A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes
In this paper, we develop a version of the bundle method to solve unconstrained difference of convex (DC) programming problems. It is assumed that a DC representation of the objective function is available. Our main idea is to utilize subgradients of both the first and second components in the DC re...
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| Veröffentlicht in: | Journal of global optimization 2017-07, Vol.68 (3), p.501-535 |
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| container_title | Journal of global optimization |
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| creator | Joki, Kaisa Bagirov, Adil M. Karmitsa, Napsu Mäkelä, Marko M. |
| description | In this paper, we develop a version of the bundle method to solve unconstrained difference of convex (DC) programming problems. It is assumed that a DC representation of the objective function is available. Our main idea is to utilize subgradients of both the first and second components in the DC representation. This subgradient information is gathered from some neighborhood of the current iteration point and it is used to build separately an approximation for each component in the DC representation. By combining these approximations we obtain a new nonconvex cutting plane model of the original objective function, which takes into account explicitly both the convex and the concave behavior of the objective function. We design the proximal bundle method for DC programming based on this new approach and prove the convergence of the method to an
ε
-critical point. The algorithm is tested using some academic test problems and the preliminary numerical results have shown the good performance of the new bundle method. An interesting fact is that the new algorithm finds nearly always the global solution in our test problems. |
| doi_str_mv | 10.1007/s10898-016-0488-3 |
| format | Article |
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ε
-critical point. The algorithm is tested using some academic test problems and the preliminary numerical results have shown the good performance of the new bundle method. An interesting fact is that the new algorithm finds nearly always the global solution in our test problems.</description><identifier>ISSN: 0925-5001</identifier><identifier>EISSN: 1573-2916</identifier><identifier>DOI: 10.1007/s10898-016-0488-3</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algorithms ; Analysis ; Approximation ; Bundling ; Computer Science ; Convergence ; Critical point ; Cutting ; Iterative methods ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Methods ; Operations Research/Decision Theory ; Optimization ; Planes ; Programming ; Real Functions</subject><ispartof>Journal of global optimization, 2017-07, Vol.68 (3), p.501-535</ispartof><rights>Springer Science+Business Media New York 2016</rights><rights>COPYRIGHT 2017 Springer</rights><rights>Journal of Global Optimization is a copyright of Springer, 2017.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c355t-9668d1407a4ef69ba8f85880481072d947c4e923aa04b2e1333cf946834697143</citedby><cites>FETCH-LOGICAL-c355t-9668d1407a4ef69ba8f85880481072d947c4e923aa04b2e1333cf946834697143</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10898-016-0488-3$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10898-016-0488-3$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>Joki, Kaisa</creatorcontrib><creatorcontrib>Bagirov, Adil M.</creatorcontrib><creatorcontrib>Karmitsa, Napsu</creatorcontrib><creatorcontrib>Mäkelä, Marko M.</creatorcontrib><title>A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes</title><title>Journal of global optimization</title><addtitle>J Glob Optim</addtitle><description>In this paper, we develop a version of the bundle method to solve unconstrained difference of convex (DC) programming problems. It is assumed that a DC representation of the objective function is available. Our main idea is to utilize subgradients of both the first and second components in the DC representation. This subgradient information is gathered from some neighborhood of the current iteration point and it is used to build separately an approximation for each component in the DC representation. By combining these approximations we obtain a new nonconvex cutting plane model of the original objective function, which takes into account explicitly both the convex and the concave behavior of the objective function. We design the proximal bundle method for DC programming based on this new approach and prove the convergence of the method to an
ε
-critical point. The algorithm is tested using some academic test problems and the preliminary numerical results have shown the good performance of the new bundle method. An interesting fact is that the new algorithm finds nearly always the global solution in our test problems.</description><subject>Algorithms</subject><subject>Analysis</subject><subject>Approximation</subject><subject>Bundling</subject><subject>Computer Science</subject><subject>Convergence</subject><subject>Critical point</subject><subject>Cutting</subject><subject>Iterative methods</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Methods</subject><subject>Operations Research/Decision Theory</subject><subject>Optimization</subject><subject>Planes</subject><subject>Programming</subject><subject>Real 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proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes</title><author>Joki, Kaisa ; Bagirov, Adil M. ; Karmitsa, Napsu ; Mäkelä, Marko M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c355t-9668d1407a4ef69ba8f85880481072d947c4e923aa04b2e1333cf946834697143</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Algorithms</topic><topic>Analysis</topic><topic>Approximation</topic><topic>Bundling</topic><topic>Computer Science</topic><topic>Convergence</topic><topic>Critical point</topic><topic>Cutting</topic><topic>Iterative methods</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Methods</topic><topic>Operations Research/Decision Theory</topic><topic>Optimization</topic><topic>Planes</topic><topic>Programming</topic><topic>Real Functions</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Joki, Kaisa</creatorcontrib><creatorcontrib>Bagirov, Adil M.</creatorcontrib><creatorcontrib>Karmitsa, Napsu</creatorcontrib><creatorcontrib>Mäkelä, Marko M.</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>ABI/INFORM Collection</collection><collection>ABI/INFORM Global (PDF only)</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>ABI/INFORM Global (Alumni Edition)</collection><collection>Science Database (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest Pharma Collection</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology 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ε
-critical point. The algorithm is tested using some academic test problems and the preliminary numerical results have shown the good performance of the new bundle method. An interesting fact is that the new algorithm finds nearly always the global solution in our test problems.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10898-016-0488-3</doi><tpages>35</tpages></addata></record> |
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| subjects | Algorithms Analysis Approximation Bundling Computer Science Convergence Critical point Cutting Iterative methods Mathematical analysis Mathematics Mathematics and Statistics Methods Operations Research/Decision Theory Optimization Planes Programming Real Functions |
| title | A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes |
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