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Feasible Direction Methods for Constrained Nonlinear Optimization: Suggestions for ImprovementsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2007 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Matematiska institutionen , 2007. , 29 p.
##### Series

Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1095
##### Keyword [en]

constrained nonlinear optimization, feasible direction methods, conjugate directions, traffic equilibrium problem, sequential linear programming algorithm, stochastic transportation problem
##### National Category

Computational Mathematics
##### Identifiers

URN: urn:nbn:se:liu:diva-8811ISBN: 978-91-85715-11-4OAI: oai:DiVA.org:liu-8811DiVA: diva2:23509
##### Public defence

2007-05-25, Alan Turing, Hus E, Campus Valla, Linköping University, Linköping, 10:15 (English)
##### Opponent

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##### Supervisors

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#####

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##### Note

The articles are note published due to copyright rextrictions.Available from: 2007-04-27 Created: 2007-04-27 Last updated: 2015-01-14
##### List of papers

This thesis concerns the development of novel feasible direction type algorithms for constrained nonlinear optimization. The new algorithms are based upon enhancements of the search direction determination and the line search steps.

The Frank-Wolfe method is popular for solving certain structured linearly constrained nonlinear problems, although its rate of convergence is often poor. We develop improved Frank--Wolfe type algorithms based on conjugate directions. In the conjugate direction Frank-Wolfe method a line search is performed along a direction which is conjugate to the previous one with respect to the Hessian matrix of the objective. A further refinement of this method is derived by applying conjugation with respect to the last two directions, instead of only the last one.

The new methods are applied to the single-class user traffic equilibrium problem, the multi-class user traffic equilibrium problem under social marginal cost pricing, and the stochastic transportation problem. In a limited set of computational tests the algorithms turn out to be quite efficient. Additionally, a feasible direction method with multi-dimensional search for the stochastic transportation problem is developed.

We also derive a novel sequential linear programming algorithm for general constrained nonlinear optimization problems, with the intention of being able to attack problems with large numbers of variables and constraints. The algorithm is based on inner approximations of both the primal and the dual spaces, which yields a method combining column and constraint generation in the primal space.

1. The Stiff is Moving - Conjugate Direction Frank -Wolfe Methods with Applications to Traffic Assignment$(function(){PrimeFaces.cw("OverlayPanel","overlay23504",{id:"formSmash:j_idt423:0:j_idt427",widgetVar:"overlay23504",target:"formSmash:j_idt423:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Multi-Class User Equilibria under Social Marginal Cost Pricing$(function(){PrimeFaces.cw("OverlayPanel","overlay23505",{id:"formSmash:j_idt423:1:j_idt427",widgetVar:"overlay23505",target:"formSmash:j_idt423:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. A Conjugate Direction Frank-Wolfe Method for Nonconvex Problems$(function(){PrimeFaces.cw("OverlayPanel","overlay23506",{id:"formSmash:j_idt423:2:j_idt427",widgetVar:"overlay23506",target:"formSmash:j_idt423:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. A Comparison of Feasible Direction Methods for the Stochastic Transportation Problem$(function(){PrimeFaces.cw("OverlayPanel","overlay23507",{id:"formSmash:j_idt423:3:j_idt427",widgetVar:"overlay23507",target:"formSmash:j_idt423:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. A Sequential Linear Programming Algorithm with Multi-dimensional Search: Derivation and Convergence$(function(){PrimeFaces.cw("OverlayPanel","overlay23508",{id:"formSmash:j_idt423:4:j_idt427",widgetVar:"overlay23508",target:"formSmash:j_idt423:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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