| 1. |
- Daneva (Mitradjieva), Maria, et al.
(author)
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A Comparison of Feasible Direction Methods for the Stochastic Transportation Problem
- 2010
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In: Computational optimization and applications. - : Springer Science and Business Media LLC. - 0926-6003 .- 1573-2894. ; 46:3, s. 451-466
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Journal article (peer-reviewed)abstract
- The feasible direction method of Frank and Wolfe has been claimed to be efficient for solving the stochastic transportation problem. While this is true for very moderate accuracy requirements, substantially more efficient algorithms are otherwise diagonalized Newton and conjugate Frank–Wolfe algorithms, which we describe and evaluate. Like the Frank–Wolfe algorithm, these two algorithms take advantage of the structure of the stochastic transportation problem. We also introduce a Frank–Wolfe type algorithm with multi-dimensional search; this search procedure exploits the Cartesian product structure of the problem. Numerical results for two classic test problem sets are given. The three new methods that are considered are shown to be superior to the Frank–Wolfe method, and also to an earlier suggested heuristic acceleration of the Frank–Wolfe method.
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| 2. |
- Daneva (Mitradjieva), Maria, et al.
(author)
-
A Sequential Linear Programming Algorithm with Multi-dimensional Search : Derivation and Convergence
- 2007
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Journal article (other academic/artistic)abstract
- We present a sequential linear programming, SLP, algorithm in which the traditional line-search step is replaced by a multi-dimensional search. The algorithm is based on inner approximations of both the primal and dual spaces, which yields a method which in the primal space combines column and constraint generation. The algorithm does not use a merit function, and the linear programming subproblem of the algorithm differs from the one obtained in traditional methods of this type, in the respect that linearized constraints are taken into account only implicitly in a Lagrangiandual fashion. Convergence to a point that satisfies the Karush-Kuhn-Tucker conditions is established. We apply the new method to a selection of the Hoch-Schittkowski’s nonlinear test problems and report a preliminary computational study in a Matlab environment. Since the proposed algorithmcombines column and constraint generation, it should be advantageous with large numbers of variables and constraints.
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| 3. |
- Daneva (Mitradjieva), Maria, 1974-
(author)
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Improved Frank-Wolfe directions with applications to traffic problems
- 2003
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Licentiate thesis (other academic/artistic)abstract
- The main contribution of this thesis is the development of some new efficient algorithms for solving structured linearly constrained optimization problems. The conventional Frank-Wolfe method is one of the most frequently used methods for solving such problems. We develop algorithms based on conjugate directions methods and aim to improve the performance of the pure Frank-Wolfe method by choosing better search directions.In the conjugate direction Frank-Wolfe method for linearly constrained convex optimization problems, one performs line search along a direction, which is conjugate to the previous one with respect to the hessian of the objective function at the current point. The new method is applied to the single-class traffic equilibrium problem. The convergence of the presented method is also proved. In a limited set of computational tests the algorithm turns out to be quite efficient, outperforming the pure and "PARTANized" Frank-Wolfe methods.One further refinement of the conjugate direction Frank-Wolfe methods. is derived by applying conjugation with respect to the last two directions instead of only the last one.We also extend the conjugate direction Frank-Wolfe method to nonconvex optimization problems with linear constraints. We apply this extension to the multi-class traffic equilibrium problem under social marginal cost pricing.
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| 4. |
- Lindberg, Per Olov, 1942-, et al.
(author)
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The Stiff is Moving - Conjugate Direction Frank-Wolfe Methods with Applications to Traffic Assignment
- 2012
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In: Transportation Science. - : Institute for Operations Research and the Management Sciences (INFORMS). - 0041-1655 .- 1526-5447. ; 47:2, s. 280-293
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Journal article (peer-reviewed)abstract
- We present versions of the Frank-Wolfe method for linearly constrained convex programs, in which consecutive search directions are made conjugate. Preliminary computational studies in a MATLAB environment applying pure Frank-Wolfe, Conjugate direction Frank-Wolfe (CFW), Bi-conjugate Frank-Wolfe (BFW) and ”PARTANized” Frank-Wolfe methods to some classical Traffic Assignment Problems show that CFW and BFW compare favorably to the other methods. This spurred a more detailed study, comparing our methods to Bar-Gera’s origin-based algorithm. This study indicates that our methods are competitive for accuracy requirements suggested by Boyce et al. We further show that CFW is globally convergent. We moreover point at independent studies by other researchers that show that our methods compare favourably with recent bush-based and gradient projection algorithms on computers with several cores.
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| 5. |
- Mitradjieva-Daneva, Maria, 1974-
(author)
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Feasible Direction Methods for Constrained Nonlinear Optimization : Suggestions for Improvements
- 2007
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Doctoral thesis (other academic/artistic)abstract
- 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.
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