Approach for Solution of Nonlinear Optimization Problems with Blocks Structure and Coupling Constraint

Олена Євстафіївна Кірік

Abstract


Background. Non-linear optimization problems for the operation of large space or functionally distributed systems with independently functioning subsystems and restrictions on some of the common resources are considered.

Objective. The aim is to build an effective approach to solving nonlinear optimization problems of block structure with coupling constraints on the basis of the combination of approximating nonlinear optimization methods and              decomposition techniques.

Methods. The three-step iterative scheme has been proposed. On the upper level the original problem is replaced by    a sequence of approximating problems with additively-separable objective function and linear constraints. On the        second level, coordinating problems, which are formed by the information received at the lowest level from local block problems, are solved.

Results. The algorithm, which is a combination of the linearization method of B.Pshenichniy and the dual decomposition, has been built. The structure of the dual coordination problem has been described. Estimation of the convergence rate has been carried out.

Сonclusions. The proposed approach and the algorithm constructed based on it can be applied to a wide range of problems associated with optimal allocation of limited resources in the large block-structured systems.

Keywords


Resource distribution problems; Optimization models; Non-linear programming methods; Decomposition algorithms

References


L.S. Lasdon, Optimization Theory for Large Systems. Moscow, USSR: Nauka, 1975, 432 p. (in Russian).

V.I. Tsurkov, Decomposition in Problems of Large Dimension. Moscow, USSR: Nauka, 1981, 352 p. (in Russian).

P. Purkayastha and J. Baras, “An optimal distributed routing algorithm using dual decomposition techniques”, Commun. Inform. Systems, vol. 8, no. 3, pp. 277–302, 2008.

D. Goldfarb and S. Ma, “Fast multiple splitting algorithms for convex optimization”, SIAM J. Optimization, vol. 22, no. 2, pp. 533–556, 2012.

N. Komodakis et al., “MRF energy minimization & beyond via dual decomposition”, IEEE Trans. Pattern Anal. Mach. Intell., vol. 33, is. 3, pp. 531–552, 2011.

B.N. Pshenichnyi and Y.M. Danilin, Numerical Methods in Extremal Problems. Moscow, USSR: Nauka, 1975, 319 p. (in Russian).

O.E. Kirik, “The linearization and the conjugate gradient algorithms for nonlinear network flow distribution problems”, Naukovi Visti NTUU “KPI”, no. 3, pp. 67–73, 2007 (in Ukrainian).

B.N. Pshenichnyi, Convex Analysis and Extremal Problems. Moscow, USSR: Nauka, 1980, 320 p. (in Russian).

O.E. Kirik and V.V. Ostapenko, “Optimal hydro resource distribution in the network structure irrigation systems”, Systemni Doslidzhennya ta Inform. Tekhnolohiyi, no. 4, pp. 79–90, 2010 (in Ukrainian).

V.M. Alexandrova and O.E. Kirik, “Optimization models and algorithms for network problems of resours’ distribution”, Naukovi Visti NTUU “KPI”, no. 5. pp. 39–45, 2014 (in Ukrainian).


GOST Style Citations


  1. Лэсдон Л.C. Оптимизация больших систем. – М.: Наука, 1975. – 432 с.
  2. Цурков В.И. Декомпозиция в задачах большой размерности. – М.: Наука, 1981. – 352 с.

  3. Purkayastha P., Baras J. An optimal distributed routing algorithm using dual decomposition techniques // Commun. Inform. Systems. – 2008. – 8, № 3. – P. 277–302.

  4. Goldfarb D., Ma S. Fast multiple splitting algorithms for convex optimization // SIAM J. Optimization. – 2012. – 22, № 2. –  P. 533–556.

  5. Komodakis N., Paragios N., Tziritas G. MRF energy minimization and beyond via dual decomposition // IEEE Trans. Pattern Anal. Mach. Intell. – 2011. – 33, is. 3. – P. 531–552.

  6. Пшеничный Б.Н., Данилин Ю.М. Численные методы в экстремальных задачах. – М.: Наука, 1975. – 319 с.

  7. Кірік О.Є. Алгоритми лінеаризації та спряжених градієнтів для нелінійних задач розподілу потоків // Наукові вісті НТУУ “КПІ”. – 2007. – № 3. – С. 67–73.

  8. Пшеничный В.Н. Выпуклый анализ и экстремальные задачи. – М.: Наука, 1980. – 320 с.

  9. Кірік О.Є., Остапенко В.В. Оптимальний розподіл гідроресурсів у зрошувальних системах мережевої структури // Системні дослідження та інформ. технології. – 2010. – № 4. – С. 79–90.

  10. Александрова В.М., Кірік О.Є. Оптимізаційні моделі й алгоритми для мережевих задач розподілу ресурсів // Наукові вісті НТУУ “КПІ”. – 2014. – № 5. – 39–45. 




DOI: https://doi.org/10.20535/1810-0546.2015.5.65113

Refbacks

  • There are currently no refbacks.




Copyright (c) 2017 NTUU KPI