Skip to main content
SpringerLink
Log in

Inexact stochastic mirror descent for two-stage nonlinear stochastic programs

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

We introduce an inexact variant of stochastic mirror descent (SMD), called inexact stochastic mirror descent (ISMD), to solve nonlinear two-stage stochastic programs where the second stage problem has linear and nonlinear coupling constraints and a nonlinear objective function which depends on both first and second stage decisions. Given a candidate first stage solution and a realization of the second stage random vector, each iteration of ISMD combines a stochastic subgradient descent using a prox-mapping with the computation of approximate (instead of exact for SMD) primal and dual second stage solutions. We provide two convergence analysis of ISMD, under two sets of assumptions. The first convergence analysis is based on the formulas for inexact cuts of value functions of convex optimization problems shown recently in Guigues (SIAM J. Optim. 30(1), 407–438, 2020). The second convergence analysis provides a convergence rate (the same as SMD) and relies on new formulas that we derive for inexact cuts of value functions of convex optimization problems assuming that the dual function of the second stage problem for all fixed first stage solution and realization of the second stage random vector, is strongly concave. We show that this assumption of strong concavity is satisfied for some classes of problems and present the results of numerical experiments on two simple two-stage problems which show that solving approximately the second stage problem for the first iterations of ISMD can help us obtain a good approximate first stage solution quicker than with SMD.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. Using the equivalence between norms in \(\mathbb {R}^n\), we can derive a valid constant of strong concavity for other norms, for instance \(\Vert \cdot \Vert _{\infty }\) and \(\Vert \cdot \Vert _1\).

  2. Note that we used (A4) to ensure that \(x(\lambda _*, \mu _*) = x_*\), which is also used in the proof of Theorem 10 in [19].

  3. Note that (H1) and (H2) imply the convexity of \(\mathcal {Q}\) given by (3.1). Indeed, let \(x_1, x_2 \in X\), \(0 \le t \le 1\), and \(y_1 \in S( x_1), y_2 \in S( x_2),\) such that \(\mathcal {Q}( x_1)=f(y_1, x_1)\) and \(\mathcal {Q}( x_2)=f(y_2, x_2)\). By convexity of g and Y, we have that have \(ty_1 +(1-t)y_2 \in S(tx_1 + (1-t)x_2)\) and therefore \(\mathcal {Q}(tx_1 +(1-t)x_2) \le f(ty_1 + (1-t)y_2,t x_1 + (1-t)x_2) \le t f(y_1,x_1) +(1-t)f(y_2,x_2)=t\mathcal {Q}(x_1)+(1-t)\mathcal {Q}(x_2)\) where for the last inequality we have used the convexity of f.

  4. The proof is similar to the proof of Proposition 4.6 in [6].

  5. According to current Mosek documentation, it is not possible to use absolute errors. Therefore, early termination of the solver can either be obtained limiting the number of iterations or defining relative errors.

  6. The deterministic equivalents of these instances are already large size quadratic programs. For instance, for \(n=10\), the deterministic equivalent of Problem (5.3)–(5.4) is a quadratically constrained quadratic program with 200,010 variables and 200,001 quadratic constraints.

  7. Naturally, after running \(t-1\) of the \(N-1\) total iterations, the approximate optimal value computed by SMD is \(\displaystyle \frac{1}{\sum _{\tau =1}^t \gamma _\tau (N)} \sum \nolimits _{\tau =1}^t \gamma _\tau ( N) \Big ( f_1(x_1^{N,\tau }) + f_2( x_2^{N,\tau }, x_1^{N,\tau }, \xi _2^{N,\tau }) \Big )\) obtained on the basis of sample \(\xi _2^{N,1},\ldots ,\xi _2^{N,t}\) of \(\xi _2\).

  8. Due to the increase in computational time when N increases, we do not take the largest sample size \(N=2000\) for all instances. However, for all instances and values of N chosen, we observe a stabilization of the approximate optimal value before stopping the algorithm, which indicates a good solution has been found at termination.

  9. When SMD (and similarly for ISMD) is run on samples of \(\xi _2\) of size N, we have seen how to compute at iteration \(t-1\) an estimation \(\displaystyle \frac{1}{\sum _{\tau =1}^t \gamma _\tau (N)} \sum \nolimits _{\tau =1}^t \gamma _\tau ( N) \Big ( f_1(x_1^{N,\tau }) + f_2( x_2^{N,\tau }, x_1^{N,\tau }, \xi _2^{N,\tau }) \Big )\) of the optimal value on the basis of sample \(\xi _2^{N,1},\ldots ,\xi _2^{N,t}\) of \(\xi _2\). The mean approximate optimal value after \(t-1\) iterations is obtained running SMD on 10 independent samples of \(\xi _2\) of size N and computing the mean of these values on these samples.

