Document Zbl 1421.49015

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Malays. Math. Sci. Soc. (2)</a> <a href="/?q=in%3A436758" title="Articles in this Issue">42, No. 4, 1853-1863 (2019)</a>. </div> <div class="abstract">In the mathematical modeling based on the set-valued optimization objective maps, the ways to identify or, at least, to approximate the best solutions are usually measured by the corresponding velocities. These are controlled by the different concepts of the derivatives, like this example devoted to the general epiderivatives.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A6295">Vasile Postolică (Piatra Neamt)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A49J53" title="MSC2020">49J53</a> </td> <td class="space"> Set-valued and variational analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A54C60" title="MSC2020">54C60</a> </td> <td class="space"> Set-valued maps in general topology </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C56" title="MSC2020">90C56</a> </td> <td class="space"> Derivative-free methods and methods using generalized derivatives </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C46" title="MSC2020">90C46</a> </td> <td class="space"> Optimality conditions and duality in mathematical programming </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aset-valued+optimization">set-valued optimization</a>; <a href="/?q=ut%3Aepiderivative">epiderivative</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 1421.49015" data-ciurl="/ci/07093462" data-biburl="/bibtex/07093462.bib" data-amsurl="/amsrefs/07093462.bib" data-xmlurl="/xml/07093462.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07093462.pdf" title="Zbl 1421.49015 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/s40840-017-0573-x" aria-label="DOI for “Higher-order generalized radial epiderivative and its applications to set-valued optimization problems”" title="10.1007/s40840-017-0573-x">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">Anh, N.L.H.: Mixed type duality for set-valued optimization problems via higher-order radial epiderivatives. Numer. Funct. Anal. Optim. 37, 823-838 (2016) · <a href="/1352.49016" class="nowrap">Zbl 1352.49016</a> · <a href="https://doi.org/10.1080/01630563.2016.1179202" class="nowrap">doi:10.1080/01630563.2016.1179202</a></td> </tr><tr> <td>[2]</td> <td class="space">Anh, N.L.H.: Higher-order optimality conditions for strict and weak efficient solutions in set-valued optimization. Positivity 20, 499-514 (2016) · <a href="/1373.90172" class="nowrap">Zbl 1373.90172</a> · <a href="https://doi.org/10.1007/s11117-015-0369-x" class="nowrap">doi:10.1007/s11117-015-0369-x</a></td> </tr><tr> <td>[3]</td> <td class="space">Anh, N.L.H., Khanh, P.Q., Tung, L.T.: Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization. Nonlinear Anal. Theory Methods Appl. 74, 7365-7379 (2011) · <a href="/1251.90386" class="nowrap">Zbl 1251.90386</a> · <a href="https://doi.org/10.1016/j.na.2011.07.055" class="nowrap">doi:10.1016/j.na.2011.07.055</a></td> </tr><tr> <td>[4]</td> <td class="space">Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990) · <a href="/0713.49021" class="nowrap">Zbl 0713.49021</a></td> </tr><tr> <td>[5]</td> <td class="space">Bednarczuk, E.M., Song, W.: Contingent epiderivative and its applications to set-valued optimization. Control Cybern. 