Convexification Techniques for Complementarity and Multilinear Constraints

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Material Information

Title:
Convexification Techniques for Complementarity and Multilinear Constraints
Physical Description:
1 online resource (189 p.)
Language:
english
Creator:
Nguyen, Trang Thi Le
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
RICHARD,JEAN-PHILIPPE P
Committee Co-Chair:
SMITH,JONATHAN COLE
Committee Members:
LAN,GUANGHUI
HAGER,WILLIAM WARD
TAWARMALANI,MOHIT

Subjects

Subjects / Keywords:
complementarity -- convexification -- multilinear
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre:
Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
In this Dissertation, we develop tighter convex relaxations for polyhedral sets with complementarity constraints and multilinear constraints using tools inspired from integer programming. Firstly, we develop convexification techniques for mathematical programs with complementarity constraints. In particular, we generalize the reformulation-linearization technique to problems with linear complementarity constraints and discuss how it reduces to the standard technique for binary mixed-integer programs. Then, we consider specially structured complementarity sets that appear in KKT systems with linear constraints and show that their convex hulls can be obtained using integer programming convexification techniques. For these sets, we study further the case where a single complementarity constraint is imposed and show that all nontrivial facet-defining inequalities can be obtained through a simple ``cancel-and-relax" procedure. We use this result to identify special cases where standard factorable relaxation techniques produce convex hulls and other cases where they do not. We then discuss various tools that can be used to extend these results to sets with multiple complementarity constraints. We conclude by illustrating, on a set of randomly generated problems, that the relaxations produced by the techniques we propose can be significantly stronger than those described in the literature. Secondly, we discuss the problem of generating strong cutting planes for linear programs with linear complementarity constraints (LPCCs). In particular, we exploit complementarity constraints to derive cuts from the optimal simplex tableaux of LP relaxations of the problem. We introduce cmax procedure to derive convex hull for the corner relaxation and compare the strength of the cuts obtained by disjunctive programming. We also introduce the notions of split cut and split closure for LPCC problems with bounded variables. We compare these notions with their integer programming counterpart and present preliminary numerical results. Lastly, we consider optimization problem containing multilinear functions and investigate how the technique of lifting inequalities help generating strong cuts for these problems. We intend to use the technique of lifting inequalities to generate strong cuts and/or convex hulls description for $S$ when $n$ is small. We believe that such results would have direct and important applications in global solvers based on factorable programming principles.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Trang Thi Le Nguyen.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: RICHARD,JEAN-PHILIPPE P.
Local:
Co-adviser: SMITH,JONATHAN COLE.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-12-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045295:00001