Geodesic
Reduction of a~minimization problem of a~separable convex function under linear constraints to a~fixed point problem
Diskretnyj analiz i issledovanie operacij, Tome 25 (2018) no. 1, pp. 75-97.

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The paper is devoted to studying a constrained nonlinear optimization problem of a special kind. The objective functional of the problem is a separable convex function whose minimum is sought for on a set of linear constraints in the form of equalities. It is proved that, for this type of optimization problems, the explicit form can be obtained of a projection operator based on a generalized projection matrix. The projection operator allows us to represent the initial problem as a fixed point problem. The explicit form of the fixed point problem makes it possible to run a process of simple iteration. We prove the linear convergence of the obtained iterative method and, under rather natural additional conditions, its quadratic convergence. It is shown that an important application of the developed method is the flow assignment in a network of an arbitrary topology with one pair of source and sink. Bibliogr. 10.
Mots-clés : constrained nonlinear optimization, fixed point problem, generalized projection matrix, network flow assignment. 
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A. Yu. Krylatov. Reduction of a~minimization problem of a~separable convex function under linear constraints to a~fixed point problem. Diskretnyj analiz i issledovanie operacij, Tome 25 (2018) no. 1, pp. 75-97. https://geodesic-test.mathdoc.fr/item/DA_2018_25_1_a3/

[1] F. R. Gantmakher, Theory of Matrices, Nauka, Moscow, 1966 (Russian) | MR

[2] I. V. Konnov, Nonlinear Optimization and Variational Inequalities, Kazan. Univ., Kazan, 2013 (Russian)

[3] A. Yu. Krylatov, “Network flow assignment as a fixed point problem”, J. Appl. Ind. Math., 10:2 (2016), 243–256 | DOI | DOI | MR

[4] M. N. S. Swamy, K. Thulasiraman, Graphs, Networks, and Algorithms, Wiley, New York, 1981 | MR

[5] Beckmann M. J., McGuire C. B., Winsten C. B., Studies in the economics of transportation, Yale Univ. Press, New Haven, CT, 1956, 359 pp.

[6] Bertsekas D. P., Nonlinear programming, Athena Scientific, Belmont, MA, 1999 | MR

[7] Chen R.-J., Meyer R. R., “Parallel optimization for traffic assignment”, Math. Program., 42:1–3 (1988), 327–345 | DOI | MR

[8] Patriksson M., “A unified description of iterative algorithms for traffic equilibria”, Eur. J. Oper. Res., 71:2 (1993), 154–176 | DOI | MR

[9] Patriksson M., The traffic assignment problem: models and methods, Dover Publ., Mineola, NY, 2015, 240 pp.

[10] Wardrop J. G., “Some theoretical aspects of road traffic research”, Proc. Inst. Civil Eng., 1:3 (1952), 325–362