Patriksson, M.: `A unified description of iterative algorithms for traffic equilibria', European Journal of Operational Research 71 (1993), 154--176.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[39] to describe the construction of the subproblem of a class of iterative methods in mathematical programming. In order to explain the notion of CA, we will consider the following conceptual problem (the full generality of the algorithm is explained in detail in the book [45] and in the papers [37, 38, 40, 42, 43, 44, 46]) minimize T (x) f(x) u(x) 1a) subject to x 2 X; 1b) where X n is nonempty, closed and convex, u : n 7 [ f 1g is lower semi continuous (l.s.c. proper and convex, and f : n 7 [ f 1g is continuously differentiable (for short, in C 1 ) on domu X, where dom denotes ....

Patriksson, M.: `A unified description of iterative algorithms for traffic equilibria', European Journal of Operational Research 71 (1993), 154--176.

....entry, the left column contains a reference to the article(s) in which the algorithm class is proposed and established convergent. The right column contains the class description as a cost approximation algorithm in the taxonomy (3) these descriptions are derived from analyses made in [27, 22, 23, 24, 13, 25, 28] In the table, fl k is a positive parameter. The reader should note that in addition to the assumptions stated in Section 1 on the parameters in the eight tuple (3) some technical assumptions are necessary in order to establish convergence of the algorithms given in Table 1. 5 ....

....is made. In the table, Delta denotes the identity mapping, fl k a positive parameter, C an index set, and B k a positive semidefinite matrix of appropriate dimension. The descriptions are derived from relations between these algorithms and the cost approximation algorithm investigated in [22, 23, 24, 25] 6 Remarks An immediate result of the classification is that we find, from Table 1, that most of the algorithm frameworks that have been developed for the solution of [VIP] and its special cases have been established both with stronger assumptions than necessary and with weaker results ....

M. Patriksson, A unified description of iterative algorithms for traffic equilibria, European Journal of Operational Research, 71 (1993), pp. 154--176.

....the Convergence of Descent Methods for Monotone Variational Inequalities M. Patriksson 1 Abstract. Recently, Zhu and Marcotte [15] established the convergence of a modified descent algorithm for monotone variational inequalities. Using algorithmic equivalence results due to Patriksson [10, 11] and Larsson and Patriksson [7] we show that this convergence result may be used to establish the convergence of slightly modified versions of the classical successive approximation algorithms of Dafermos [2] and Cohen [1] and of the descent algorithms of Wu et al. 14] Patriksson [10, 11] and ....

.... [10, 11] and Larsson and Patriksson [7] we show that this convergence result may be used to establish the convergence of slightly modified versions of the classical successive approximation algorithms of Dafermos [2] and Cohen [1] and of the descent algorithms of Wu et al. 14] Patriksson [10, 11], and Larsson and Patriksson [7] under assumptions that are both much milder and much easier to verify than those for their original statements. Key Words. Monotone variational inequalities, successive approximation algorithms, merit functions, descent methods, algorithmic equivalence, ....

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M. Patriksson, A unified description of iterative algorithms for traffic equilibria, European Journal of Operational Research, 71 (1993), pp. 154--176.

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M. Patriksson, "A unified description of iterative algorithms for traffic equilibria," Report LiTH-MAT-R-91-34, Linkoping Institute of Technology (Linkoping, Sweden, 1991). Forthcoming in European Journal of Operational Research.

....set of f at x 0 2 X with respect to X is the set L(x 0 ) f x 2 X j f(x) f(x 0 ) g: 1. 3 The Cost Approximation Algorithm Derivation The iterative algorithms studied in this paper are applications to [CDP] of the class of cost approximation (CA) algorithms, introduced and analyzed in [50, 51, 52, 53]. These algorithms are designed to find stationary points of [CDP] that is, solutions x 2 X to the variational inequality problem [VIP] rf(x ) T (x Gamma x ) 0; 8x 2 X: We let Omega denote the set of stationary points of [CDP] A stationary point is a global minimizer of f ....

.... i ) T (y i Gamma y i (x) 0; 8y i 2 X i : If Phi i j r i for some convex function i : X i 7 in C 1 on X i , then [VIP Phi i ] is the optimality conditions for [CDP i ] min y i 2X i f i (y i ) i (y i ) r i f(x) Gamma r i (x i ) T y i : In the convergence analysis in [50, 51, 52, 53] of the CA algorithm, two forms of the cost approximating mapping have been used. The most general case is the choice of a sequence f Phi t g of continuous and monotone mappings on X; each mapping may be chosen based on the current iterate x t (or, more generally, on the whole sequence of ....

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M. Patriksson, A unified description of iterative algorithms for traffic equilibria, European J. Oper. Res., 71 (1993), pp. 154--176.

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Patriksson, M., A unified description of iterative algorithms for traffic equilibria, European Journal of Operational Research, 71 (1993) 154--176.

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Patriksson, M., A Unified Description of Iterative Algorithms for Traffic Equilibria, European Journal of Operational Research, forthcoming.

....[NDP] expressing the objective T of [NDP] as [ u] f Gamma ] the objective T of [NDP ] is obtained from a firstorder approximation (or linearization) of the second term at x. This interpretation of the cost approximation algorithm as a partial linearization method was first made in [76, 77, 78]. Thus, by applying the cost approximation concept, the generalized equation [GE] can be solved through a sequence of convex optimization problems. Indeed, in most of the iterative methods for the solution of [GE] that we will identify as special cases from the class of cost approximation ....

