Document Zbl 1461.65113

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Math. Artif. Intell.</a> <a href="/?q=in%3A446326" title="Articles in this Issue">88, No. 1-3, 3-37 (2020)</a>. </div> <div class="abstract">Summary: A detailed review of the dynamic search trajectory methods for global optimization is given. In addition, a family of dynamic search trajectories methods that are created using numerical methods for solving autonomous ordinary differential equations is presented. Furthermore, a strategy for developing globally convergent methods that is applicable to the proposed family of methods is given and the corresponding theorem is proved. Finally, theoretical results for obtaining nonmonotone convergent methods that exploit the accumulated information with regard to the most recent values of the objective function are given.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7166064">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65K05" title="MSC2020">65K05</a> </td> <td class="space"> Numerical mathematical programming methods </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65K10" title="MSC2020">65K10</a> </td> <td class="space"> Numerical optimization and variational techniques </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65L05" title="MSC2020">65L05</a> </td> <td class="space"> Numerical methods for initial value problems involving ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68T05" title="MSC2020">68T05</a> </td> <td class="space"> Learning and adaptive systems in artificial intelligence </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68T20" title="MSC2020">68T20</a> </td> <td class="space"> Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adynamic+search+trajectories">dynamic search trajectories</a>; <a href="/?q=ut%3Atrajectory+methods">trajectory methods</a>; <a href="/?q=ut%3Aautonomous+initial+value+problems">autonomous initial value problems</a>; <a href="/?q=ut%3Aglobally+convergent+algorithms">globally convergent algorithms</a>; <a href="/?q=ut%3Anonmonotone+convergent+strategies">nonmonotone convergent strategies</a>; <a href="/?q=ut%3Aglobal+optimization">global optimization</a>; <a href="/?q=ut%3Aneural+networks+training">neural networks training</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1461.65113" data-ciurl="/ci/07166064" data-biburl="/bibtex/07166064.bib" data-amsurl="/amsrefs/07166064.bib" data-xmlurl="/xml/07166064.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07166064.pdf" title="Zbl 1461.65113 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s10472-019-09661-7" aria-label="DOI for “Dynamic search trajectory methods for global optimization”" title="10.1007/s10472-019-09661-7">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Adam, Stavros P.; Alexandropoulos, Stamatios-Aggelos N.; Pardalos, Panos M.; Vrahatis, Michael N., No Free Lunch Theorem: A Review, Approximation and Optimization, 57-82 (2019), Cham: Springer International Publishing, Cham · <a href="/1425.90111" class="nowrap">Zbl 1425.90111</a></td> </tr><tr> <td>[2]</td> <td class="space">Armijo, L., Minimization of functions having Lipschitz continuous first partial derivatives, Pac. J. Math., 16, 1, 1-3 (1966) · <a href="/0202.46105" class="nowrap">Zbl 0202.46105</a></td> </tr><tr> <td>[3]</td> <td class="space">Askarzadeh, A., A novel metaheuristic method for solving constrained engineering optimization problems: Crow search algorithm, Comput. Struct., 169, 1-12 (2016)</td> </tr><tr> <td>[4]</td> <td class="space">Back, T.: