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Linear and nonlinear semidefinite re...

Zhu, Tao.

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Linear and nonlinear semidefinite relaxations of some NP-hard problems.

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Linear and nonlinear semidefinite relaxations of some NP-hard problems./
作者:	Zhu, Tao.
面頁冊數:	114 p.
附註:	Source: Dissertation Abstracts International, Volume: 75-07(E), Section: B.
Contained By:	Dissertation Abstracts International75-07B(E).
標題:	Operations Research. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3614760
ISBN:	9781303803611

Linear and nonlinear semidefinite relaxations of some NP-hard problems.
Zhu, Tao.

Linear and nonlinear semidefinite relaxations of some NP-hard problems. - 114 p.

Source: Dissertation Abstracts International, Volume: 75-07(E), Section: B.

Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2013.

Semidefinite relaxation (SDR) is a powerful tool to estimate bounds and obtain approximate solutions for NP-hard problems. This thesis introduces and studies several novel linear and nonlinear semidefinite relaxation models for some NP-hard problems. We first study the semidefinite relaxation of Quadratic Assignment Problem (QAP) based on matrix splitting. We characterize an optimal subset of all valid matrix splittings and propose a method to find them by solving a tractable auxiliary problem. A new matrix splitting scheme called sum-matrix splitting is also proposed and its numerical performance is evaluated.

ISBN: 9781303803611Subjects--Topical Terms:

626629
Operations Research.

Linear and nonlinear semidefinite relaxations of some NP-hard problems.
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245 1 0 $a Linear and nonlinear semidefinite relaxations of some NP-hard problems.
300 $a 114 p.
500 $a Source: Dissertation Abstracts International, Volume: 75-07(E), Section: B.
500 $a Adviser: Jiming Peng.
502 $a Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2013.
520 $a Semidefinite relaxation (SDR) is a powerful tool to estimate bounds and obtain approximate solutions for NP-hard problems. This thesis introduces and studies several novel linear and nonlinear semidefinite relaxation models for some NP-hard problems. We first study the semidefinite relaxation of Quadratic Assignment Problem (QAP) based on matrix splitting. We characterize an optimal subset of all valid matrix splittings and propose a method to find them by solving a tractable auxiliary problem. A new matrix splitting scheme called sum-matrix splitting is also proposed and its numerical performance is evaluated.
520 $a We next consider the so-called Worst-case Linear Optimization (WCLO) problem which has applications in systemic risk estimation and stochastic optimization. We show that WCLO is NP-hard and a coarse linear SDR is presented. An iterative procedure is introduced to sequentially refine the coarse SDR model and it is shown that the sequence of refined models converge to a nonlinear semidefinite relaxation (NLSDR) model. We then propose a bisection algorithm to solve the NLSDR in polynomial time. Our preliminary numerical results show that the NLSDR can provide very tight bounds, even the exact global solution, for WCLO.
520 $a Motivated by the NLSDR model, we introduce a new class of relaxation called conditionally quasi-convex relaxation (CQCR). The new CQCR model is obtained by augmenting the objective with a special kind of penalty function. The general CQCR model has an undetermined nonnegative parameter alpha and the CQCR model with alpha = 0 (denoted by CQCR(0)) is the strongest of all CQCR models. We next propose an iterative procedure to approximately solve CQCR(0) and a bisection procedure to solve CQCR(0) under some assumption. Preliminary numerical experiments illustrate the proposed algorithms are effective and the CQCR(0) model outperforms classic relaxation models.
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