....x is a direction of descent at noncritical points. In the more general case of generalized equation which corresponds to an asymmetric variational inequality problem, the merit function may be chosen as the gap function associated with the symmetric cost approximation subproblem [NDP ] see [75, 77, 78, 55, 79, 80] for further details. At the new point, the original mapping is again approximated perhaps with another cost approximating mapping Phi and the algorithm proceeds until some stopping criterion is fulfilled. 3.3 The conceptual algorithm A sequence f Phi k g of monotone cost approximating ....

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M. Patriksson, A unified description of iterative algorithms for traffic equilibria, European J. Oper. Res., 71 (1993), pp. 154--176.

....The paper concludes with some observations and remarks. 2 A merit function The class of merit functions used defines an extension to primal dual variational inequality problems of the differentiable gap functions for primal variational inequality problems which have a long history (see, e.g. [Auc89, Fuk92, Pat93a, Pat93b, WFM93, ZhM93, LaP94, Pat94, ZhM94, Fuk96, MaZ95, Pat97, Pat98]) Let : X Theta X 7 n be a C 1 function on X Theta X of the form (y; x) strictly convex in y for each fixed x 2 X, and with the additional property that (x; x) 0 and r y (x; x) 0 n for all x 2 X. Let ff 0, and consider the function ff (x) maximum y2X L ff (y; x) ....

....in r. The result follows. 3 A modified descent algorithm for variational inequalities We introduce a class of search direction subproblems in the class of cost approximation methods ( Pat93b] for the solution (3) which extend the search direction finding subproblems of the algorithms of [Pat93a, Pat93b, WFM93, ZhM93, Pat94, ZhM94, MaZ95] to the explicit treatment of constraints and to the present primal dual setting. For any ff 0, let Phi ff : X Theta X Theta m 7 n be a continuous mapping on X Theta X Theta m of the form Phi ff (y; x; p) monotone and in C 1 on X with respect to y. Given (x; p) 2 X ....

M. Patriksson (1993), "A unified description of iterative algorithms for traffic equilibria," European Journal of Operational Research, vol. 71, pp. 154--176.

....that the result holds, thanks to the properties of the mapping A. The rest of the proof follows that of Theorem 4.1. 2 A natural sufficient condition for the boundedness of fy t g is that dom u is bounded; the special case u j ffi X for a nonempty, compact and convex set X is covered in [Pat93b]. 4.3 Implementable step length rules Theorem 4.3 (Convergence under Rule A ) Let Phi : dom u Theta dom u 7 n be a continuous mapping on dom u Theta dom u of the form Phi(y; x) maximal and strictly monotone on dom u with respect to y. Assume that the point x 0 is chosen so that the ....

.... defined by the CA method, and the convergence analyses are, in all cases, more limited and weaker than ours; examples of such frameworks include those in [DFL86, LaM90, Tse91, Mig94, ZhM95] Applications of the convergence analysis of the CA method to differentiable optimization can be found in [Pat93b, Pat93c]. 5.3 Concluding remarks The class of CA algorithms is very general; some examples of methods that are included have been mentioned. What we have not mentioned is that the generality present allows one to adapt the algorithm to problem structures (such as choosing Phi to adapt to a separability ....

M. Patriksson, A unified description of iterative algorithms for traffic equilibria, European J. Oper. Res., 71 (1993), pp. 154--176.

.... new solution h k 1 and the corresponding demand solution d k 1 , and the process is repeated with k : k 1, until some convergence criterion terminates the solution of [CDA SUE R] The algorithm described briefly above is an instance of the partial linearization algorithm, which is analyzed in [15, 16, 17]. According to [16, Lem. 1] and [15, Thm. 2.1] h k = h k holds if and only if h k solves [CDA SUE R] otherwise, the direction of h k Gamma h k is a feasible direction of descent with respect to the objective function W of [CDA SUER ] Since W is strictly convex in h, the sequence fh ....

M. Patriksson, A unified description of iterative algorithms for traffic equilibria, European Journal of Operational Research, 17 (1993) 154--176.

....iterative algorithms for the solution of [CDP] which are adapted to the Cartesian product structure of its constraints. The basis for the development is the convergence analysis of the cost approximation (CA) algorithm [54, 55] and the closely related family of partial linearization algorithms [52, 53]. These are designed to find stationary points of [CDP] that is, points x 2 X satisfying the variational inequality [VIP] rf(x ) T (x Gamma x ) 0; 8x 2 X: We let Omega denote the set of stationary points of [CDP] A stationary point is a global minimizer of f on X whenever f is ....

.... T (y i Gamma y i (x) 0; 8y i 2 X i : If Phi i j r i for some convex function i : X i 7 in C 1 on X i , then [VIP Phi i ] is the optimality conditions for [CDP i ] minimize y i 2X i f i (y i ) i (y i ) r i f(x) Gamma r i (x i ) T y i : In the convergence analysis in [52, 53, 54, 55] of the CA algorithm, two forms of the cost approximating mapping have been used. The most general one is a sequence f Phi t g of continuous and monotone mappings on X; each mapping may be chosen based on the current iterate x t (or, more generally, on the whole sequence of iterates up to x ....

[Article contains additional citation context not shown here]

M. Patriksson, A unified description of iterative algorithms for traffic equilibria, European J. Oper. Res., 71 (1993), pp. 154--176.

No context found.

Patriksson. M. (1991), A unified description of iterative algorithms for traffic equilibria. Report LiTH-MAT-R-91-35, Linkoping Institute of Technology, Department of Mathematics, Linkoping, Sweden. Forthcoming in European Journal of Operational Research.

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