Skip to main content

A finite \(\epsilon \)-convergence algorithm for two-stage stochastic convex nonlinear programs with mixed-binary first and second-stage variables

Abstract

In this paper, we propose a generalized Benders decomposition-based branch and bound algorithm (GBDBAB) to solve two-stage convex mixed-binary nonlinear stochastic programs with mixed-binary variables in both first and second-stage decisions. In order to construct the convex hull of the MINLP subproblem for each scenario in closed-form, we first represent each MINLP subproblem as a generalized disjunctive program in conjunctive normal form (CNF). Second, we apply basic steps to convert the CNF of the MINLP subproblem into disjunctive normal form to obtain the convex hull of the MINLP subproblem. We prove that GBD is able to converge for the problems with pure binary variables given that the convex hull of each subproblem is constructed in closed-form. However, for problems with mixed-binary first and second-stage variables, we propose an algorithm, GBDBAB, where we may have to branch and bound on the continuous first-stage variables to obtain an optimal solution. We prove that the algorithm GBDBAB can converge to \(\epsilon \)-optimality in a finite number of steps. Since constructing the convex hull can be expensive, we propose a sequential convexification scheme that progressively applies basic steps to the CNF. Computational results on a problem with quadratic constraints, a constrained layout problem, and a planning problem, demonstrate the effectiveness of the algorithm.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. 1.

    Angulo, G., Ahmed, S., Dey, S.S.: Improving the integer L-shaped method. INFORMS J. Comput. 28(3), 483–499 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Atakan, S., Sen, S.: A progressive hedging based branch-and-bound algorithm for mixed-integer stochastic programs. Comput. Manag. Sci. 15, 1–40 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Balas, E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebraic Discrete Methods 6(3), 466–486 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Birge, J.R., Louveaux, F.: Introduction to stochastic programming. Springer, New York (2011)

    Book  MATH  Google Scholar 

  5. 5.

    Brunaud, B., Bassett, M.H., Agarwal, A., Wassick, J.M., Grossmann, I.E.: Efficient formulations for dynamic warehouse location under discrete transportation costs. Comput. Chem. Eng. (2017). https://doi.org/10.1016/j.compchemeng.2017.05.011

    Article  Google Scholar 

  6. 6.

    Bussieck, M.R., Drud, A.: SBB: A new solver for mixed integer nonlinear programming. Talk, OR (2001). http://ftp.gamsworld.org/presentations/present_sbb.pdf. Accessed 20 Aug 2019

  7. 7.

    Byrd, R.H., Nocedal, J., Waltz, R.A.: Knitro: an integrated package for nonlinear optimization. In: Large-Scale Nonlinear Optimization, pp. 35–59. Springer, New York (2006)

    MATH  Google Scholar 

  8. 8.

    Carøe, C.C., Schultz, R.: Dual decomposition in stochastic integer programming. Oper. Res. Lett. 24(1), 37–45 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Program. 86(3), 595–614 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    CPLEX, I.I.: V12. 1: User’s manual for CPLEX, Vol. 46, No. 53, p. 157. International Business Machines Corporation (2009)

  11. 11.

    Dunning, I., Huchette, J., Lubin, M.: JuMP: a modeling language for mathematical optimization. SIAM Rev. 59(2), 295–320 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Furman, K.C., Sawaya, N., Grossmann, I.: A computationally useful algebraic representation of nonlinear disjunctive convex sets using the perspective function. Optimization Online (2016). http://www.optimization-online.org/DB_FILE/2016/07/5544.pdf. Accessed 20 Aug 2019

  13. 13.

    Gade, D., Hackebeil, G., Ryan, S.M., Watson, J.P., Wets, R.J.B., Woodruff, D.L.: Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs. Math. Program. 157(1), 47–67 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Gade, D., Küçükyavuz, S., Sen, S.: Decomposition algorithms with parametric Gomory cuts for two-stage stochastic integer programs. Math. Program. 144(1–2), 39–64 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Geoffrion, A.M.: Generalized Benders decomposition. J. Optim. Theory Appl. 10(4), 237–260 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Grossmann, I.E., Trespalacios, F.: Systematic modeling of discrete-continuous optimization models through generalized disjunctive programming. AIChE J. 59(9), 3276–3295 (2013)

    Article  Google Scholar 

  17. 17.