Evolutionary Algorithms in Theory and Practice: Evolution Strategies,Evolutionary Programming, Genetic Algorithms. Oxford University Press (1996) · <a href="/0877.68060" class="nowrap">Zbl 0877.68060</a></td> </tr><tr> <td>[5]</td> <td class="space">Barbieri, C.; Cocco, S.; Monasson, R., On the trajectories and performance of infotaxis, an information-based greedy search algorithm, EPL (Europhysics Letters), 94, 2, 20005-p1-p6 (2011)</td> </tr><tr> <td>[6]</td> <td class="space">Battiti, R., First-and second-order methods for learning: Between steepest descent and Newton’s method, Neur. Comput., 4, 2, 141-166 (1992)</td> </tr><tr> <td>[7]</td> <td class="space">Bhaya, A.; Kaszkurewicz, E., Steepest descent with momentum for quadratic functions is a version of the conjugate gradient method, Neural Netw., 17, 1, 65-71 (2004) · <a href="/1082.68093" class="nowrap">Zbl 1082.68093</a></td> </tr><tr> <td>[8]</td> <td class="space">Bhaya, A., Kaszkurewicz, E.: Control Perspectives on Numerical Algorithms and Matrix Problems, vol. 10. SIAM (2006) · <a href="/1128.93001" class="nowrap">Zbl 1128.93001</a></td> </tr><tr> <td>[9]</td> <td class="space">Bhaya, A., Pazos, F., Kaszkurewicz, E.: The controlled conjugate gradient type trajectory-following neural net for minimization of nonconvex functions. In: The 2010 International Joint Conference on Neural Networks (IJCNN), pp 1-8. IEEE (2010)</td> </tr><tr> <td>[10]</td> <td class="space">Boggs, Pt, An algorithm, based on singular perturbation theory, for ill-conditioned minimization problems, SIAM J. Numer. Anal., 14, 5, 830-843 (1977) · <a href="/0375.65033" class="nowrap">Zbl 0375.65033</a></td> </tr><tr> <td>[11]</td> <td class="space">Branin, Fh, Widely convergent method for finding multiple solutions of simultaneous nonlinear equations, IBM J. Res. Dev., 16, 5, 504-522 (1972) · <a href="/0271.65034" class="nowrap">Zbl 0271.65034</a></td> </tr><tr> <td>[12]</td> <td class="space">Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods. Wiley-Interscience (1987) · <a href="/0616.65072" class="nowrap">Zbl 0616.65072</a></td> </tr><tr> <td>[13]</td> <td class="space">Butcher, J.C.: Numerical Analysis of Ordinary Differential Equations, 2nd edn. Wiley (2008) · <a href="/1167.65041" class="nowrap">Zbl 1167.65041</a></td> </tr><tr> <td>[14]</td> <td class="space">Cesari, L.: Optimization Theory and Applications: Problems with Ordinary Differential Equations, vol. 17. Springer Science & Business Media (2012)</td> </tr><tr> <td>[15]</td> <td class="space">Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Classics in Applied Mathematics. SIAM (1996) · <a href="/0847.65038" class="nowrap">Zbl 0847.65038</a></td> </tr><tr> <td>[16]</td> <td class="space">Dixon, L.C.W.: Neural networks and unconstrained optimization. In: Algorithms for Continuous Optimization, pp 513-530. Springer (1994) · <a href="/0829.90112" class="nowrap">Zbl 0829.90112</a></td> </tr><tr> <td>[17]</td> <td class="space">Dixon, Lcw; Szego, Gp, The global optimization problem: An introduction, Toward Global Optim., 2, 1-15 (1978)</td> </tr><tr> <td>[18]</td> <td class="space">D’yakonov, E.G.: Optimization in Solving Elliptic Problems. CRC Press (2018)</td> </tr><tr> <td>[19]</td> <td class="space">Farkas, J., Jármai, K.: Design and Optimization of Metal Structures. Elsevier (2008)</td> </tr><tr> <td>[20]</td> <td class="space">Farkas, J.; Jármai, K.; Snyman, Ja, Global minimum cost design of a welded square stiffened plate supported at four corners, Struct. Multidiscip. Optim., 