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Valid Inequalities and Reformulation Techniques for Mixed Integer Nonlinear Programming

Modaresi, Sina (2016) Valid Inequalities and Reformulation Techniques for Mixed Integer Nonlinear Programming. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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One of the most important breakthroughs in the area of Mixed Integer Linear Programming (MILP) is the characterization of the convex hull of specially structured non-convex polyhedral sets in order to develop valid inequalities or cutting planes. Development of strong valid inequalities such as Split cuts, Gomory Mixed Integer (GMI) cuts, and Mixed Integer Rounding (MIR) cuts has resulted in highly effective branch-and-cut algorithms. While such cuts are known to be equivalent, each of their characterizations provides different advantages and insights.

The study of cutting planes for Mixed Integer Nonlinear Programming (MINLP) is still much more limited than that for MILP, since characterizing cuts for MINLP requires the study of the convex hull of a non-convex and non-polyhedral set, which has proven to be significantly harder than the polyhedral case. However, there has been significant work on the computational use of cuts in MINLP. Furthermore, there has recently been a significant interest in extending the associated theoretical results from MILP to the realm of MINLP.

This dissertation is focused on the development of new cuts and extended formulations for Mixed Integer Nonlinear Programs. We study the generalization of split, k-branch split, and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split, k-branch split, and intersection cuts for several classes of non-polyhedral sets. We also study the relation between the introduced cuts and some known classes of cutting planes from MILP. Furthermore, we show how an aggregation technique can be easily extended to characterize the convex hull of sets defined by two quadratic or by a conic quadratic and a quadratic inequality. We also computationally evaluate the performance of the introduced cuts and extended formulations on two classes of MINLP problems.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Modaresi, Sinasim23@pitt.eduSIM230000-0003-0175-5029
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairRajgopal, Jayantrajgopal@pitt.eduRAJGOPAL
Committee MemberProkopyev, Oleg A.droleg@pitt.eduDROLEG
Committee MemberSchaefer, Andrew
Committee MemberVielma, Juan
Date: 25 January 2016
Date Type: Publication
Defense Date: 12 November 2015
Approval Date: 25 January 2016
Submission Date: 18 November 2015
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 120
Institution: University of Pittsburgh
Schools and Programs: Swanson School of Engineering > Industrial Engineering
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Mixed Integer Nonlinear Programming, Valid Inequality, Split Cut, Intersection Cut, Branch-and-Cut, Extended Formulation
Date Deposited: 25 Jan 2016 21:50
Last Modified: 15 Nov 2016 14:30


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