    Guignard, M.: Lagrangean relaxation. Top 11(2), 151–200 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Jiang, R., Guan, Y., Watson, J.P.: Cutting planes for the multistage stochastic unit commitment problem. Math. Program. 157(1), 121–151 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Kim, K., Zavala, V.M.: Algorithmic innovations and software for the dual decomposition method applied to stochastic mixed-integer programs. In: Mathematical Programming Computation, pp. 1–42 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Küçükyavuz, S., Sen, S.: An introduction to two-stage stochastic mixed-integer programming. In: Leading Developments from INFORMS Communities, pp. 1–27. INFORMS (2017)

    Google Scholar 

  21. 21.

    Laporte, G., Louveaux, F.V.: The integer L-shaped method for stochastic integer programs with complete recourse. Oper. Res. Lett. 13(3), 133–142 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Li, C., Grossmann, I.E.: An improved L-shaped method for two-stage convex 0–1 mixed integer nonlinear stochastic programs. Comput. Chem. Eng. 112, 165–179 (2018)

    Article  Google Scholar 

  23. 23.

    Li, X., Tomasgard, A., Barton, P.I.: Nonconvex generalized Benders decomposition for stochastic separable mixed-integer nonlinear programs. J. Optim. Theory Appl. 151(3), 425 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Li, X., Tomasgard, A., Barton, P.I.: Decomposition strategy for the stochastic pooling problem. J. Global Optim. 54(4), 765–790 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Lubin, M., Yamangil, E., Bent, R., Vielma, J.P.: Extended Formulations in Mixed-Integer Convex Programming, pp. 102–113. Springer, Cham (2016)

    MATH  Google Scholar 

  26. 26.

    Mijangos, E.: An algorithm for two-stage stochastic mixed-integer nonlinear convex problems. Ann. Oper. Res. 235(1), 581–598 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Ntaimo, L., Tanner, M.W.: Computations with disjunctive cuts for two-stage stochastic mixed 0–1 integer programs. J. Global Optim. 41(3), 365–384 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Özaltın, O.Y., Prokopyev, O.A., Schaefer, A.J.: Two-stage quadratic integer programs with stochastic right-hand sides. Math. Program. 133(1–2), 121–158 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Qi, Y., Sen, S.: The ancestral Benders cutting plane algorithm with multi-term disjunctions for mixed-integer recourse decisions in stochastic programming. Math. Program. 161(1–2), 193–235 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Rockafellar, R.T., Wets, R.J.B.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16(1), 119–147 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Ruiz, J.P., Grossmann, I.E.: A hierarchy of relaxations for nonlinear convex generalized disjunctive programming. Eur. J. Oper. Res. 218(1), 38–47 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Sawaya, N.: Reformulations, relaxations and cutting planes for generalized disjunctive programming. Ph.D. thesis, Carnegie Mellon University, Pittsburgh (2006)

  33. 33.

    Sen, S., Higle, J.L.: The C-3 theorem and a D-2 algorithm for large scale stochastic mixed-integer programming: set convexification. Math. Program. 104(1), 1–20 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Sen, S., Sherali, H.D.: Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming. Math. Program. 106(2), 203–223 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Sherali, H.D., Fraticelli, B.M.: A modification of Benders decomposition algorithm for discrete subproblems: an approach for stochastic programs with integer recourse. J. Global Optim. 22(1–4), 319–342 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Sherali, H.D., Zhu, X.: On solving discrete two-stage stochastic programs having mixed-integer first-and second-stage variables. Math. Program. 108(2), 597–616 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Van Slyke, R.M., Wets, R.: L-shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math. 17(4), 638–663 (1969)

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Viswanathan, J., Grossmann, I.E.: A combined penalty function and outer-approximation method for MINLP optimization. Comput. Chem. Eng. 14(7), 769–782 (1990)

    Article  Google Scholar 

  39. 39.