40, 1-6, 477 (2010) · <a href="/1274.74224" class="nowrap">Zbl 1274.74224</a></td> </tr><tr> <td>[21]</td> <td class="space">Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Classics in Applied Mathematics. SIAM (1990) · <a href="/0713.90043" class="nowrap">Zbl 0713.90043</a></td> </tr><tr> <td>[22]</td> <td class="space">Fletcher, R.: Fortran subroutines for minimization by quasi-Newton methods. Technical Report AERE-R-7125. Atomic Energy Research Establishment Harwell, England (1972)</td> </tr><tr> <td>[23]</td> <td class="space">Fletcher, R.: Practical Methods of Optimization. Wiley (2013) · <a href="/0988.65043" class="nowrap">Zbl 0988.65043</a></td> </tr><tr> <td>[24]</td> <td class="space">Floudas, C.A., Pardalos, P.M.: A Collection of Test Problems for Constrained Global Optimization Algorithms, pp. 455. Springer Science & Business Media (1990) · <a href="/0718.90054" class="nowrap">Zbl 0718.90054</a></td> </tr><tr> <td>[25]</td> <td class="space">Gill, Pe; Murray, W., Quasi-Newton methods for unconstrained optimization, IMA J. Appl. Math., 9, 1, 91-108 (1972) · <a href="/0264.49026" class="nowrap">Zbl 0264.49026</a></td> </tr><tr> <td>[26]</td> <td class="space">Griewank, A.O.: A Generalized Descent Method for Global Optimization. Master Thesis The Australian National University (1977)</td> </tr><tr> <td>[27]</td> <td class="space">Griewank, Ao, Generalized descent for global optimization, J. Optim. Theory Appl., 34, 1, 11-39 (1981) · <a href="/0431.49036" class="nowrap">Zbl 0431.49036</a></td> </tr><tr> <td>[28]</td> <td class="space">Grippo, L.; Lampariello, F.; Lucidi, S., A nonmonotone line search technique for Newton’s method, SIAM J. Numer. Anal., 23, 4, 707-716 (1986) · <a href="/0616.65067" class="nowrap">Zbl 0616.65067</a></td> </tr><tr> <td>[29]</td> <td class="space">Groenwold, A.A., Snyman, J.A.: Global optimization using dynamic search trajectories. Combinatorial and global optimization. In: Pardalos, P.M., Migdalas, A., Burkard, R. (eds.) Series on Applied Mathematics, vol. 14, pp 123-132. World Scientific Publishing Co (2002) · <a href="/1042.90034" class="nowrap">Zbl 1042.90034</a></td> </tr><tr> <td>[30]</td> <td class="space">Groenwold, Aa; Snyman, Ja; Stander, N., Modified trajectory method for practical global optimization problems, AIAA J., 34, 10, 2126-2131 (1996) · <a href="/0899.73321" class="nowrap">Zbl 0899.73321</a></td> </tr><tr> <td>[31]</td> <td class="space">Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations I: Nonstiff problems. Springer Series in Computational Mathematics, p. 1. Springer (2009) · <a href="/0789.65048" class="nowrap">Zbl 0789.65048</a></td> </tr><tr> <td>[32]</td> <td class="space">Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley (1962) · <a href="/0112.34901" class="nowrap">Zbl 0112.34901</a></td> </tr><tr> <td>[33]</td> <td class="space">Horst, P., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, 2nd edn. Kluwer Academic Publishers (2000) · <a href="/0966.90073" class="nowrap">Zbl 0966.90073</a></td> </tr><tr> <td>[34]</td> <td class="space">Himmelblau, D.M.: Applied Nonlinear Programming. McGraw-Hill Companies (1972) · <a href="/0241.90051" class="nowrap">Zbl 0241.90051</a></td> </tr><tr> <td>[35]</td> <td class="space">Incerti, S.; Parisi, V.; Zirilli, F., A new method for solving nonlinear simultaneous equations, SIAM J. Numer. Anal., 16, 5, 779-789 (1979) · <a href="/0411.65028" class="nowrap">Zbl 0411.65028</a></td> </tr><tr> <td>[36]</td> <td class="space">Inomata, S.; Cumada, M., On the golf