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Blocks of coordinates, stochastic programming, and markets

  • Sjur Didrik Flåm
Original Paper
  • 25 Downloads

Abstract

Considered here are extremal convolutions concerned with allocative efficiency, risk sharing, or market equilibrium. Each additive term is upper semicontinuous, proper concave, maybe non-smooth, and possibly extended-valued. In a leading interpretation, each term, alongside its block of coordinates, is controlled by an independent economic agent. Vectors are construed as contingent claims or as bundles of commodities. These are diverse, divisible, and perfectly transferable. At every stage two randomly selected agents make bilateral direct exchanges. The amounts transferred between the two parties depend on the difference between their generalized gradients. The resulting process—and the associated convergence analysis—fits the frames of stochastic programming. Motivation stems from exchange markets.

Keywords

Block-coordinate methods Convolution Projected gradients Stochastic programming Bilateral matching Market equilibrium 

Notes

Acknowledgements

Thanks for support are due the department and Røwdes Fond.

References

  1. Benveniste A, Métivier M, Priouret P (1990) Adaptive algorithms and stochastic approximations. Springer, BerlinCrossRefGoogle Scholar
  2. Cockerell HAL, Green E (1976) The British Insurance Business. Sheffield Academic Press, SheffieldGoogle Scholar
  3. Eeckhoudt L, Gollier C, Schlesinger H (2005) Economic and financial decisions under risk. Princeton University Press, PrincetonGoogle Scholar
  4. Feldman AM (1973) Bilateral trading processes, pair-wise optimality, and Pareto optimality. Rev Econ Stud 4:463–473CrossRefGoogle Scholar
  5. Flåm SD (2016a) Bilateral exchange and competitive equilibrium. Set-Valued Var Anal 24:1–11CrossRefGoogle Scholar
  6. Flåm SD (2016b) Borch’s theorem, equal margins, and efficient allocations. Insur Math Econ 70:162–168CrossRefGoogle Scholar
  7. Flåm SD, Gramstad K (2012) Direct exchange in linear economies. Int Game Theory Rev 14:4CrossRefGoogle Scholar
  8. Gaivoronski AA (1994) Convergence properties of backpropagation for neural nets via theory of stochastic gradient methods. Optim Methods Softw 4(2):117–134CrossRefGoogle Scholar
  9. Hiriart-Urruty J-B (2012) Bases, outils et principes pour l’analyse variationelle. Springer, BerlinGoogle Scholar
  10. Hiriart-Urruty J-B, Marechal C (1993) Convex analysis and minimization algorithms I. Springer, BerlinCrossRefGoogle Scholar
  11. Hua X, Yamashita N (2016) Block coordinate proximal gradient methods for nonsmooth separable optimization. Math Program Ser A 160:1–32CrossRefGoogle Scholar
  12. Lengwiler Y (2004) Microfoundations of financial economics. Princeton University Press, PrincetonGoogle Scholar
  13. Necoara I (2013) Random coordinate descent algorithms for multi-agent convex optimization over networks. IEEE Trans Autom Control 58(8):2001–2013CrossRefGoogle Scholar
  14. Necoara I, Patrascu A (2014) A random coordinate descent algorithm for optimization problems with composite objective function and linear coupled constraints. Comput Optim Appl 57(2):307–337CrossRefGoogle Scholar
  15. Necoara I, Nesterov Yu, Glineur F (2017) Random block coordinate descent methods for linearly constrained optimization over networks. J Optim Theory Appl 173:227–2354CrossRefGoogle Scholar
  16. Nesterov Yu (2012) Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM J Optim 22(2):341–362CrossRefGoogle Scholar
  17. Nesterov Yu, Shikhman V (2017) Distributed price adjustment based on convex analysis. J Optim Theory Appl 172:594–622CrossRefGoogle Scholar
  18. Pichler A (2017) A quantitative comparison of risk measures. Ann Oper Res.  https://doi.org/10.1007/s10479-017-2397-3 Google Scholar
  19. Polak E (1997) Optimization. Springer, BerlinCrossRefGoogle Scholar
  20. Rockafellar RT (1970) Convex analysis. Princeton University Press, PrincetonCrossRefGoogle Scholar
  21. Rockafellar RT, Wets RJ-B (1998) Variational analysis. Springer, BerlinCrossRefGoogle Scholar
  22. Sutton RS, Barto AG (1998) Reinforcement learning. The MIT Press, LondonGoogle Scholar
  23. Tseng P, Yun S (2009) Block-coordinate gradient descent method for linearly constrained nonsmooth separable optimization. J Optim Theory Appl 140:513–535CrossRefGoogle Scholar
  24. Tseng P (2009) A coordinate gradient descent method for nonsmooth separable minimization. Math Program Ser B 117:387–423CrossRefGoogle Scholar
  25. Varian H (1992) Microeconomic analysis. Norton, New YorkGoogle Scholar
  26. Xiao L, Boyd S (2006) Optimal scaling of a gradient method for distributed resource allocation. J Optim Theory Appl 129(3):469–88CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Informatics DepartmentUniversity of BergenBergenNorway

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