math. Phys., 43 (2003), 40. Google Scholar

[62]

D. E. Kvasov and Ya. D. Sergeyev, A univariate global search working with a set of Lipschitz constants for the first derivative,, Optim. Lett., 3 (2009), 303. doi: 10.1007/s11590-008-0110-9. Google Scholar

[63]

D. Lera and Ya. D. Sergeyev, An information global minimization algorithm using the local improvement technique,, J. Global Optim., 48 (2010), 99. doi: 10.1007/s10898-009-9508-x. Google Scholar

[64]

D. Lera and Ya. D. Sergeyev, Lipschitz and {H\"older global optimization using space-filling curves},, Appl. Numer. Math., 60 (2010), 115. doi: 10.1016/j.apnum.2009.10.004. Google Scholar

[65]

G. Liuzzi, S. Lucidi and V. Piccialli, A DIRECT-based approach exploiting local minimizations for the solution of large-scale global optimization problems,, Comput. Optim. Appl., 45 (2010), 353. doi: 10.1007/s10589-008-9217-2. Google Scholar

[66]

K. Ljungberg, S. Holmgren and Ö. Carlborg, Simultaneous search for multiple QTL using the global optimization algorithm DIRECT,, Bioinformatics, 20 (2004), 1887. doi: 10.1093/bioinformatics/bth175. Google Scholar

[67]

D. MacLagan, T. Sturge and W. Baritompa, Equivalent methods for global optimization,, In, (1996), 201. doi: 10.1007/978-1-4613-3437-8_13. Google Scholar

[68]

O. L. Mangasarian, "Nonlinear Programming,", McGraw-Hill, (1969). Google Scholar

[69]

C. D. Maranas and C. A. Floudas, Global minimum potential energy conformations of small molecules,, J. Global Optim., 4 (1994), 135. doi: 10.1007/BF01096720. Google Scholar

[70]

C. C. Meewella and D. Q. Mayne, An algorithm for global optimization of Lipschitz continuous functions,, J. Optim. Theory Appl., 57 (1988), 307. doi: 10.1007/BF00938542. Google Scholar

[71]

C. C. Meewella and D. Q. Mayne, Efficient domain partitioning algorithms for global optimization of rational and Lipschitz continuous functions,, J. Optim. Theory Appl., 61 (1989), 247. doi: 10.1007/BF00962799. Google Scholar

[72]

R. H. Mladineo, An algorithm for finding the global maximum of a multimodal multivariate function,, Math. Program., 34 (1986), 188. doi: 10.1007/BF01580583. Google Scholar

[73]

J. Mockus, W. Eddy, A. Mockus, L. Mockus and G. Reklaitis, "Bayesian Heuristic Approach to Discrete and Global Optimization,", Kluwer Academic Publishers, (1996). Google Scholar

[74]

J. Mockus, "Bayesian Approach to Global Optimization,", Kluwer Academic Publishers, (1989). Google Scholar

[75]

C. G. Moles, P. Mendes and J. R. Banga, Parameter estimation in biochemical pathways: A comparison of global optimization methods,, Genome Res., 13 (2003), 2467. doi: 10.1101/gr.1262503. Google Scholar

[76]

A. Molinaro, C. Pizzuti and Ya. D. Sergeyev, Acceleration tools for diagonal information global optimization algorithms,, Comput. Optim. Appl., 18 (2001), 5. doi: 10.1023/A:1008719926680. Google Scholar

[77]

A. Molinaro and Ya. D. Sergeyev, Finding the minimal root of an equation with the multiextremal and nondifferentiable left-hand part,, Numer. Algorithms, 28 (2001), 255. doi: 10.1023/A:1014063303984. Google Scholar

[78]

V. N. Nefedov, Some problems of solving Lipschitzian global optimization problems using the branch and bound method,, Comput. Math. Math. Phys., 32 (1992), 433. Google Scholar

[79]

Yu. I. Neimark and R. G. Strongin, The information approach to the problem of search of extrema of functions,, Engineering Cybernetics, 1 (1966), 17. Google Scholar

[80]

A. Neumaier, Complete search in continuous global optimization and constraint satisfaction,, In, 13 (2004), 271. doi: 10.1017/CBO9780511569975.004. Google Scholar

[81]

J. Nocedal and S. J. Wright, "Numerical Optimization,", Springer-Verlag, (1999). Google Scholar

[82]

P. M. Pardalos and M. G. C. Resende, "Handbook of Applied Optimization,", Oxford, (2002). Google Scholar

[83]

P. M. Pardalos and H. E. Romeijn, "Handbook of Optimization in Medicine,", Springer, (2009). Google Scholar

[84]

R. Paulavičius, J. Žilinskas and A. Grothey, Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds,, Optim. Lett., 4 (2010), 173. doi: 10.1007/s11590-009-0156-3. Google Scholar

[85]

J. D. Pintér, "Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications),", Kluwer Academic Publishers, (1996). Google Scholar

[86]

J. D. Pintér, Global Optimization: Scientific and Engineering Case Studies,, Nonconvex Optimization and Its Applications, 85 (2006). Google Scholar