    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Watson, J.P., Woodruff, D.L.: Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems. Comput. Manag. Sci. 8(4), 355–370 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Watson, J.P., Woodruff, D.L., Hart, W.E.: PySP: modeling and solving stochastic programs in Python. Math. Program. Comput. 4(2), 109–149 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Westerlund, T., Lundqvist, K.: Alpha-ECP, version 5.01: An interactive MINLP-solver based on the extended cutting plane method. Åbo Akademi (2001)

Download references

Acknowledgements

The authors gratefully acknowledge financial support from the Center of Advanced Process Decision-making at Carnegie Mellon University and from the Department of Energy as part of the IDAES Project.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ignacio E. Grossmann.

Additional information

Publisher's Note

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

Appendices

Appendix 1: Benders cuts for the illustrative example

The Benders cuts that are generated in each node of the branch-and-bound tree for the illustrative example in Sect. 4 are shown in Tables 9, 10 and 11.

Table 9 Benders cuts that are generated at the root node
Table 10 Benders cuts that are generated at node 1
Table 11 Benders cuts that are generated at node 2

Appendix 2: Constrained layout problem under price uncertainty

The constrained layout problem is adapted from the PhD thesis of Sawaya [32]. In this problem, we need to decide the layout of some units represented by rectangles with known lengths and widths. The units have to be manufactured before we install them at certain locations. In the manufacturing process, we need to provide the relative position of the units in order to produce the pipes that connect those units. The first-stage decisions include the designed relative position of the units. After the units are manufactured, they can be installed at several locations. In order to install some unit(s) at a given location, we need to purchase a circled area around the center of that location. The rectangle(s) installed at that location has to be constrained within the circle that is purchased. The price of each area is uncertain and the designed units are installed according to the prices.

The sets, parameters, and variables that are used in the model is defined as follows: Sets

  • \(i\in N=\) rectangles (units)

  • \(q\in Q=\) areas

  • \(\omega \in \varOmega =\) scenarios

Parameters

  • \(L_{i}=\) length of rectangle i

  • \(H_{i}=\) width of rectangle i

  • \(xbar_{q}=\) the x coordinate of area q

  • \(ybar_{q}=\) the y coordinate of area q

  • \(\tau _{\omega }=\) probability of scenario \(\omega \)

  • \(d_{q\omega }=\) unit price of area q in scenario \(\omega \)

first-stage decisions

  • \(x_{i}=\) the designed x coordinate of the center of rectangle i

  • \(y_{i}=\) the designed y coordinate of the center of rectangle i

  • \(delx_{ij}=\) the designed distance of the center of rectangle i and the center of rectangle j along the x axis

  • \(dely_{ij}=\) the designed distance of the center of rectangle i and the center of rectangle j along the y axis

  • \(Z_{ij}^{1}\), \(Z_{ij}^{2}\), \(Z_{ij}^{3}\), \(Z_{ij}^{4}\in \{True,False\}\) decides the relative position of rectangle i and rectangle j

second-stage decisions

  • \(xs_{i\omega }=\) the installed x coordinate of the center of rectangle i in scenario \(\omega \)

  • \(ys_{i\omega }=\) the installed y coordinate of the center of rectangle i in scenario \(\omega \)

  • \(W_{qi\omega }=\{True, False\}\) whether to position rectangle i in area q or not

  • \(S_{q\omega }=\) the size of area q that is purchased in scenario \(\omega \)

Constraint (62) relates the designed relative distance of rectangle i and rectangle j in both x and y direction to the designed x and y coordinate of rectangle i and rectangle j. There are variable costs associated with the relative designed distances, which corresponds to the pipes that connect those units. Constraint (63) prevents rectangle i and rectangle j from overlapping with each other. After the design decisions are made, the actual position to install each unit \(i\in N\) are decided. Constraint (64) enforces that the actual installed positions of rectangle i and rectangle j in each scenario \(\omega \in \varOmega \) have to coincide with the designed relative position. Moreover, in each scenario, each rectangle i has to be constrained in exactly one circle \(q\in Q\) that is purchased, which is enforced by constraint (65). The objective also includes the expected cost of the purchase of circles. Note that the problem is expressed as a GDP and can be reformulated with big-M or hull reformulation (see Table 7 in [16]) as a convex MINLP, which is a two-stage stochastic program with mixed integer variables in both first and second stage.