27, 376-386 (1998) · <a href="/0951.49022" class="nowrap">Zbl 0951.49022</a></td> </tr><tr> <td>[6]</td> <td class="space">Borwein, J.M.: On the existence of Pareto efficient points. Math. Oper. Res. 8, 64-73 (1983) · <a href="/0508.90080" class="nowrap">Zbl 0508.90080</a> · <a href="https://doi.org/10.1287/moor.8.1.64" class="nowrap">doi:10.1287/moor.8.1.64</a></td> </tr><tr> <td>[7]</td> <td class="space">Chen, C.R., Li, S.J., Teo, K.L.: Higher order weak epiderivatives and applications to duality and optimality conditions. Comput. Math. Appl. 57, 1389-1399 (2009) · <a href="/1186.90122" class="nowrap">Zbl 1186.90122</a> · <a href="https://doi.org/10.1016/j.camwa.2009.01.012" class="nowrap">doi:10.1016/j.camwa.2009.01.012</a></td> </tr><tr> <td>[8]</td> <td class="space">Chuong, T.D., Yao, J.C.: Generalized Clarke epiderivatives of parametric vector optimization problems. J. Optim. Theory Appl. 146, 77-94 (2010) · <a href="/1206.90158" class="nowrap">Zbl 1206.90158</a> · <a href="https://doi.org/10.1007/s10957-010-9646-9" class="nowrap">doi:10.1007/s10957-010-9646-9</a></td> </tr><tr> <td>[9]</td> <td class="space">Corley, H.W.: Optimality conditions for maximizations of set-valued functions. J. Optim. Theory Appl. 58, 1-10 (1988) · <a href="/0956.90509" class="nowrap">Zbl 0956.90509</a> · <a href="https://doi.org/10.1007/BF00939767" class="nowrap">doi:10.1007/BF00939767</a></td> </tr><tr> <td>[10]</td> <td class="space">Crepsi, G.P., Ginchev, I., Rocca, M.: First-order optimality conditions in set-valued optimization. Math. Methods Oper. Res. 63, 87-106 (2006) · <a href="/1103.90089" class="nowrap">Zbl 1103.90089</a> · <a href="https://doi.org/10.1007/s00186-005-0023-7" class="nowrap">doi:10.1007/s00186-005-0023-7</a></td> </tr><tr> <td>[11]</td> <td class="space">Durea, M.: First and second order optimality conditions for set-valued optimization problems. Rend. Circ. Math. Palermo 2(53), 451-468 (2004) · <a href="/1096.49012" class="nowrap">Zbl 1096.49012</a> · <a href="https://doi.org/10.1007/BF02875738" class="nowrap">doi:10.1007/BF02875738</a></td> </tr><tr> <td>[12]</td> <td class="space">Durea, M.: Optimality conditions for weak and firm efficiency in set-valued optimization. J. Math. Anal. Appl. 344, 1018-1028 (2008) · <a href="/1170.49015" class="nowrap">Zbl 1170.49015</a> · <a href="https://doi.org/10.1016/j.jmaa.2008.03.053" class="nowrap">doi:10.1016/j.jmaa.2008.03.053</a></td> </tr><tr> <td>[13]</td> <td class="space">Flores-Bazán, F.: Optimality conditions in nonconvex set-valued optimization. Math. Methods Oper. Res. 53, 403-417 (2001) · <a href="/1031.90032" class="nowrap">Zbl 1031.90032</a> · <a href="https://doi.org/10.1007/s001860100130" class="nowrap">doi:10.1007/s001860100130</a></td> </tr><tr> <td>[14]</td> <td class="space">Flores-Bazán, F.: Radial epiderivatives and asymptotic function in nonconvex vector optimization. SIAM J. Optim. 14, 284-305 (2003) · <a href="/1046.46035" class="nowrap">Zbl 1046.46035</a> · <a href="https://doi.org/10.1137/S1052623401392111" class="nowrap">doi:10.1137/S1052623401392111</a></td> </tr><tr> <td>[15]</td> <td class="space">Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2004) · <a href="/1055.90065" class="nowrap">Zbl 1055.90065</a> · <a href="https://doi.org/10.1007/978-3-540-24828-6" class="nowrap">doi:10.1007/978-3-540-24828-6</a></td> </tr><tr> <td>[16]</td> <td class="space">Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Methods Oper. Res. 46, 193-211 (1997) · <a href="/0889.90123" class="nowrap">Zbl 0889.90123</a> · <a href="https://doi.org/10.1007/BF01217690" class="nowrap">doi:10.1007/BF01217690</a></td> </tr><tr> <td>[17]</td> <td class="space">Jahn, J., Khan, A.A.: Generalized contingent epiderivatives in set-valued optimization: optimality conditions. Numer. Funct. Anal. Optim. 