[87]

S. A. Piyavskij, An algorithm for finding the absolute minimum of a function,, In, 2 (1967), 13. Google Scholar

[88]

S. A. Piyavskij, An algorithm for finding the absolute extremum of a function,, USSR Comput. Math. Math. Phys., 12 (1972), 57. doi: 10.1016/0041-5553(72)90115-2. Google Scholar

[89]

S. Rebennack, P. M. Pardalos, M. V. F. Pereira and N. A. Iliadis, "Handbook of Power Systems I,", Springer, (2010). Google Scholar

[90]

M. G. C. Resende and P. M. Pardalos, "Handbook of Optimization in Telecommunications,", Springer, (2006). Google Scholar

[91]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970). Google Scholar

[92]

F. Schoen, On a sequential search strategy in global optimization problems,, Calcolo, 19 (1982), 321. Google Scholar

[93]

Ya. D. Sergeyev, P. Daponte, D. Grimaldi and A. Molinaro, Two methods for solving optimization problems arising in electronic measurements and electrical engineering,, SIAM J. Optim., 10 (1999), 1. doi: 10.1137/S1052623496312393. Google Scholar

[94]

Ya. D. Sergeyev and V. A. Grishagin, Parallel asynchronous global search and the nested optimization scheme,, J. Comput. Anal. Appl., 3 (2001), 123. doi: 10.1023/A:1010185125012. Google Scholar

[95]

Ya. D. Sergeyev and D. E. Kvasov, Global search based on efficient diagonal partitions and a set of Lipschitz constants,, SIAM J. Optim., 16 (2006), 910. doi: 10.1137/040621132. Google Scholar

[96]

Ya. D. Sergeyev and D. E. Kvasov, "Diagonal Global Optimization Methods,", FizMatLit, (2008). Google Scholar

[97]

Ya. D. Sergeyev and D. L. Markin, An algorithm for solving global optimization problems with nonlinear constraints,, J. Global Optim., 7 (1995), 407. doi: 10.1007/BF01099650. Google Scholar

[98]

Ya. D. Sergeyev, "Divide the best" algorithms for global optimization,, Technical Report 2-94, (1994), 2. Google Scholar

[99]

Ya. D. Sergeyev, Global optimization algorithms using smooth auxiliary functions,, Technical Report 5, (1994). Google Scholar

[100]

Ya. D. Sergeyev, A global optimization algorithm using derivatives and local tuning,, Technical Report 1, (1994). Google Scholar

[101]

Ya. D. Sergeyev, An information global optimization algorithm with local tuning,, SIAM J. Optim., 5 (1995), 858. doi: 10.1137/0805041. Google Scholar

[102]

Ya. D. Sergeyev, A one-dimensional deterministic global minimization algorithm,, Comput. Math. Math. Phys., 35 (1995), 705. Google Scholar

[103]

Ya. D. Sergeyev, A method using local tuning for minimizing functions with Lipschitz derivatives,, In, (1997), 199. Google Scholar

[104]

Ya. D. Sergeyev, Global one-dimensional optimization using smooth auxiliary functions,, Math. Program., 81 (1998), 127. doi: 10.1007/BF01584848. Google Scholar

[105]

Ya. D. Sergeyev, On convergence of "Divide the Best" global optimization algorithms,, Optimization, 44 (1998), 303. doi: 10.1080/02331939808844414. Google Scholar

[106]

Ya. D. Sergeyev, Multidimensional global optimization using the first derivatives,, Comput. Math. Math. Phys., 39 (1999), 711. Google Scholar

[107]

Ya. D. Sergeyev, Univariate global optimization with multiextremal non-differentiable constraints without penalty functions,, Comput. Optim. Appl., 34 (2006), 229. doi: 10.1007/s10589-005-3906-x. Google Scholar

[108]

Z. Shen and Y. Zhu, An interval version of Shubert's iterative method for the localization of the global maximum,, Computing, 38 (1987), 275. doi: 10.1007/BF02240102. Google Scholar

[109]

B. O. Shubert, A sequential method seeking the global maximum of a function,, SIAM J. Numer. Anal., 9 (1972), 379. doi: 10.1137/0709036. Google Scholar

[110]

C. P. Stephens and W. Baritompa, Global optimization requires global information,, J. Optim. Theory Appl., 96 (1998), 575. doi: 10.1023/A:1022612511618. Google Scholar

[111]

A. S. Strekalovsky, "Elements of Nonconvex Optimization,", Nauka, (2003). Google Scholar

[112]

R. G. Strongin and D. L. Markin, Minimization of multiextremal functions with nonconvex constraints,, Cybernetics, 22 (1986), 486. doi: 10.1007/BF01075079. Google Scholar

[113]

R. G. Strongin and Ya. D. Sergeyev, "Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms,", Kluwer Academic Publishers, (2000). Google Scholar

[114]

R. G. Strongin, Multiextremal minimization for measurements with interference,, Engineering Cybernetics, 16 (1969), 105. Google Scholar

[115]

