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Incremental constraint projection methods for variational inequalities

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Abstract

We consider the solution of strongly monotone variational inequalities of the form \(F(x^*)'(x-x^*)\ge 0\), for all \(x\in X\). We focus on special structures that lend themselves to sampling, such as when \(X\) is the intersection of a large number of sets, and/or \(F\) is an expected value or is the sum of a large number of component functions. We propose new methods that combine elements of incremental constraint projection and stochastic gradient. These methods are suitable for problems involving large-scale data, as well as problems with certain online or distributed structures. We analyze the convergence and the rate of convergence of these methods with various types of sampling schemes, and we establish a substantial rate of convergence advantage for random sampling over cyclic sampling.

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References

  1. Bauschke, H.H.: Projection algorithms and monotone operators. Ph.D. thesis, Simon Frazer University, Canada (1996)

  2. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bauschke, H., Borwein, J.M., Lewis, A.S.: The method of cyclic projections for closed convex sets in hilbert space. Contemp. Math. 204, 1–38 (1997)

    Article  MathSciNet  Google Scholar 

  4. Bertsekas, D.P.: Approximate policy iteration: a survey and some new methods. J. Control Theory Appl. 9, 310–335 (2010)

    Article  MathSciNet  Google Scholar 

  5. Bertsekas, D.P.: Incremental proximal methods for large scale convex optimization. Math. Program. Ser. B 129, 163–195 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bertsekas, D.P.: Temporal difference methods for general projected equations. IEEE Trans. Autom. Control 56, 2128–2139 (2011)

    Article  MathSciNet  Google Scholar 

  7. Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. II: Approximate Dynamic Programming. Athena Scientific, Belmont (2012)

    Google Scholar 

  8. Bertsekas, D.P., Gafni, E.M.: Projection methods for variational inequalities with application to the traffic assignment problem. Math. Program. Study 17, 139–159 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bertsekas, D.P., Nedić, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)

    MATH  Google Scholar 

  10. Borkar, V.S.: Stochastic Approximation: A Dynamical Systems Viewpoint. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  11. Bertsekas, D.P., Tsitsiklis, J.N.: Neuro-dynamic Programming. Athena Scientific, Belmont (1996)

    MATH  Google Scholar 

  12. Censor, Y., Gibali, A.: Projections onto super-half-spaces for monotone variational inequality problems in finite-dimensional spaces. J. Nonlinear Convex Anal 9, 461–475 (2008)

    MATH  MathSciNet  Google Scholar 

  13. Censor, Y., Gibali, A., Reich, S.: A von neumann alternating method for finding common solutions to variational inequalities. Nonlinear Anal. Ser. A Theory Methods Appl. 75, 4596–4603 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Censor, Y., Gibali, A., Reich, S., Sabach, S.: Common solutions to variational inequalities. Set-Valued Var. Anal. 20, 229–247 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cegielski, A., Suchocka, A.: Relaxed alternating projection methods. SIAM J. Optim. 19, 1093–1106 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Desai, V.V., Farias, V.F., Moallemi, C.C.: Approximate dynamic programming via a smoothed approximate linear program. Oper. Res. 60, 655–674 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. de Farias, D.P., Van Roy, B.: The linear programming approach to approximate dynamic programming. Oper. Res. 51, 850–865 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. de Farias, D.P., Van Roy, B.: On constraint sampling in the linear programming approach to approximate dynamic programming. Math. Oper. Res. 29, 462–478 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Deutsch, F., Hundal, H.: The rate of convergence for the cyclic projections algorithm i: angles between convex sets. J. Approx. Theory 142, 36–55 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Deutsch, F., Hundal, H.: The rate of convergence for the cyclic projections algorithm ii: norms of nonlinear operators. J. Approx. Theory 142, 56–82 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Deutsch, F., Hundal, H.: The rate of convergence for the cyclic projections algorithm iii: regularity of convex sets. J. Approx. Theory 155, 155–184 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)

    Google Scholar 

  23. Fukushima, M.: A relaxed projection method for variational inequalities. Math. Program. 35, 58–70 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gurkan, G., Ozge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999)

    Article  MathSciNet  Google Scholar 

  25. Gubin, L.G., Polyak, B.T., Raik, E.V.: The method of projections for finding the common point of convex sets. USSR Comput. Math. Math. Phys. 7, 1211–1228 (1967)

    Article  Google Scholar 

  26. Halperin, I.: The product of projection operators. Acta Sci. Math. 23, 96–99 (1962)

    MATH  MathSciNet  Google Scholar 

  27. Jiang, H., Xu, F.: Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Trans. Autom. Control 53, 1462–1475 (2008)

    Article  MathSciNet  Google Scholar 

  28. Koshal, J., Nedić, A., Shanbhag, U.V.: Regularized iterative stochastic approximation methods for stochastic variational inequality problems. IEEE Trans. Autom. Control 58, 594–608 (2013)