References

  1. Andersen, E.D., Andersen, K.D.: The MOSEK optimization toolbox for MATLAB manual. Version 7.0, (2013). https://www.mosek.com/

  2. Birge, J., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)

    MATH  Google Scholar 

  3. Dantzig, G.B., Glynn, P.W.: Parallel processors for planning under uncertainty. Ann. Oper. Res. 22, 1–21 (1990)

    Article  MathSciNet  Google Scholar 

  4. Guigues, V.: Convergence analysis of sampling-based decomposition methods for risk-averse multistage stochastic convex programs. SIAM J. Optim. 26, 2468–2494 (2016)

    Article  MathSciNet  Google Scholar 

  5. Guigues, V.: Multistep stochastic mirror descent for risk-averse convex stochastic programs based on extended polyhedral risk measures. Math. Program. 163, 169–212 (2017)

    Article  MathSciNet  Google Scholar 

  6. Guigues, V.: Inexact cuts in Stochastic Dual Dynamic Programming. SIAM J. Optim. 30(1), 407–438 (2020)

    Article  MathSciNet  Google Scholar 

  7. Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1996)

    MATH  Google Scholar 

  8. Infanger, G.: Monte Carlo (importance) sampling within a benders decomposition algorithm for stochastic linear programs. Ann. Oper. Res. 39, 69–95 (1992)

    Article  MathSciNet  Google Scholar 

  9. Juditsky, A., Nesterov, Y.: Primal–dual subgradient methods for minimizing uniformly convex functions. arXiv:1401.1792 (2010)

  10. Lan, G., Nemirovski, A., Shapiro, A.: Validation analysis of mirror descent stochastic approximation method. Math. Program. 134, 425–458 (2012)

    Article  MathSciNet  Google Scholar 

  11. Lan, G., Zhou, Z.: Dynamic stochastic approximation for multi-stage stochastic optimization. arXiv (2017)

  12. Lemaréchal, C., Nemirovski, A., Nesterov, Y.: New variants of bundle methods. Math. Program. 69, 111–148 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Pereira, M.V.F., Pinto, L.M.V.G.: Multi-stage stochastic optimization applied to energy planning. Math. Program. 52, 359–375 (1991)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Grundlehren der Mathematischen Wissenschaften. Springer, New York (1997)

    MATH  Google Scholar 

  17. Ruszczyński, A.: A multicut regularized decomposition method for minimizing a sum of polyhedral functions. Math. Program. 35, 309–333 (1986)

    Article  Google Scholar 

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

    Book  Google Scholar 

  19. Yu, H., Neely, J.: On the convergence time of the drift-plus-penalty algorithm for strongly convex programs. arXiv:1503.06235 (2015)

Download references

Acknowledgements

The author’s research was partially supported by an FGV Grant, CNPq Grant 311289/2016-9, and FAPERJ Grant E-26/201.599/2014. The author would like to thank Alberto Seeger for helpful discussions.

Author information

Authors and Affiliations

  1. School of Applied Mathematics, FGV, Praia de Botafogo, Rio de Janeiro, Brazil

    Vincent Guigues

Authors
  1. Vincent Guigues
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vincent Guigues.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

See Figs. 6, 7 and Table 5.

Fig. 6
figure 6

Dual function \(\theta _{{\bar{x}}}\) of problem (3.30) for some \({\bar{x}}\) randomly drawn in ball \(\{x \in \mathbb {R}^n: \Vert x-x_0 \Vert _2 \le 1\}\), \(S=AA^T + \lambda I_{2 n}\) for some random matrix A with random entries in \([-20,20]\), and several values of the pair \((n,\lambda )\). The dual iterates are represented by red diamonds

Full size image
Fig. 7
figure 7

Plots of \(\eta _{1}(\varepsilon _k, {\bar{x}})\) and \(\eta _2(\varepsilon _k, {\bar{x}})\) as a function of iteration k where \(\varepsilon _k\) is the duality gap at iteration k for problem (3.30) for some \({\bar{x}}\) randomly drawn in ball \(\{x \in \mathbb {R}^n: \Vert x-x_0 \Vert _2 \le 1\}\), \(S=AA^T + \lambda I_{2 n}\) for some random matrix A with random entries in \([-20,20]\), and several values of the pair \((n,\lambda )\)

Full size image
Table 5 Maximal number of iterations for Mosek interior point solver used to solve second stage problems as a function of the iteration number \(i=1,\ldots ,N\), of ISMD and the maximal number of iterations \(I_{\max }\) allowed for Mosek solver to solve subproblems with SMD
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guigues, V. Inexact stochastic mirror descent for two-stage nonlinear stochastic programs. Math. Program. 187, 533–577 (2021). https://doi.org/10.1007/s10107-020-01490-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-020-01490-5

Keywords

Mathematics Subject Classification

Search

Navigation