method, Bull. Electron. Laboratory, 25, 3, 495-512 (1964)</td> </tr><tr> <td>[37]</td> <td class="space">Jain, Mk, Numerical Solution of Differential Equations (1979), Eastern New Delhi: Wiley, Eastern New Delhi · <a href="/0409.65002" class="nowrap">Zbl 0409.65002</a></td> </tr><tr> <td>[38]</td> <td class="space">Kam, Ty; Chang, Rr, Optimal design of laminated composite plates with dynamic and static considerations, Comput. Struct., 32, 2, 387-393 (1989) · <a href="/0702.73045" class="nowrap">Zbl 0702.73045</a></td> </tr><tr> <td>[39]</td> <td class="space">Kam, Ty; Lai, Md, Multilevel optimal design of laminated composite plate structures, Comput. Struct., 31, 2, 197-202 (1989)</td> </tr><tr> <td>[40]</td> <td class="space">Kam, Ty; Snyman, Ja, Optimal design of laminated composite plates using a global optimization technique, Compos. Struct., 19, 4, 351-370 (1991)</td> </tr><tr> <td>[41]</td> <td class="space">Kan, Ar; Timmer, Gt, Stochastic methods for global optimization, Am. J. Math. Manag. Sci., 4, 1-2, 7-40 (1984) · <a href="/0556.90073" class="nowrap">Zbl 0556.90073</a></td> </tr><tr> <td>[42]</td> <td class="space">Kazarlis, Sa; Papadakis, Se; Theocharis, J.; Petridis, V., Microgenetic algorithms as generalized hill-climbing operators for GA optimization, IEEE Trans. Evol. Comput., 5, 3, 204-217 (2001)</td> </tr><tr> <td>[43]</td> <td class="space">Lambert, J.D.: Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Wiley (1991) · <a href="/0745.65049" class="nowrap">Zbl 0745.65049</a></td> </tr><tr> <td>[44]</td> <td class="space">Laskari, Ec; Parsopoulos, Ke; Vrahatis, Mn, Evolutionary operators in global optimization with dynamic search trajectories, Numer. Algor., 34, 2-4, 393-403 (2003) · <a href="/1062.90050" class="nowrap">Zbl 1062.90050</a></td> </tr><tr> <td>[45]</td> <td class="space">Leung, Yw; Wang, Y., An orthogonal genetic algorithm with quantization for global numerical optimization, IEEE Trans. Evol. Comput., 5, 1, 41-53 (2001)</td> </tr><tr> <td>[46]</td> <td class="space">Li, H.; Zhang, Q., Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II, IEEE Trans. Evol. Comput., 13, 2, 284-302 (2009)</td> </tr><tr> <td>[47]</td> <td class="space">Massard, Tn, Computer sizing of composite laminates for strength, J. Reinforced Plastics Compos., 3, 4, 300-345 (1984)</td> </tr><tr> <td>[48]</td> <td class="space">Migdalas, A.; Pardalos, P. M., A Note on Open Problems and Challenges in Optimization Theory and Algorithms, Open Problems in Optimization and Data Analysis, 1-8 (2018), Cham: Springer International Publishing, Cham · <a href="/1421.90002" class="nowrap">Zbl 1421.90002</a></td> </tr><tr> <td>[49]</td> <td class="space">Pardalos, Panos M.; Migdalas, Athanasios, Open Problems in Optimization and Data Analysis (2018), Cham: Springer International Publishing, Cham · <a href="/1408.90003" class="nowrap">Zbl 1408.90003</a></td> </tr><tr> <td>[50]</td> <td class="space">Parsopoulos, Ke; Vrahatis, Mn, Particle swarm optimization method for constrained optimization problems, Intelligent Technologies-Theory and Application: New Trends in Intelligent Technologies, 76, 1, 214-220 (2002)</td> </tr><tr> <td>[51]</td> <td class="space">Parsopoulos, K.E., Vrahatis, M.N.: Particle swarm optimization method in multiobjective problems. In: Proceedings of the ACM Symposium on Applied Computing (SAC 2002), pp. 603-607 (2002)</td> </tr><tr> <td>[52]</td> <td class="space">Parsopoulos, K.E., Vrahatis, M.N.: Particle Swarm Optimization and Intelligence: Advances and Applications. Information