R. G. Strongin, "Numerical Methods in Multiextremal Problems (Information-Statistical Algorithms),", Nauka, (1978). Google Scholar

[116]

A. G. Sukharev, "Minimax Algorithms in Problems of Numerical Analysis,", Nauka, (1989). Google Scholar

[117]

L. N. Timonov, An algorithm for search of a global extremum,, Engineering Cybernetics, 15 (1977), 38. Google Scholar

[118]

A. Törn and A. Žilinskas, "Global Optimization,", Lecture Notes in Computer Science, 350 (1989). Google Scholar

[119]

R. J. Vanderbei, Extension of Piyavskii's algorithm to continuous global optimization,, J. Global Optim., 14 (1999), 205. doi: 10.1023/A:1008395413111. Google Scholar

[120]

L. T. Watson and C. Baker, A fully-distributed parallel global search algorithm,, Engineering Computations, 18 (2001), 155. doi: 10.1108/02644400110365851. Google Scholar

[121]

G. R. Wood and B. P. Zhang, Estimation of the Lipschitz constant of a function,, J. Global Optim., 8 (1996), 91. doi: 10.1007/BF00229304. Google Scholar

[122]

G. R. Wood, Multidimensional bisection applied to global optimisation,, Comput. Math. Appl., 21 (1991), 161. doi: 10.1016/0898-1221(91)90170-9. Google Scholar

[123]

A. A. Zhigljavsky and A. Žilinskas, "Stochastic Global Optimization,", Springer, (2008). Google Scholar

[124]

A. Žilinskas, Axiomatic approach to statistical models and their use in multimodal optimization theory,, Math. Program., 22 (1982), 104. doi: 10.1007/BF01581029. Google Scholar

[125]

A. Žilinskas, "Global Optimization. Axiomatics of Statistical Models, Algorithms, and Applications,", Mokslas, (1986). Google Scholar

show all references

References:
[1]

C. S. Adjiman, S. Dallwig, C. A. Floudas and A. Neumaier, A global optimization method, $\alpha$BB, for general twice-differentiable constrained NLPs - I., Theoretical advances,, Comput. Chem. Engng., 22 (1998), 1137. doi: 10.1016/S0098-1354(98)00027-1. Google Scholar

[2]

M. Yu. Andramonov, A. M. Rubinov and B. M. Glover, Cutting angle methods in global optimization,, Appl. Math. Lett., 12 (1999), 95. doi: 10.1016/S0893-9659(98)00179-7. Google Scholar

[3]

I. P. Androulakis, C. D. Maranas and C. A. Floudas, $\alpha$BB: A global optimization method for general constrained nonconvex problems,, J. Global Optim., 7 (1995), 337. doi: 10.1007/BF01099647. Google Scholar

[4]

C. Audet, P. Hansen and G. Savard, "Essays and Surveys in Global Optimization,", GERAD 25th Anniversary. Springer-Verlag, (2005). Google Scholar

[5]

A. M. Bagirov, A. M. Rubinov and J. Zhang, Local optimization method with global multidimensional search,, J. Global Optim., 32 (2005), 161. doi: 10.1007/s10898-004-2700-0. Google Scholar

[6]

W. Baritompa and A. Cutler, Accelerations for global optimization covering methods using second derivatives,, J. Global Optim., 4 (1994), 329. doi: 10.1007/BF01098365. Google Scholar

[7]

W. Baritompa, Customizing methods for global optimization - A geometric viewpoint,, J. Global Optim., 3 (1993), 193. doi: 10.1007/BF01096738. Google Scholar

[8]

W. Baritompa, Accelerations for a variety of global optimization methods,, J. Global Optim., 4 (1994), 37. doi: 10.1007/BF01096533. Google Scholar

[9]

K. A. Barkalov and R. G. Strongin, A global optimization technique with an adaptive order of checking for constraints,, Comput. Math. Math. Phys., 42 (2002), 1289. Google Scholar

[10]

M. C. Bartholomew-Biggs, Z. J. Ulanowski and S. Zakovic, Using global optimization for a microparticle identification problem with noisy data,, J. Global Optim., 32 (2005), 325. doi: 10.1007/s10898-004-1943-0. Google Scholar

[11]

P. Basso, Iterative methods for the localization of the global maximum,, SIAM J. Numer. Anal., 19 (1982), 781. doi: 10.1137/0719054. Google Scholar

[12]

G. Beliakov and A. Ferrer, Bounded lower subdifferentiability optimization techniques: Applications,, J. Global Optim., 47 (2010), 211. doi: 10.1007/s10898-009-9467-2. Google Scholar

[13]

D. P. Bertsekas, "Nonlinear Programming,", Athena Scientific, (1999). Google Scholar

[14]

B. Betrò, Bayesian methods in global optimization,, J. Global Optim., 1 (1991), 1. doi: 10.1007/BF00120661. Google Scholar

[15]

M. Björkman and K. Holmström, Global optimization of costly nonconvex functions using radial basis functions,, Optim. Eng., 1 (2000), 373. doi: 10.1023/A:1011584207202. Google Scholar

[16]