23, 807-832 (2002) · <a href="/1134.90466" class="nowrap">Zbl 1134.90466</a> · <a href="https://doi.org/10.1081/NFA-120016271" class="nowrap">doi:10.1081/NFA-120016271</a></td> </tr><tr> <td>[18]</td> <td class="space">Kasimbeyli, R.: Radial epiderivatives and set-valued optimization. Optimization 58, 521-534 (2009) · <a href="/1175.90316" class="nowrap">Zbl 1175.90316</a> · <a href="https://doi.org/10.1080/02331930902928310" class="nowrap">doi:10.1080/02331930902928310</a></td> </tr><tr> <td>[19]</td> <td class="space">Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization: An Introduction with Applications. Springer, Heidelberg (2015) · <a href="/1308.49004" class="nowrap">Zbl 1308.49004</a> · <a href="https://doi.org/10.1007/978-3-642-54265-7" class="nowrap">doi:10.1007/978-3-642-54265-7</a></td> </tr><tr> <td>[20]</td> <td class="space">Li, S.J., Chen, C.R.: Higher-order optimality conditions for Henig efficient solutions in set-valued optimization. J. Math. Anal. Appl. 323, 1184-1200 (2006) · <a href="/1144.90481" class="nowrap">Zbl 1144.90481</a> · <a href="https://doi.org/10.1016/j.jmaa.2005.11.035" class="nowrap">doi:10.1016/j.jmaa.2005.11.035</a></td> </tr><tr> <td>[21]</td> <td class="space">Li, S.J., Teo, K.L., Yang, X.Q.: Higher-order Mond-Weir duality for set-valued optimization. J. Comput. Appl. Math. 217, 339-349 (2008) · <a href="/1171.90011" class="nowrap">Zbl 1171.90011</a> · <a href="https://doi.org/10.1016/j.cam.2007.02.011" class="nowrap">doi:10.1016/j.cam.2007.02.011</a></td> </tr><tr> <td>[22]</td> <td class="space">Li, S.J., Yang, X.Q., Chen, G.Y.: Nonconvex vector optimization of set-valued mappings. J. Math. Anal. Appl. 283, 337-350 (2003) · <a href="/1137.90642" class="nowrap">Zbl 1137.90642</a> · <a href="https://doi.org/10.1016/S0022-247X(02)00410-9" class="nowrap">doi:10.1016/S0022-247X(02)00410-9</a></td> </tr><tr> <td>[23]</td> <td class="space">Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989) · <a href="https://doi.org/10.1007/978-3-642-50280-4" class="nowrap">doi:10.1007/978-3-642-50280-4</a></td> </tr><tr> <td>[24]</td> <td class="space">Luc, D.T.: Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50, 99-111 (1991) · <a href="/0718.90080" class="nowrap">Zbl 0718.90080</a> · <a href="https://doi.org/10.1007/BF01594928" class="nowrap">doi:10.1007/BF01594928</a></td> </tr><tr> <td>[25]</td> <td class="space">Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory. Springer, Berlin (2006) · <a href="https://doi.org/10.1007/3-540-31246-3" class="nowrap">doi:10.1007/3-540-31246-3</a></td> </tr><tr> <td>[26]</td> <td class="space">Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. II: Applications. Springer, Berlin (2006) · <a href="https://doi.org/10.1007/3-540-31246-3" class="nowrap">doi:10.1007/3-540-31246-3</a></td> </tr><tr> <td>[27]</td> <td class="space">Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, 3rd edn. Springer, Berlin (2009) · <a href="/0888.49001" class="nowrap">Zbl 0888.49001</a></td> </tr><tr> <td>[28]</td> <td class="space">Sun, X.K., Li, S.J.: Generalized second-order contingent epiderivatives in parametric vector optimization problems. J. Optim. Theory Appl. 58, 351-363 (2014) · <a href="/1297.49027" class="nowrap">Zbl 1297.49027</a></td> </tr><tr> <td>[29]</td> <td class="space">Wang, Q.L., Li, S.J.: Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency. Numer. Funct. Anal. Optim. 30, 849-869 (2009) · <a href="/1180.90254" class="nowrap">Zbl 1180.90254</a> · <a href="https://doi.org/10.1080/01630560903139540" class="nowrap">doi:10.1080/01630560903139540</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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