Science Publishing, IGI Global (2010)</td> </tr><tr> <td>[53]</td> <td class="space">Petalas, Y.G., Tasoulis, D.K., Vrahatis, M.N.: Trajectory methods for neural network training. In: Hamza, M.H. (ed.) Artificial Intelligence and Applications, vol. 1, pp 400-408. IASTED/ACTA Press, USA (2004) · <a href="/1058.68596" class="nowrap">Zbl 1058.68596</a></td> </tr><tr> <td>[54]</td> <td class="space">Petalas, Yg; Tasoulis, Dk; Vrahatis, Mn, Dynamic search trajectory methods for neural network training, Lect. Notes Comput. Sci. (LNAI), 3070, 241-246 (2004) · <a href="/1058.68596" class="nowrap">Zbl 1058.68596</a></td> </tr><tr> <td>[55]</td> <td class="space">Petalas, Y.G., Vrahatis, M.N.: Trajectory methods for supervised learning. In: Proceedings of the First International Conference from Scientific Computing to Computational Engineering (IC-SCCE 2004), September 8-10, Athens (2004)</td> </tr><tr> <td>[56]</td> <td class="space">Plagianakos, V.P., Magoulas, G.D., Vrahatis, M.N.: Nonmonotone learning rules for backpropagation networks. In:: Proceedings of the Sixth IEEE International Conference on Electronics, Circuits and Systems (ICECS 1999), September 5-8, 1999, Pafos, Cyprus, vol. 1, art.no. 812280, pp. 291-294. IEEE (1999)</td> </tr><tr> <td>[57]</td> <td class="space">Plagianakos, Vp; Magoulas, Gd; Vrahatis, Mn, Deterministic nonmonotone strategies for effective training of multilayer perceptrons, IEEE Trans. Neural Netw., 13, 6, 1268-1284 (2002)</td> </tr><tr> <td>[58]</td> <td class="space">Plagianakos, V.P., Vrahatis, M.N., Magoulas, G.D.: Nonmonotone methods for backpropagation training with adaptive learning rate. In: IEEE Proceedings of the International Joint Conference on Neural Networks (IJCNN 1999), July 10-16, 1999, Washington DC, USA, vol. 3, art.no. 832644, pp. 1762-1767, IEEE (1999)</td> </tr><tr> <td>[59]</td> <td class="space">Polyak, Bt, Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys., 4, 5, 1-17 (1964) · <a href="/0147.35301" class="nowrap">Zbl 0147.35301</a></td> </tr><tr> <td>[60]</td> <td class="space">Powell, M.J.: A fast algorithm for nonlinearly constrained optimization calculations. In: Numerical Analysis, pp 144-157. Springer (1978) · <a href="/0374.65032" class="nowrap">Zbl 0374.65032</a></td> </tr><tr> <td>[61]</td> <td class="space">Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C. Cambridge University Press (2002) · <a href="/1078.65500" class="nowrap">Zbl 1078.65500</a></td> </tr><tr> <td>[62]</td> <td class="space">Price, W., Global optimization by controlled random search, J. Optim. Theory Appl., 40, 3, 333-348 (1983) · <a href="/0494.90063" class="nowrap">Zbl 0494.90063</a></td> </tr><tr> <td>[63]</td> <td class="space">Raydan, M., The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim., 7, 1, 26-33 (1997) · <a href="/0898.90119" class="nowrap">Zbl 0898.90119</a></td> </tr><tr> <td>[64]</td> <td class="space">Roughgarden, T., Beyond worst-case analysis, Commun. ACM, 62, 3, 88-96 (2019)</td> </tr><tr> <td>[65]</td> <td class="space">Salvadori, L.: Famiglie ad un parametro di funzioni di Liapunov nello studio della stabilita. In: Symposia Mathematica, vol. 6, pp 309-330. Academic Press (1971) · <a href="/0243.34099" class="nowrap">Zbl 0243.34099</a></td> </tr><tr> <td>[66]</td> <td class="space">Schäffler, S.; Warsitz, H., A trajectory-following method for unconstrained optimization, J. Optim. Theory Appl., 67, 1, 133-140 (1990) · <a href="/0687.90075" class="nowrap">Zbl 0687.90075</a></td> </tr><tr> <td>[67]</td> <td