Integer programming approaches for semicontinuous and stochastic optimization

Angulo Olivares, Gustavo, I

Title:

Integer programming approaches for semicontinuous and stochastic optimization

dc.contributor.advisor	Ahmed, Shabbir
dc.contributor.advisor	Dey, Santanu S.
dc.contributor.author	Angulo Olivares, Gustavo, I
dc.contributor.committeeMember	Nemhauser, George L.
dc.contributor.committeeMember	Kaibel, Volker
dc.contributor.committeeMember	Cheon, Myun-Seok
dc.contributor.department	Industrial and Systems Engineering
dc.date.accessioned	2014-05-22T15:30:16Z
dc.date.available	2014-05-22T15:30:16Z
dc.date.created	2014-05
dc.date.issued	2014-04-07
dc.date.submitted	May 2014
dc.date.updated	2014-05-22T15:30:16Z
dc.description.abstract	This thesis concerns the application of mixed-integer programming techniques to solve special classes of network flow problems and stochastic integer programs. We draw tools from complexity and polyhedral theory to analyze these problems and propose improved solution methods. In the first part, we consider semi-continuous network flow problems, that is, a class of network flow problems where some of the variables are required to take values above a prespecified minimum threshold whenever they are not zero. These problems find applications in management and supply chain models where orders in small quantities are undesirable. We introduce the semi-continuous inflow set with variable upper bounds as a relaxation of general semi-continuous network flow problems. Two particular cases of this set are considered, for which we present complete descriptions of the convex hull in terms of linear inequalities and extended formulations. We also consider a class of semi-continuous transportation problems where inflow systems arise as substructures, for which we investigate complexity questions. Finally, we study the computational efficacy of the developed polyhedral results in solving randomly generated instances of semi-continuous transportation problems. In the second part, we introduce and study the forbidden-vertices problem. Given a polytope P and a subset X of its vertices, we study the complexity of optimizing a linear function on the subset of vertices of P that are not contained in X. This problem is closely related to finding the k-best basic solutions to a linear problem and finds applications in stochastic integer programming. We observe that the complexity of the problem depends on how P and X are specified. For instance, P can be explicitly given by its linear description, or implicitly by an oracle. Similarly, X can be explicitly given as a list of vectors, or implicitly as a face of P. While removing vertices turns to be hard in general, it is tractable for tractable 0-1 polytopes, and compact extended formulations can be obtained. Some extensions to integral polytopes are also presented. The third part is devoted to the integer L-shaped method for two-stage stochastic integer programs. A widely used model assumes that decisions are made in a two-step fashion, where first-stage decisions are followed by second-stage recourse actions after the uncertain parameters are observed, and we seek to minimize the expected overall cost. In the case of finitely many possible outcomes or scenarios, the integer L-shaped method proposes a decomposition scheme akin to Benders' decomposition for linear problems, but where a series of mixed-integer subproblems have to be solved at each iteration. To improve the performance of the method, we devise a simple modification that alternates between linear and mixed-integer subproblems, yielding significant time savings in instances from the literature. We also present a general framework to generate optimality cuts via a cut-generating problem. Using an extended formulation of the forbidden-vertices problem, we recast our cut-generating problem as a linear problem and embed it within the integer L-shaped method. Our numerical experiments suggest that this approach can prove beneficial when the first-stage set is relatively complicated.
dc.description.degree	Ph.D.
dc.format.mimetype	application/pdf
dc.identifier.uri	http://hdl.handle.net/1853/51862
dc.language.iso	en_US
dc.publisher	Georgia Institute of Technology
dc.subject	Integer programming
dc.subject	Stochastic programming
dc.subject	Forbidden vertices
dc.subject.lcsh	Mathematical optimization
dc.subject.lcsh	Dynamic programming
dc.subject.lcsh	Integer programming
dc.title	Integer programming approaches for semicontinuous and stochastic optimization
dc.type	Text
dc.type.genre	Dissertation
dspace.entity.type	Publication
local.contributor.advisor	Dey, Santanu S.
local.contributor.corporatename	H. Milton Stewart School of Industrial and Systems Engineering
local.contributor.corporatename	College of Engineering
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relation.isOrgUnitOfPublication	29ad75f0-242d-49a7-9b3d-0ac88893323c
relation.isOrgUnitOfPublication	7c022d60-21d5-497c-b552-95e489a06569
thesis.degree.level	Doctoral