L. Breiman and A. Cutler, A deterministic algorithm for global optimization,, Math. Program., 58 (1993), 179. doi: 10.1007/BF01581266. Google Scholar

[17]

R. G. Carter, J. M. Gablonsky, A. Patrick, C. T. Kelley and O. J. Eslinger, Algorithms for noisy problems in gas transmission pipeline optimization,, Optim. Eng., 2 (2001), 139. doi: 10.1023/A:1013123110266. Google Scholar

[18]

L. G. Casado, I. García and Ya. D. Sergeyev, Interval algorithms for finding the minimal root in a set of multiextremal non-differentiable one-dimensional functions,, SIAM J. Sci. Comput., 24 (2002), 359. doi: 10.1137/S1064827599357590. Google Scholar

[19]

M. H. Chang, Y. C. Park, and T. Y. Lee, A new global optimization method for univariate constrained twice-differentiable NLP problems,, J. Global Optim., 39 (2007), 79. doi: 10.1007/s10898-006-9121-1. Google Scholar

[20]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley & Sons, (1983). Google Scholar

[21]

J. J. Cochran, "Wiley Encyclopedia of Operations Research and Management Science (8 Volumes),", Wiley, (2011). doi: 10.1002/9780470400531. Google Scholar

[22]

A. R. Conn, K. Scheinberg and L. N. Vicente, "Introduction to Derivative-Free Optimization,", SIAM, (2009). doi: 10.1137/1.9780898718768. Google Scholar

[23]

S. E. Cox, R. T. Haftka, C. A. Baker, B. Grossman, W. H. Mason and L. T. Watson, A comparison of global optimization methods for the design of a high-speed civil transport,, J. Global Optim., 21 (2001), 415. doi: 10.1023/A:1012782825166. Google Scholar

[24]

A. E. Csallner, T. Csendes and M. Cs. Markót, Multisection in interval branch-and-bound methods for global optimization - I. Theoretical results,, J. Global Optim., 16 (2000), 371. doi: 10.1023/A:1008354711345. Google Scholar

[25]

Yu. M. Danilin, Estimation of the efficiency of an absolute-minimum-finding algorithm,, USSR Comput. Math. Math. Phys., 11 (1971), 261. doi: 10.1016/0041-5553(71)90020-6. Google Scholar

[26]

V. F. Demyanov and V. N. Malozemov, "Introduction to Minimax,", John Wiley & Sons, (1974). Google Scholar

[27]

V. F. Demyanov and A. M. Rubinov, "Quasidifferential Calculus,", Optimization Software Inc., (1986). Google Scholar

[28]

S. M. Elsakov and V. I. Shiryaev, Homogeneous algorithms for multiextremal optimization,, Comput. Math. Math. Phys., 50 (2010), 1642. doi: 10.1134/S0965542510100027. Google Scholar

[29]

Yu. G. Evtushenko, V. U. Malkova and A. A. Stanevichyus, Parallel global optimization of functions of several variables,, Comput. Math. Math. Phys., 49 (2009), 246. doi: 10.1134/S0965542509020055. Google Scholar

[30]

Yu. G. Evtushenko, M. A. Posypkin and I. Kh. Sigal, A framework for parallel large-scale global optimization,, Comp. Sci. - Res. Dev., 23 (2009), 211. doi: 10.1007/s00450-009-0083-7. Google Scholar

[31]

Yu. G. Evtushenko and M. A. Posypkin, Coverings for global optimization of partial-integer nonlinear problems,, Doklady Mathematics, 83 (2011), 268. doi: 10.1134/S1064562411020074. Google Scholar

[32]

Yu. G. Evtushenko, Numerical methods for finding global extrema (Case of a non-uniform mesh),, USSR Comput. Math. Math. Phys., 11 (1971), 38. doi: 10.1016/0041-5553(71)90065-6. Google Scholar

[33]

Yu. G. Evtushenko, "Numerical Optimization Techniques,", Translations Series in Mathematics and Engineering. Springer-Verlag, (1985). doi: 10.1007/978-1-4612-5022-7. Google Scholar

[34]

D. E. Finkel and C. T. Kelley, Additive scaling and the DIRECT algorithm,, J. Global Optim., 36 (2006), 597. doi: 10.1007/s10898-006-9029-9. Google Scholar

[35]

R. Fletcher, "Practical Methods of Optimization,", John Wiley & Sons, (2000). Google Scholar

[36]

C. A. Floudas and C. E. Gounaris, A review of recent advances in global optimization,, J. Global Optim., 45 (2009), 3. doi: 10.1007/s10898-008-9332-8. Google Scholar

[37]

C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. Esposito, Z. Gümüs, S. Harding, J. Klepeis, C. Meyer and C. Schweiger, "Handbook of Test Problems in Local and Global Optimization,", Kluwer Academic Publishers, (1999). Google Scholar

[38]

C. A. Floudas and P. M. Pardalos, "Encyclopedia of Optimization (6 Volumes),", Kluwer Academic Publishers, (2001). Google Scholar

[39]