class="space">Sengupta, S.; Basak, S.; Peters, R., Particle swarm optimization: A survey of historical and recent developments with hybridization perspectives, Mach. Learn. Knowl. Extract., 1, 1, 157-191 (2018)</td> </tr><tr> <td>[68]</td> <td class="space">Shang, Y.; Wah, Bw, Global optimization for neural network training, Computer, 29, 3, 45-54 (1996)</td> </tr><tr> <td>[69]</td> <td class="space">Snyman, Ja, A new and dynamic method for unconstrained minimization, Appl. Math. Model., 6, 6, 449-462 (1982) · <a href="/0501.65026" class="nowrap">Zbl 0501.65026</a></td> </tr><tr> <td>[70]</td> <td class="space">Snyman, Ja, An improved version of the original leap-frog dynamic method for unconstrained minimization: LFOP1(b), Appl. Math. Model., 7, 3, 216-218 (1983) · <a href="/0548.65046" class="nowrap">Zbl 0548.65046</a></td> </tr><tr> <td>[71]</td> <td class="space">Snyman, Ja; Fatti, Lp, A multi-start global minimization algorithm with dynamic search trajectories, J. Optim. Theory Appl., 54, 1, 121-141 (1987) · <a href="/0595.90073" class="nowrap">Zbl 0595.90073</a></td> </tr><tr> <td>[72]</td> <td class="space">Snyman, J.A., Geerthsen, K.A.: The practical application of a dynamic search-trajectory method for constrained global optimization. In: IUTAM Symposium on Optimization of Mechanical Systems, pp 285-292. Springer (1996) · <a href="/0874.73050" class="nowrap">Zbl 0874.73050</a></td> </tr><tr> <td>[73]</td> <td class="space">Snyman, Ja; Hay, Am, The dynamic-q optimization method: An alternative to SQP?, Comput. Math. Appl., 44, 12, 1589-1598 (2002) · <a href="/1044.90078" class="nowrap">Zbl 1044.90078</a></td> </tr><tr> <td>[74]</td> <td class="space">Snyman, Ja; Kok, S., A reassessment of the Snyman-Fatti dynamic search trajectory method for unconstrained global optimization, J. Glob. Optim., 43, 1, 67-82 (2009) · <a href="/1169.90459" class="nowrap">Zbl 1169.90459</a></td> </tr><tr> <td>[75]</td> <td class="space">Snyman, Ja; Stander, N.; Roux, Wj, A dynamic penalty function method for the solution of structural optimization problems, Appl. Math. Model., 18, 8, 453-460 (1994) · <a href="/0807.73046" class="nowrap">Zbl 0807.73046</a></td> </tr><tr> <td>[76]</td> <td class="space">Snyman, J.A., Wilke, D.N.: New Gradient-Based Trajectory and Approximation Methods. Springer Optimization and its Applications. In: Pardalos, P.M. (ed.) , vol. 133, pp 197-250. Springer International Publishing AG, Cham (2018) · <a href="/1401.90007" class="nowrap">Zbl 1401.90007</a></td> </tr><tr> <td>[77]</td> <td class="space">Snyman, J.A., Wilke, D.N.: Practical mathematical optimization: Basic optimization theory and gradient-based algorithms. In: Pardalos, P.M. (ed.) Springer Optimization and its Applications. 2nd edn., vol. 133. Springer International Publishing AG, Cham (2018) · <a href="/1401.90007" class="nowrap">Zbl 1401.90007</a></td> </tr><tr> <td>[78]</td> <td class="space">Soliman, S.A.H., Mantawy, A.A.H.: Modern Optimization Techniques with Applications in Electric Power Systems. Springer Science & Business Media (2011) · <a href="/1229.90003" class="nowrap">Zbl 1229.90003</a></td> </tr><tr> <td>[79]</td> <td class="space">Sörensen, K., Metaheuristics-the metaphor exposed, Int. Trans. Oper. Res., 22, 1, 3-18 (2015) · <a href="/1309.90127" class="nowrap">Zbl 1309.90127</a></td> </tr><tr> <td>[80]</td> <td class="space">Storn, R.; Price, K., Differential evolution: A simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim., 11, 4, 341-359 (1997) · <a href="/0888.90135" class="nowrap">Zbl 