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Accelerated gradient methods for nonconvex nonlinear and stochastic programming

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Abstract

In this paper, we generalize the well-known Nesterov’s accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first-order information, similarly to the gradient descent method. We then consider an important class of composite optimization problems and show that the AG method can solve them uniformly, i.e., by using the same aggressive stepsize policy as in the convex case, even if the problem turns out to be nonconvex. We demonstrate that the AG method exhibits an optimal rate of convergence if the composite problem is convex, and improves the best known rate of convergence if the problem is nonconvex. Based on the AG method, we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization. To the best of our knowledge, this is the first time that the convergence of the AG method has been established for solving nonconvex nonlinear programming in the literature.

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Notes

  1. A function\(\varPsi (\cdot )\) satisfying (1.2) is denoted by\(\varPsi \in {\mathcal {C}}_{L_\varPsi }^{1,1}(\mathbb {R}^n)\).

References

  1. Andradóttir, S.: A review of simulation optimization techniques. In: Proceedings of the 1998 Winter Simulation Conference, pp. 151–158 (1998)

  2. Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algorithm and Analysis. Springer, New York (2000)

    Google Scholar 

  3. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci.2, 183–202 (2009)

    Article MathSciNet MATH  Google Scholar 

  4. Cartis, C., Gould, N.I.M., Toint, P.L.: On the complexity of steepest descent, Newton’s and regularized Newton’s methods for nonconvex unconstrained optimization. SIAM J. Optim.20(6), 2833–2852 (2010)

    Article MathSciNet MATH  Google Scholar 

  5. Chen, X., Ge, D., Wang, Z., Ye, Y.: Complexity of unconstrained\(l_2-l_p\) minimization. Math. Program. (2012). doi:10.1007/s10107-012-0613-0

  6. Dang, C.D., Lan, G.: Stochastic block mirror descent methods for nonsmooth and stochastic optimization. Manuscript, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA (2013)

  7. Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc.96, 1348–1360 (2001)

    Article MathSciNet MATH  Google Scholar 

  8. Feng, M., Mitchell, J.E., Pang, J.-S., Shen, X., Wächter, A.: Complementarity formulations of\(l_0\)-norm optimization problems. Manuscript (2013)

  9. Fu, M.: Optimization for simulation: theory vs. practice. INFORMS J. Comput.14, 192–215 (2002)

    Article MathSciNet MATH  Google Scholar 

  10. Ghadimi, S., Lan, G.: Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization, I: a generic algorithmic framework. SIAM J. Optim.22, 1469–1492 (2012)

    Article MathSciNet MATH  Google Scholar 

  11. Ghadimi, S., Lan, G.: Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization, II: shrinking procedures and optimal algorithms. SIAM J. Optim.23, 2061–2089 (2013)

    Article MathSciNet MATH  Google Scholar 

  12. Ghadimi, S., Lan, G.: Stochastic first- and zeroth-order methods for nonconvex stochastic programming. SIAM J. Optim.23(4), 2341–2368 (2013)

    Article MathSciNet MATH  Google Scholar 

  13. Ghadimi, S., Lan, G., Zhang, H.: Mini-batch stochastic approximation methods for constrained nonconvex stochastic programming. Manuscript, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA. Mathematical Programming (Under second-round review) (2014)

  14. Lemarechal, C., Hiriart-Urruty, J.-B.: Convex Analysis and Minimization Algorithms II. A Series of Comperhensive Studies in Mathematics. Springer Science & Business Media, Heidelberg (1993)

  15. Lan, G.: An optimal method for stochastic composite optimization. Math. Program.133(1), 365–397 (2012)

    Article MathSciNet MATH  Google Scholar 

  16. Lan, G.: The complexity of large-scale convex programming under a linear optimization oracle. Manuscript, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA.http://www.optimization-online.org/ (2013)

  17. Lan, G., Monteiro, R.D.C.: Iteration-complexity of first-order penalty methods for convex programming. Math. Program.138, 115–139 (2013)

    Article MathSciNet MATH  Google Scholar 

  18. Lan, G., Monteiro, R.D.C.: Iteration-complexity of first-order augmented lagrangian methods for convex programming. Technical report, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA. Mathematical Programming (Under second-round review) (2013)

  19. Law, A.M.: Simulation Modeling and Analysis. McGraw Hill, New York (2007)

    Google Scholar 

  20. Lewis, A.S., Wright, S.J.: A proximal method for composite minimization. Manuscript, Cornell University, Ithaca, NY (2009)

  21. Mairal, J., Bach, F., Ponce, J., Sapiro, G.: Online dictionary learning for sparse coding. In: ICML, pp. 689–696 (2009)

  22. Monteiro, R.D.C., Svaiter, B.F.: An accelerated hybrid proximal extragradient method for convex optimization and its implications to second-order methods. Manuscript, School of ISyE, Georgia Tech, Atlanta, GA, 30332, USA (2011)

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

    Article MathSciNet MATH  Google Scholar 

  24. Nemirovski, A.S., Yudin, D.: Problem complexity and method efficiency in optimization. In: Graham, R.L., Lenstra, J.K. (eds.) Wiley-Interscience Series in Discrete Mathematics. Wiley, XV (1983)

  25. Nesterov, Y.E.: A method for unconstrained convex minimization problem with the rate of convergence\(O(1/k^2)\). Doklady AN SSSR269, 543–547 (1983)

    MathSciNet  Google Scholar 

  26. Nesterov, Y.E.: Introductory Lectures on Convex Optimization: A Basic Course. Kluwer Academic Publishers, Massachusetts (2004)

    Book  Google Scholar 

  27. Nesterov, Y.E.: Smooth minimization of nonsmooth functions. Math. Program.103, 127–152 (2005)

    Article MathSciNet MATH  Google Scholar 

  28. Nesterov, Y.E.: How to make gradients small. Optima88, 10–11 (2012)

    Google Scholar 

  29. Nesterov, Y.E.: Gradient methods for minimizing composite objective functions. Math. Program. Ser. B140, 125–161 (2013)

    Article MathSciNet MATH  Google Scholar 

  30. Polyak, B.T., Juditsky, A.B.: Acceleration of stochastic approximation by averaging. SIAM J. Control Optim.30, 838–855 (1992)

    Article MathSciNet MATH  Google Scholar 

  31. Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat.22, 400–407 (1951)

    Article MathSciNet MATH  Google Scholar 

  32. Sartenaer, A., Gratton, S., Toint, P.L.: Recursive trust-region methods for multiscale nonlinear optimization. SIAM J. Optim.19, 414–444 (2008)

    Article MathSciNet MATH  Google Scholar 

  33. Spall, J.C.: Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control. Wiley, Hoboken (2003)

    Book  Google Scholar 

  34. Tseng, P.: On accelerated proximal gradient methods for convex–concave optimization. Manuscript, University of Washington, Seattle (May 2008)

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611, USA

    Saeed Ghadimi & Guanghui Lan

Authors
  1. Saeed Ghadimi

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  2. Guanghui Lan

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Correspondence toGuanghui Lan.

Additional information

This research was partially supported by NSF Grants CMMI-1000347, CMMI-1254446, DMS-1319050, and ONR Grant N00014-13-1-0036.

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Ghadimi, S., Lan, G. Accelerated gradient methods for nonconvex nonlinear and stochastic programming.Math. Program.156, 59–99 (2016). https://doi.org/10.1007/s10107-015-0871-8

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