K. R. Fowler, J. P. Reese, C. E. Kees, J. E. Dennis Jr., C. T. Kelley, C. T. Miller, C. Audet, A. J. Booker, G. Couture, R. W. Darwin, M. W. Farthing, D. E. Finkel, J. M. Gablonsky, G. Gray and T. G. Kolda, Comparison of derivative-free optimization methods for groundwater supply and hydraulic capture community problems,, Adv. Water Res., 31 (2008), 743. doi: 10.1016/j.advwatres.2008.01.010. Google Scholar

[40]

J. M. Gablonsky and C. T. Kelley, A locally-biased form of the DIRECT algorithm, J. Global Optim., 21 (2001), 27. doi: 10.1023/A:1017930332101. Google Scholar

[41]

D. Y. Gao and H. D. Sherali, Canonical duality theory: Connection between nonconvex mechanics and global optimization,, In, (2009), 257. doi: 10.1007/978-0-387-75714-8_8. Google Scholar

[42]

D. Y. Gao, "Duality Principles in Nonconvex Systems: Theory, Methods, and Applications,", Kluwer Academic Publishers, (2000). Google Scholar

[43]

V. P. Gergel, A global search algorithm using derivatives,, In, (1992), 161. Google Scholar

[44]

V. A. Grishagin, Operating characteristics of some global search algorithms,, In, 7 (1978), 198. Google Scholar

[45]

I. E. Grossmann, "Global Optimization in Engineering Design,", Kluwer Academic Publishers, (1996). Google Scholar

[46]

H.-M. Gutmann, A radial basis function method for global optimization,, J. Global Optim., 19 (2001), 201. doi: 10.1023/A:1011255519438. Google Scholar

[47]

P. Hansen and B. Jaumard, Lipschitz optimization,, In, 1 (1995), 407. Google Scholar

[48]

E. M. T. Hendrix and B. G.-Toth, "Introduction to Nonlinear and Global Optimization,", Springer, (2010). Google Scholar

[49]

J. He, L. T. Watson, N. Ramakrishnan, C. A. Shaffer, A. Verstak, J. Jiang, K. Bae and W. H. Tranter, Dynamic data structures for a direct search algorithm,, Comput. Optim. Appl., 23 (2002), 5. doi: 10.1023/A:1019992822938. Google Scholar

[50]

J. B. Hiriart-Urruty and C. Lemaréchal, "Convex Analysis and Minimization Algorithms (Parts I and II),", Springer-Verlag, (1996). Google Scholar

[51]

R. Horst, P. M. Pardalos and N. V. Thoai, "Introduction to Global Optimization,", Kluwer Academic Publishers, (1995). Google Scholar

[52]

R. Horst and P. M. Pardalos, "Handbook of Global Optimization,", volume 1. Kluwer Academic Publishers, (1995). Google Scholar

[53]

R. Horst and H. Tuy, "Global Optimization - Deterministic Approaches,", Springer-Verlag, (1996). Google Scholar

[54]

R. Horst, Deterministic global optimization with partition sets whose feasibility is not known: Application to concave minimization, reverse convex constraints, DC-programming, and Lipschitzian optimization,, J. Optim. Theory Appl., 58 (1988), 11. doi: 10.1007/BF00939768. Google Scholar

[55]

V. V. Ivanov, On optimal algorithms for the minimization of functions of certain classes,, Cybernetics, 4 (1972), 81. Google Scholar

[56]

D. R. Jones, C. D. Perttunen and B. E. Stuckman, Lipschitzian optimization without the Lipschitz constant,, J. Optim. Theory Appl., 79 (1993), 157. doi: 10.1007/BF00941892. Google Scholar

[57]

D. R. Jones, M. Schonlau and W. J. Welch, Efficient global optimization of expensive black-box functions,, J. Global Optim., 13 (1998), 455. doi: 10.1023/A:1008306431147. Google Scholar

[58]

O. V. Khamisov, Global optimization of functions with a concave support minorant,, Comput. Math. Math. Phys., 44 (2004), 1473. Google Scholar

[59]

A. G. Korotchenko, An algorithm for seeking the maximum value of univariate functions,, USSR Comput. Math. Math. Phys., 18 (1978), 34. doi: 10.1016/0041-5553(78)90162-3. Google Scholar

[60]

D. E. Kvasov, C. Pizzuti and Ya. D. Sergeyev, Local tuning and partition strategies for diagonal GO methods,, Numer. Math., 94 (2003), 93. doi: 10.1007/s00211-002-0419-8. Google Scholar

[61]

D. E. Kvasov and Ya. D. Sergeyev, Multidimensional global optimization algorithm based on adaptive diagonal curves,, Comput. Math. Math. Phys., 43 (2003), 40. Google Scholar

[62]

D. E. Kvasov and Ya. D. Sergeyev, A univariate global search working with a set of Lipschitz constants for the first derivative,, Optim. Lett., 3 (2009), 303. doi: 10.1007/s11590-008-0110-9. Google Scholar

[63]

D. Lera and Ya. D. Sergeyev, An information global minimization algorithm using the local improvement technique,, J. Global Optim., 48 (2010), 99. doi: 10.1007/s10898-009-9508-x. Google Scholar

