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Conic Optimization: Optimal Partitio...

Lehigh University.

Conic Optimization: Optimal Partition, Parametric, and Stability Analysis.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Conic Optimization: Optimal Partition, Parametric, and Stability Analysis.
Author:	Mohammad-Nezhad, Ali.
Published:	Ann Arbor : ProQuest Dissertations & Theses, 2019
Description:	294 p.
Notes:	Source: Dissertations Abstracts International, Volume: 80-09, Section: B.
Notes:	Publisher info.: Dissertation/Thesis.
Notes:	Advisor: Terlaky, Tamas.
Contained By:	Dissertations Abstracts International80-09B.
Subject:	Applied Mathematics.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10933941
ISBN:	9780438889354

Conic Optimization: Optimal Partition, Parametric, and Stability Analysis.
Mohammad-Nezhad, Ali.

Conic Optimization: Optimal Partition, Parametric, and Stability Analysis. - Ann Arbor : ProQuest Dissertations & Theses, 2019 - 294 p.

Source: Dissertations Abstracts International, Volume: 80-09, Section: B.

Thesis (Ph.D.)--Lehigh University, 2019.

This item must not be sold to any third party vendors.

A linear conic optimization problem consists of the minimization of a linear objective function over the intersection of an affine space and a closed convex cone. In recent years, linear conic optimization has received significant attention, partly due to the fact that we can take advantage of linear conic optimization to reformulate and approximate intractable optimization problems. Steady advances in computational optimization have enabled us to approximately solve a wide variety of linear conic optimization problems in polynomial time. Nevertheless, preprocessing methods, rounding procedures and sensitivity analysis tools are still the missing parts of conic optimization solvers. Given the output of a conic optimization solver, we need methodologies to generate approximate complementary solutions or to speed up the convergence to an exact optimal solution. A preprocessing method reduces the size of a problem by finding the minimal face of the cone which contains the set of feasible solutions. However, such a preprocessing method assumes the knowledge of an exact solution. More importantly, we need robust sensitivity and post-optimal analysis tools for an optimal solution of a linear conic optimization problem. Motivated by the vital importance of linear conic optimization, we take active steps to fill this gap. This thesis is concerned with several aspects of a linear conic optimization problem, from algorithm through solution identification, to parametric analysis, which have not been fully addressed in the literature. We specifically focus on three special classes of linear conic optimization problems, namely semidefinite and second-order conic optimization, and their common generalization, symmetric conic optimization. We propose a polynomial time algorithm for symmetric conic optimization problems. We show how to approximate/identify the optimal partition of semidefinite optimization and second-order conic optimization, a concept which has its origin in linear optimization. Further, we use the optimal partition information to either generate an approximate optimal solution or to speed up the convergence of a solution identification process to the unique optimal solution of the problem. Finally, we study the parametric analysis of semidefinite and second-order conic optimization problems. We investigate the behavior of the optimal partition and the optimal set mapping under perturbation of the objective function vector.

ISBN: 9780438889354Subjects--Topical Terms:

530992
Applied Mathematics.

Conic Optimization: Optimal Partition, Parametric, and Stability Analysis.
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300 $a 294 p.
500 $a Source: Dissertations Abstracts International, Volume: 80-09, Section: B.
500 $a Publisher info.: Dissertation/Thesis.
500 $a Advisor: Terlaky, Tamas.
502 $a Thesis (Ph.D.)--Lehigh University, 2019.
506 $a This item must not be sold to any third party vendors.
520 $a A linear conic optimization problem consists of the minimization of a linear objective function over the intersection of an affine space and a closed convex cone. In recent years, linear conic optimization has received significant attention, partly due to the fact that we can take advantage of linear conic optimization to reformulate and approximate intractable optimization problems. Steady advances in computational optimization have enabled us to approximately solve a wide variety of linear conic optimization problems in polynomial time. Nevertheless, preprocessing methods, rounding procedures and sensitivity analysis tools are still the missing parts of conic optimization solvers. Given the output of a conic optimization solver, we need methodologies to generate approximate complementary solutions or to speed up the convergence to an exact optimal solution. A preprocessing method reduces the size of a problem by finding the minimal face of the cone which contains the set of feasible solutions. However, such a preprocessing method assumes the knowledge of an exact solution. More importantly, we need robust sensitivity and post-optimal analysis tools for an optimal solution of a linear conic optimization problem. Motivated by the vital importance of linear conic optimization, we take active steps to fill this gap. This thesis is concerned with several aspects of a linear conic optimization problem, from algorithm through solution identification, to parametric analysis, which have not been fully addressed in the literature. We specifically focus on three special classes of linear conic optimization problems, namely semidefinite and second-order conic optimization, and their common generalization, symmetric conic optimization. We propose a polynomial time algorithm for symmetric conic optimization problems. We show how to approximate/identify the optimal partition of semidefinite optimization and second-order conic optimization, a concept which has its origin in linear optimization. Further, we use the optimal partition information to either generate an approximate optimal solution or to speed up the convergence of a solution identification process to the unique optimal solution of the problem. Finally, we study the parametric analysis of semidefinite and second-order conic optimization problems. We investigate the behavior of the optimal partition and the optimal set mapping under perturbation of the objective function vector.
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