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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 64 (1994), S. 53-79 
    ISSN: 1436-4646
    Keywords: Finite-dimensional variational inequalities ; Merit functions ; Variational principles ; Successive approximation algorithms ; Descent algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Recently Auchmuty (1989) has introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. We develop and generalize Auchmuty's results, and relate his class of merit functions to other works done in this field. Especially, we investigate differentiability and convexity properties, and present characterizations of the set of solutions to variational inequalities. We then present new descent algorithms for variational inequalities within this framework, including approximate solutions of the direction finding and line search problems. The new class of merit functions include the primal and dual gap functions, introduced by Zuhovickii et al. (1969a, 1969b), and the differentiable merit function recently presented by Fukushima (1992); also, the descent algorithm proposed by Fukushima is a special case from the class of descent methods developed in this paper. Through a generalization of Auchmuty's class of merit functions we extend those inherent in the works of Dafermos (1983), Cohen (1988) and Wu et al. (1991); new algorithmic equivalence results, relating these algorithm classes to each other and to Auchmuty's framework, are also given.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 86 (1999), S. 283-312 
    ISSN: 1436-4646
    Keywords: Key words: convex programming – Lagrangean duality – Lagrangean relaxation – subgradient optimization – ergodic convergence – primal convergence – traffic equilibrium assignment – road pricing
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract. Lagrangean dualization and subgradient optimization techniques are frequently used within the field of computational optimization for finding approximate solutions to large, structured optimization problems. The dual subgradient scheme does not automatically produce primal feasible solutions; there is an abundance of techniques for computing such solutions (via penalty functions, tangential approximation schemes, or the solution of auxiliary primal programs), all of which require a fair amount of computational effort. We consider a subgradient optimization scheme applied to a Lagrangean dual formulation of a convex program, and construct, at minor cost, an ergodic sequence of subproblem solutions which converges to the primal solution set. Numerical experiments performed on a traffic equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained, compared to those generated in the basic subgradient scheme.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Computational optimization and applications 9 (1998), S. 5-42 
    ISSN: 1573-2894
    Keywords: Cartesian product sets ; decomposition ; cost approximation ; sequential algorithm ; parallel processing ; partially asynchronous algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract This paper presents a unified analysis of decomposition algorithms for continuously differentiable optimization problems defined on Cartesian products of convex feasible sets. The decomposition algorithms are analyzed using the framework of cost approx imation algorithms. A convergence analysis is made for three decomposition algorithms: a sequential algorithm which extends the classical Gauss-Seidel scheme, a synchronized parallel algorithm which extends the Jacobi method, and a partially asynchronous parallel algorithm. The analysis validates inexact computations in both the subproblem and line search phases, and includes convergence rate results. The range of feasible step lengths within each algorithm is shown to have a direct correspondence to the increasing degree of parallelism and asynchronism, and the resulting usage of more outdated information in the algorithms.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Annals of operations research 82 (1998), S. 309-330 
    ISSN: 1572-9338
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract The combined distribution and assignment problem is the problem of the simultaneousdetermination of the distribution of trips between origins and destinations in a transportationnetwork and the assignment of trips to routes in each origin - destination pair. In the modelmost widely used, the distribution is assumed to follow a gravity model with a negativeexponential deterrence function and the assignment is made according to the deterministicuser equilibrium principle. In this paper, we describe an extension of this model in whichthe allocation of trips to routes is made according to the principle of stochastic user equilibrium.We discuss the behavioural foundations of trip assignment and combined modelsassuming deterministic or stochastic route choice. In particular, we describe how they can bederived using the efficiency principle from discrete choice theory; the combined distributionand stochastic assignment model is obtained as the continuous approximation of the discreteproblem of finding the most probable flow pattern under the assumption of efficient tripmaking behaviour. We outline an algorithm for the solution of the model which is based onroute (column) generation, disaggregate simplicial decomposition, and partial linearization.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 40 (1997), S. 1295-1321 
    ISSN: 0029-5981
    Keywords: optimum sheet ; unilateral contact ; non-differentiable convex programming ; subgradient optimization algorithm ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We consider the solution of finite element discretized optimum sheet problems by an iterative algorithm. The problem is that of maximizing the stiffness of a sheet subject to constraints on the admissible designs and unilateral contact conditions on the displacements. The model allows for zero design volumes, and thus constitutes a true topology optimization problem.We propose and evaluate a subgradient optimization algorithm for a reformulation into a non-differentiable, convex minimization problem in the displacement variables. The convergence of the method and its low computational complexity are established. An optimal design is derived through a simple averaging scheme which combines the solutions to the linear design problems solved within the subgradient method.To illustrate the efficiency of the algorithm and investigate the properties of the optimal designs, thealgorithm is numerically tested on some medium- and large-scale problems. © 1997 by John Wiley & Sons, Ltd.
    Additional Material: 7 Ill.
    Type of Medium: Electronic Resource
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  • 6
    Title: ¬An¬ introduction to continuous optimization /
    Author: Andréasson, Niclas
    Contributer: Evgrafov, Anton , Patriksson, Michael
    Publisher: Lund :Studentlitteratur,
    Year of publication: 2005
    Pages: [3], S. VI-XII, 389 S.
    ISBN: 91-44-04455-0
    Type of Medium: Book
    Language: Undetermined
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  • 7
    Book
    Book
    Utrecht, Netherlands :VSP,
    Title: ¬The¬ traffic assigment problem: models and methods
    Author: Patriksson, Michael
    Publisher: Utrecht, Netherlands :VSP,
    Year of publication: 1994
    Pages: 223 S.
    Series Statement: Topics in transportation
    Type of Medium: Book
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  • 8
    Title: Optimization of maintenance activities - models, methods and applications /; 224
    Contributer: Patriksson, Michael
    Year of publication: 2015
    Pages: 192 S.
    Series Statement: Annals of operations research 224
    Type of Medium: Book
    Language: English
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