    Article  Google Scholar 

  29. Korpelevich, G.M.: An extragradient method for finding saddle points and for other problems. Matecon 12, 747–756 (1976)

    MATH  Google Scholar 

  30. Krasnoselskii, M.A.: Approximate Solution of Operator Equations. D. Wolters-Noordhoff Pub, Groningen (1972)

  31. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York, London (1980)

    MATH  Google Scholar 

  32. Kannan, A., Shanbhag, U.V.: Distributed computation of equilibria in monotone nash games via iterative regularization techniques. SIAM J. Optim. 22(4), 1177–1205 (2012)

  33. Kushner, H.J., Yin, G.: Stochastic Approximation and Recursive Algorithms and Applications. Springer, New York (2003)

    MATH  Google Scholar 

  34. Leventhal, D., Lewis, A.S.: Randomized methods for linear constraints: convergence rates and conditioning. Math. Oper. Res. 35, 641–654 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  35. Lewis, A.S., Malick, J.: Alternating projections on manifolds. Math. Oper. Res. 33, 216–234 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  36. Lu, S.: Confidence regions for stochastic variational inequalities. Math. Oper. Res. 38, 545–569 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  37. Nedić, A., Bertsekas, D.P.: Convergence rate of the incremental subgradient algorithm. In: Uryasev, S., Pardalos, P.M. (eds.) Stochastic Optimization: Algorithms and Applications, pp. 263–304 (2000)

  38. Nedić, A., Bertsekas, D.P.: Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optim. 12, 109–138 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  39. Nedić, A.: Random projection algorithms for convex set intersection problems. In: The 49th IEEE conference on decision and control, Atlanta, Georgia, pp. 7655–7660 (2010)

  40. Nedić, A.: Random algorithms for convex minimization problems. Math. Program. Ser. B 129, 225–253 (2011)

    Article  MATH  Google Scholar 

  41. Nemirovski, A., Juditsky, A., Lan, G., Shapiro, A.: Robust stochastic approximation approach to stochastic programming. SIAM J. Optim. 19, 1574–1609 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  42. Noor, M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 197–277 (2004)

    Google Scholar 

  43. Patriksson, M.: Nonlinear Programing and Variational Inequality Problems: A Unified Approach. Kluwer, Dordrecht, The Netherlands (1999)

    Book  Google Scholar 

  44. Recht, B., Re, C.: Beneath the valley of the noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences. arXiv:1202.4184 (2012)

  45. Saad, Y.: Iterative Methods for Sparse Linear Systems 2nd edn. SIAM, Philadelphia (2003)

  46. Sutton, R.S., Barto, A.G.: Reinforcement Learning. MIT Press, Cambridge (1998)

    Google Scholar 

  47. Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)

    Book  Google Scholar 

  48. Shapiro, A.: Monte Carlo sampling methods. In: Stochastic Programming. Handbook in OR & MS, vol. 10, North-Holland, Amsterdam (2003)

  49. Sibony, M.: Methodes iteratives pour les equations et inequations aux derivees partielles non lineaires de type monotone. Calcolo 7, 65–183 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  50. Tseng, P.: Successive projection under a quasi-cyclic order. Laboratory for Information and Decision Systems Report LIDS-P-1938, MIT, Cambridge, MA (1990)

  51. von Neumann, J.: Functional Operators. Princeton University Press, Princeton (1950)

    Google Scholar 

  52. Wang, M., Bertsekas, D.P.: Incremental constraint projection-proximal methods for nonsmooth convex optimization. Laboratory for Information and Decision Systems, Report LIDS-P-2907, MIT (2013)

  53. Xu, H.: Sample average approximation methods for a class of stochastic variational inequality problems. Asian Pac. J. Oper. Res. 27, 103–119 (2010)

    Article  MATH  Google Scholar 

  54. Xiu, N., Zhang, J.: Some recent advances in projection-type methods for variational inequalities. J. Comput. Appl. Math. 152, 559–585 (2003)

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Authors and Affiliations

  1. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, USA

    Mengdi Wang

  2. Laboratory for Information and Decision Systems (LIDS), Department of Electrical Engineering and Computer Science, M.I.T., Cambridge, MA, USA

    Dimitri P. Bertsekas

Authors
  1. Mengdi Wang
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  2. Dimitri P. Bertsekas
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Correspondence to Mengdi Wang.

Additional information

Work supported by the Air Force Grant FA9550-10-1-0412.

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Wang, M., Bertsekas, D.P. Incremental constraint projection methods for variational inequalities. Math. Program. 150, 321–363 (2015). https://doi.org/10.1007/s10107-014-0769-x

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