[64]

D. Lera and Ya. D. Sergeyev, Lipschitz and {H\"older global optimization using space-filling curves},, Appl. Numer. Math., 60 (2010), 115. doi: 10.1016/j.apnum.2009.10.004. Google Scholar

[65]

G. Liuzzi, S. Lucidi and V. Piccialli, A DIRECT-based approach exploiting local minimizations for the solution of large-scale global optimization problems,, Comput. Optim. Appl., 45 (2010), 353. doi: 10.1007/s10589-008-9217-2. Google Scholar

[66]

K. Ljungberg, S. Holmgren and Ö. Carlborg, Simultaneous search for multiple QTL using the global optimization algorithm DIRECT,, Bioinformatics, 20 (2004), 1887. doi: 10.1093/bioinformatics/bth175. Google Scholar

[67]

D. MacLagan, T. Sturge and W. Baritompa, Equivalent methods for global optimization,, In, (1996), 201. doi: 10.1007/978-1-4613-3437-8_13. Google Scholar

[68]

O. L. Mangasarian, "Nonlinear Programming,", McGraw-Hill, (1969). Google Scholar

[69]

C. D. Maranas and C. A. Floudas, Global minimum potential energy conformations of small molecules,, J. Global Optim., 4 (1994), 135. doi: 10.1007/BF01096720. Google Scholar

[70]

C. C. Meewella and D. Q. Mayne, An algorithm for global optimization of Lipschitz continuous functions,, J. Optim. Theory Appl., 57 (1988), 307. doi: 10.1007/BF00938542. Google Scholar

[71]

C. C. Meewella and D. Q. Mayne, Efficient domain partitioning algorithms for global optimization of rational and Lipschitz continuous functions,, J. Optim. Theory Appl., 61 (1989), 247. doi: 10.1007/BF00962799. Google Scholar

[72]

R. H. Mladineo, An algorithm for finding the global maximum of a multimodal multivariate function,, Math. Program., 34 (1986), 188. doi: 10.1007/BF01580583. Google Scholar

[73]

J. Mockus, W. Eddy, A. Mockus, L. Mockus and G. Reklaitis, "Bayesian Heuristic Approach to Discrete and Global Optimization,", Kluwer Academic Publishers, (1996). Google Scholar

[74]

J. Mockus, "Bayesian Approach to Global Optimization,", Kluwer Academic Publishers, (1989). Google Scholar

[75]

C. G. Moles, P. Mendes and J. R. Banga, Parameter estimation in biochemical pathways: A comparison of global optimization methods,, Genome Res., 13 (2003), 2467. doi: 10.1101/gr.1262503. Google Scholar

[76]

A. Molinaro, C. Pizzuti and Ya. D. Sergeyev, Acceleration tools for diagonal information global optimization algorithms,, Comput. Optim. Appl., 18 (2001), 5. doi: 10.1023/A:1008719926680. Google Scholar

[77]

A. Molinaro and Ya. D. Sergeyev, Finding the minimal root of an equation with the multiextremal and nondifferentiable left-hand part,, Numer. Algorithms, 28 (2001), 255. doi: 10.1023/A:1014063303984. Google Scholar

[78]

V. N. Nefedov, Some problems of solving Lipschitzian global optimization problems using the branch and bound method,, Comput. Math. Math. Phys., 32 (1992), 433. Google Scholar

[79]

Yu. I. Neimark and R. G. Strongin, The information approach to the problem of search of extrema of functions,, Engineering Cybernetics, 1 (1966), 17. Google Scholar

[80]

A. Neumaier, Complete search in continuous global optimization and constraint satisfaction,, In, 13 (2004), 271. doi: 10.1017/CBO9780511569975.004. Google Scholar

[81]

J. Nocedal and S. J. Wright, "Numerical Optimization,", Springer-Verlag, (1999). Google Scholar

[82]

P. M. Pardalos and M. G. C. Resende, "Handbook of Applied Optimization,", Oxford, (2002). Google Scholar

[83]

P. M. Pardalos and H. E. Romeijn, "Handbook of Optimization in Medicine,", Springer, (2009). Google Scholar

[84]

R. Paulavičius, J. Žilinskas and A. Grothey, Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds,, Optim. Lett., 4 (2010), 173. doi: 10.1007/s11590-009-0156-3. Google Scholar

[85]

J. D. Pintér, "Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications),", Kluwer Academic Publishers, (1996). Google Scholar

[86]

J. D. Pintér, Global Optimization: Scientific and Engineering Case Studies,, Nonconvex Optimization and Its Applications, 85 (2006). Google Scholar

[87]

S. A. Piyavskij, An algorithm for finding the absolute minimum of a function,, In, 2 (1967), 13. Google Scholar

[88]

S. A. Piyavskij, An algorithm for finding the absolute extremum of a function,, USSR Comput. Math. Math. Phys., 12 (1972), 57. doi: 10.1016/0041-5553(72)90115-2. Google Scholar

[89]