0888.90135</a></td> </tr><tr> <td>[81]</td> <td class="space">Tseng, L.Y., Chen, C.: Multiple trajectory search for multiobjective optimization. In: IEEE Congress on Evolutionary Computation, (CEC 2007), pp 3609-3616. IEEE (2007)</td> </tr><tr> <td>[82]</td> <td class="space">Tseng, L.Y., Chen, C.: Multiple trajectory search for large scale global optimization. In: IEEE Congress on Evolutionary Computation, (CEC 2008), (IEEE World Congress on Computational Intelligence), pp 3052-3059. IEEE (2008)</td> </tr><tr> <td>[83]</td> <td class="space">Vincent, T.; Goh, B.; Teo, K., Trajectory-following algorithms for min-max optimization problems, J. Optim. Theory Appl., 75, 3, 501-519 (1992) · <a href="/0797.90099" class="nowrap">Zbl 0797.90099</a></td> </tr><tr> <td>[84]</td> <td class="space">Vogl, Tp; Mangis, J.; Rigler, A.; Zink, W.; Alkon, D., Accelerating the convergence of the back-propagation method, Biol. Cybern., 59, 4-5, 257-263 (1988)</td> </tr><tr> <td>[85]</td> <td class="space">Vavasis, S.A.: Complexity issues in global optimization: A survey. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp 27-41. Kluwer Academic (1995) · <a href="/0836.90138" class="nowrap">Zbl 0836.90138</a></td> </tr><tr> <td>[86]</td> <td class="space">Vrahatis, Mn; Magoulas, Gd; Plagianakos, Vp, From linear to nonlinear iterative methods, Appl. Numer. Math., 45, 1, 59-77 (2003) · <a href="/1022.65060" class="nowrap">Zbl 1022.65060</a></td> </tr><tr> <td>[87]</td> <td class="space">Wales, Dj; Doye, Jp, Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms, J. Phys. Chem. A, 101, 28, 5111-5116 (1997)</td> </tr><tr> <td>[88]</td> <td class="space">Walter, W.: Gewöhnliche differentialgleichungen. Springer (2000) · <a href="/0953.34002" class="nowrap">Zbl 0953.34002</a></td> </tr><tr> <td>[89]</td> <td class="space">Wolpert, Dh; Macready, Wg, No free lunch theorems for optimization, IEEE Trans. Evol. Comput., 1, 1, 67-82 (1997)</td> </tr><tr> <td>[90]</td> <td class="space">Wolfe, P., Convergence conditions for ascent methods, SIAM Rev., 11, 226-235 (1969) · <a href="/0177.20603" class="nowrap">Zbl 0177.20603</a></td> </tr><tr> <td>[91]</td> <td class="space">Wolfe, P., Convergence conditions for ascent methods II: Some corrections, SIAM Rev., 13, 185-188 (1971) · <a href="/0216.26901" class="nowrap">Zbl 0216.26901</a></td> </tr><tr> <td>[92]</td> <td class="space">Yi, J.H., Deb, S., Dong, J., Alavi, A.H., Wang, G.G.: An improved nsga-iii algorithm with adaptive mutation operator for big data optimization problems. Future Generation Computer Systems (2018)</td> </tr><tr> <td>[93]</td> <td class="space">Zabinsky, Z.B.: Stochastic Adaptive Search for Global Optimization. Springer (2003) · <a href="/1044.90001" class="nowrap">Zbl 1044.90001</a></td> </tr><tr> <td>[94]</td> <td class="space">Zidkov, N.; Siedrin, B., A certain method of search for the minimum of a function of several variables, Comput. Methods Program., 10, 203-210 (1968)</td> </tr><tr> <td>[95]</td> <td class="space">Zitzler, E.; Thiele, L., Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach, IEEE Trans. Evol. Comput., 3, 4, 257-271 (1999)</td> </tr><tr> <td>[96]</td> <td class="space">Zoutendijk, G.: Nonlinear Programming, Computational Methods. In: Abadie, J. (ed.) Integer and Nonlinear Programming, pp 37-86, North-Holland (1970) · <a href="/0336.90057" class="nowrap">Zbl 0336.90057</a></td> </tr></table> <div class="reference_disclaimer"> 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. 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