{"63985":{"#nid":"63985","#data":{"type":"event","title":"Hierarchy of Relaxations for Convex Nonlinear Generalized Disjunctive Programs and Extensions to Nonconvex Optimization","body":[{"value":"\u003Cp\u003E\u003Cstrong\u003ETITLE:\u0026nbsp; \u003C\/strong\u003EHierarchy of Relaxations for Convex Nonlinear Generalized\nDisjunctive Programs and Extensions to Nonconvex Optimization\u003C\/p\u003E\u003Cp\u003E\u003Cstrong\u003ESPEAKER:\u003C\/strong\u003E\u0026nbsp; Ignacio Grossman\u003C\/p\u003E\u003Cp\u003E\u003Cstrong\u003EABSTRACT:\u003C\/strong\u003E\u003C\/p\u003E\u003Cp\u003EThis seminar deals with the theory\nof reformulations and numerical solution of generalized disjunctive programming\n(GDP) problems, which are expressed in terms of Boolean and continuous\nvariables, and involve algebraic constraints, disjunctions and propositional\nlogic statements. We propose a framework to generate alternative MINLP\nformulations for convex nonlinear GDPs that lead to stronger relaxations by\ngeneralizing the seminal work by Egon Balas (1988) for linear disjunctive\nprograms. We define for the case of convex nonlinear GDPs an operation\nequivalent to a basic step for linear disjunctive programs that takes a\ndisjunctive set to another one with fewer conjuncts. We show that the strength\nof relaxations increases as the number of conjuncts decreases, leading to a\nhierarchy of relaxations. We prove that the tightest of these relaxations,\nallows in theory the solution of the convex GDP problem as an NLP problem. We\npresent a guide for the generation of strong relaxations without incurring in\nan exponential increase of the size of the reformulated MINLP. We apply the\nproposed theory for generating strong relaxations to a dozen convex GDPs which\nare solved with a NLP-based branch and bound method. Compared to the reformulation\nbased on the hull relaxation, the computational results show that with the\nproposed reformulations significant improvements can be obtained in the\npredicted lower bounds, which in turn translates into a smaller number of nodes\nfor the branch and bound enumeration. \u003C\/p\u003E\n\n\u003Cp\u003EWe next address the extension of\nthe above ideas to the solution of nonconvex GDPs that involve bilinear and\nconcave terms. In order to solve these nonconvex problems with a spatial branch\nand bound method, a \u003Cem\u003Econvex GDP relaxation\u003C\/em\u003E\nis obtained by using suitable under- and over-estimating functions of the nonconvex\nconstraints such as the convex envelopes proposed by McCormick (1976).\u0026nbsp; In order to predict tighter lower bounds to\nthe global optimum we exploiting the hierarchy of relaxations for linear GDP\nproblems. A family of tighter reformulations is obtained by performing a\nsequence of basic steps on the original disjunctive set. Since each\nintersection usually creates new variables and constraints some general rules\nare described to limit the size of the problem. \u0026nbsp;We illustrate\nthe application of these ideas with linear relaxations of several bilinear and\nconcave GDPs related to the optimization of process systems to demonstrate the\ncomputational savings that can be achieved with the tighter lower bounds. Finally,\nextensions for nonconvex problems involving nonlinear convex under- and\nover-estimating functions. \u003C\/p\u003E\u003Cp align=\u0022center\u0022\u003E\u0026nbsp;\u003C\/p\u003E","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":"","field_summary_sentence":[{"value":"Hierarchy of Relaxations for Convex Nonlinear Generalized Disjunctive Programs and Extensions to Nonconvex Optimization"}],"uid":"27187","created_gmt":"2011-02-01 11:49:02","changed_gmt":"2016-10-08 01:53:28","author":"Anita Race","boilerplate_text":"","field_publication":"","field_article_url":"","field_event_time":{"event_time_start":"2011-02-22T10:00:00-05:00","event_time_end":"2011-02-22T11:00:00-05:00","event_time_end_last":"2011-02-22T11:00:00-05:00","gmt_time_start":"2011-02-22 15:00:00","gmt_time_end":"2011-02-22 16:00:00","gmt_time_end_last":"2011-02-22 16:00:00","rrule":null,"timezone":"America\/New_York"},"extras":[],"groups":[{"id":"1242","name":"School of Industrial and Systems Engineering (ISYE)"}],"categories":[],"keywords":[],"core_research_areas":[],"news_room_topics":[],"event_categories":[],"invited_audience":[],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}