S. Rebennack, P. M. Pardalos, M. V. F. Pereira and N. A. Iliadis, "Handbook of Power Systems I,", Springer, (2010). Google Scholar

[90]

M. G. C. Resende and P. M. Pardalos, "Handbook of Optimization in Telecommunications,", Springer, (2006). Google Scholar

[91]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970). Google Scholar

[92]

F. Schoen, On a sequential search strategy in global optimization problems,, Calcolo, 19 (1982), 321. Google Scholar

[93]

Ya. D. Sergeyev, P. Daponte, D. Grimaldi and A. Molinaro, Two methods for solving optimization problems arising in electronic measurements and electrical engineering,, SIAM J. Optim., 10 (1999), 1. doi: 10.1137/S1052623496312393. Google Scholar

[94]

Ya. D. Sergeyev and V. A. Grishagin, Parallel asynchronous global search and the nested optimization scheme,, J. Comput. Anal. Appl., 3 (2001), 123. doi: 10.1023/A:1010185125012. Google Scholar

[95]

Ya. D. Sergeyev and D. E. Kvasov, Global search based on efficient diagonal partitions and a set of Lipschitz constants,, SIAM J. Optim., 16 (2006), 910. doi: 10.1137/040621132. Google Scholar

[96]

Ya. D. Sergeyev and D. E. Kvasov, "Diagonal Global Optimization Methods,", FizMatLit, (2008). Google Scholar

[97]

Ya. D. Sergeyev and D. L. Markin, An algorithm for solving global optimization problems with nonlinear constraints,, J. Global Optim., 7 (1995), 407. doi: 10.1007/BF01099650. Google Scholar

[98]

Ya. D. Sergeyev, "Divide the best" algorithms for global optimization,, Technical Report 2-94, (1994), 2. Google Scholar

[99]

Ya. D. Sergeyev, Global optimization algorithms using smooth auxiliary functions,, Technical Report 5, (1994). Google Scholar

[100]

Ya. D. Sergeyev, A global optimization algorithm using derivatives and local tuning,, Technical Report 1, (1994). Google Scholar

[101]

Ya. D. Sergeyev, An information global optimization algorithm with local tuning,, SIAM J. Optim., 5 (1995), 858. doi: 10.1137/0805041. Google Scholar

[102]

Ya. D. Sergeyev, A one-dimensional deterministic global minimization algorithm,, Comput. Math. Math. Phys., 35 (1995), 705. Google Scholar

[103]

Ya. D. Sergeyev, A method using local tuning for minimizing functions with Lipschitz derivatives,, In, (1997), 199. Google Scholar

[104]

Ya. D. Sergeyev, Global one-dimensional optimization using smooth auxiliary functions,, Math. Program., 81 (1998), 127. doi: 10.1007/BF01584848. Google Scholar

[105]

Ya. D. Sergeyev, On convergence of "Divide the Best" global optimization algorithms,, Optimization, 44 (1998), 303. doi: 10.1080/02331939808844414. Google Scholar

[106]

Ya. D. Sergeyev, Multidimensional global optimization using the first derivatives,, Comput. Math. Math. Phys., 39 (1999), 711. Google Scholar

[107]

Ya. D. Sergeyev, Univariate global optimization with multiextremal non-differentiable constraints without penalty functions,, Comput. Optim. Appl., 34 (2006), 229. doi: 10.1007/s10589-005-3906-x. Google Scholar

[108]

Z. Shen and Y. Zhu, An interval version of Shubert's iterative method for the localization of the global maximum,, Computing, 38 (1987), 275. doi: 10.1007/BF02240102. Google Scholar

[109]

B. O. Shubert, A sequential method seeking the global maximum of a function,, SIAM J. Numer. Anal., 9 (1972), 379. doi: 10.1137/0709036. Google Scholar

[110]

C. P. Stephens and W. Baritompa, Global optimization requires global information,, J. Optim. Theory Appl., 96 (1998), 575. doi: 10.1023/A:1022612511618. Google Scholar

[111]

A. S. Strekalovsky, "Elements of Nonconvex Optimization,", Nauka, (2003). Google Scholar

[112]

R. G. Strongin and D. L. Markin, Minimization of multiextremal functions with nonconvex constraints,, Cybernetics, 22 (1986), 486. doi: 10.1007/BF01075079. Google Scholar

[113]

R. G. Strongin and Ya. D. Sergeyev, "Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms,", Kluwer Academic Publishers, (2000). Google Scholar

[114]

R. G. Strongin, Multiextremal minimization for measurements with interference,, Engineering Cybernetics, 16 (1969), 105. Google Scholar

[115]

R. G. Strongin, "Numerical Methods in Multiextremal Problems (Information-Statistical Algorithms),", Nauka, (1978). Google Scholar

[116]

A. G. Sukharev, "Minimax Algorithms in Problems of Numerical Analysis,", Nauka, (1989). Google Scholar

[117]

L. N. Timonov, An algorithm for search of a global extremum,, Engineering Cybernetics, 15 (1977), 38. Google Scholar

[118]

A. Törn and A. Žilinskas, "Global Optimization,", Lecture Notes in Computer Science, 350 (1989). Google Scholar

[119]

R. J. Vanderbei, Extension of Piyavskii's algorithm to continuous global optimization,, J. Global Optim., 14 (1999), 205. doi: 10.1023/A:1008395413111. Google Scholar

[120]

L. T. Watson and C. Baker, A fully-distributed parallel global search algorithm,, Engineering Computations, 18 (2001), 155. doi: 10.1108/02644400110365851. Google Scholar

[121]

G. R. Wood and B. P. Zhang, Estimation of the Lipschitz constant of a function,, J. Global Optim., 8 (1996), 91. doi: 10.1007/BF00229304. Google Scholar

[122]

G. R. Wood, Multidimensional bisection applied to global optimisation,, Comput. Math. Appl., 21 (1991), 161. doi: 10.1016/0898-1221(91)90170-9. Google Scholar

[123]

A. A. Zhigljavsky and A. Žilinskas, "Stochastic Global Optimization,", Springer, (2008). Google Scholar

[124]

A. Žilinskas, Axiomatic approach to statistical models and their use in multimodal optimization theory,, Math. Program., 22 (1982), 104. doi: 10.1007/BF01581029. Google Scholar

[125]

A. Žilinskas, "Global Optimization. Axiomatics of Statistical Models, Algorithms, and Applications,", Mokslas, (1986). Google Scholar

[1]

Shenggui Zhang. A sufficient condition of Euclidean rings given by polynomial optimization over a box. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 93-101. doi: 10.3934/naco.2014.4.93

[2]

Enkhbat Rentsen, J. Zhou, K. L. Teo. A global optimization approach to fractional optimal control. Journal of Industrial & Management Optimization, 2016, 12 (1) : 73-82. doi: 10.3934/jimo.2016.12.73

[3]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407

[4]

Z.Y. Wu, H.W.J. Lee, F.S. Bai, L.S. Zhang. Quadratic smoothing approximation to $l_1$ exact penalty function in global optimization. Journal of Industrial & Management Optimization, 2005, 1 (4) : 533-547. doi: 10.3934/jimo.2005.1.533

[5]

Mahamadi Warma. Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1881-1905. doi: 10.3934/cpaa.2013.12.1881

[6]

Na Zhao, Zheng-Hai Huang. A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a $P_0$ function. Journal of Industrial & Management Optimization, 2011, 7 (2) : 467-482. doi: 10.3934/jimo.2011.7.467

[7]

Javier Fernández, Marcela Zuccalli. A geometric approach to discrete connections on principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 433-444. doi: 10.3934/jgm.2013.5.433

[8]

M. Fernández-Martínez, Yolanda Guerrero-Sánchez, Pía López-Jornet. A novel approach to improve the accuracy of the box dimension calculations: Applications to trabecular bone quality. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1527-1534. doi: 10.3934/dcdss.2019105

[9]

Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018140

[10]

Jianjun Liu, Min Zeng, Yifan Ge, Changzhi Wu, Xiangyu Wang. Improved Cuckoo Search algorithm for numerical function optimization. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2018142

[11]

Hongxiu Zhong, Guoliang Chen, Xueping Guo. Semi-local convergence of the Newton-HSS method under the center Lipschitz condition. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 85-99. doi: 10.3934/naco.2019007

[12]

Alireza Bahiraie, A.K.M. Azhar, Noor Akma Ibrahim. A new dynamic geometric approach for empirical analysis of financial ratios and bankruptcy. Journal of Industrial & Management Optimization, 2011, 7 (4) : 947-965. doi: 10.3934/jimo.2011.7.947

[13]

Weishi Liu. Geometric approach to a singular boundary value problem with turning points. Conference Publications, 2005, 2005 (Special) : 624-633. doi: 10.3934/proc.2005.2005.624

[14]

Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17

[15]

Ricai Luo, Honglei Xu, Wu-Sheng Wang, Jie Sun, Wei Xu. A weak condition for global stability of delayed neural networks. Journal of Industrial & Management Optimization, 2016, 12 (2) : 505-514. doi: 10.3934/jimo.2016.12.505

[16]

Joon Kwon, Panayotis Mertikopoulos. A continuous-time approach to online optimization. Journal of Dynamics & Games, 2017, 4 (2) : 125-148. doi: 10.3934/jdg.2017008

[17]

Renato Bruni, Gianpiero Bianchi, Alessandra Reale. A combinatorial optimization approach to the selection of statistical units. Journal of Industrial & Management Optimization, 2016, 12 (2) : 515-527. doi: 10.3934/jimo.2016.12.515

[18]

Zhifeng Dai, Fenghua Wen. A generalized approach to sparse and stable portfolio optimization problem. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1651-1666. doi: 10.3934/jimo.2018025

[19]

Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1213-1223. doi: 10.3934/jimo.2018092

[20]

Torsten Trimborn, Lorenzo Pareschi, Martin Frank. Portfolio optimization and model predictive control: A kinetic approach. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-30. doi: 10.3934/dcdsb.2019136

 Impact Factor: 

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]