$$\begin{aligned} \min&\displaystyle \sum _{i}\sum _{j,j>i}c_{ij}(delx_{ij}+dely_{ij})+\sum _{\omega \in \varOmega }\tau _{\omega }\sum _{q\in Q}(d_{q\omega }S_{q\omega }) \end{aligned}$$
(61)
$$\begin{aligned}&\displaystyle s.t. \nonumber \\&\displaystyle delx_{ij}\ge x_{i}-x_{j}\quad \forall i,j\in N, i<j \nonumber \\&\displaystyle delx_{ij}\ge x_{j}-x_{i}\quad \forall i,j\in N, i<j \nonumber \\&\displaystyle dely_{ij}\ge y_{i}-y_{j}\quad \forall i,j\in N, i<j \nonumber \\&\displaystyle dely_{ij}\ge y_{j}-y_{i}\quad \forall i,j\in N, i<j \nonumber \\ \end{aligned}$$
(62)
$$\begin{aligned}&\displaystyle \begin{bmatrix} Z_{ij}^1\\ x_{i}+L_{i}/2\le x_{j}-L_{j}/2 \end{bmatrix} \vee \begin{bmatrix} Z_{ij}^2\\ x_{j}+L_{j}/2\le x_{i}-L_{i}/2 \end{bmatrix} \nonumber \\&\displaystyle \quad \vee \begin{bmatrix} Z_{ij}^3\\ y_{i}+H_{i}/2\le y_{j}-H_{j}/2 \end{bmatrix}\vee \begin{bmatrix} Z_{ij}^4\\ y_{j}+H_{j}/2\le y_{i}-H_{i}/2 \end{bmatrix}\nonumber \\&\displaystyle \quad \forall i,j\in N, i<j\end{aligned}$$
(63)
$$\begin{aligned}&\displaystyle xs_{i\omega }-xs_{j\omega }=x_{i}-x_{j}\quad \forall i,j\in N, i<j,\omega \in \varOmega \nonumber \\&\displaystyle ys_{i\omega }-ys_{j\omega }=y_{i}-y_{j}\quad \forall i,j\in N, i<j,\omega \in \varOmega \end{aligned}$$
(64)
$$\begin{aligned}&\displaystyle \vee _{q\in Q} \begin{bmatrix} W_{qi\omega }\\ (xs_{i\omega }-L_{i}/2-xbar_{q})^2+(ys_{i\omega }+H_{i}/2-ybar_{q})^2\le S_{q\omega }\\ (xs_{i\omega }-L_{i}/2-xbar_{q})^2+(ys_{i\omega }-H_{i}/2-ybar_{q})^2\le S_{q\omega }\\ (xs_{i\omega }+L_{i}/2-xbar_{q})^2+(ys_{i\omega }+H_{i}/2-ybar_{q})^2\le S_{q\omega }\\ (xs_{i\omega }+L_{i}/2-xbar_{q})^2+(ys_{i\omega }-H_{i}/2-ybar_{q})^2\le S_{q\omega } \end{bmatrix} \quad \forall i\in N,\omega \in \varOmega \nonumber \\ \end{aligned}$$
(65)
$$\begin{aligned}&\displaystyle delx_{ij},dely_{ij},S_{q\omega }\in \mathrm{I\!R}^{1}_{+}, Z_{ij}^{1},Z_{ij}^{2},Z_{ij}^{3},Z_{ij}^{4}, W_{qi\omega }\nonumber \\&\displaystyle \quad \in \{True, False\}\quad \forall i,j\in N, i<j,q\in Q,\omega \in \varOmega \end{aligned}$$
(66)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, C., Grossmann, I.E. A finite \(\epsilon \)-convergence algorithm for two-stage stochastic convex nonlinear programs with mixed-binary first and second-stage variables. J Glob Optim 75, 921–947 (2019). https://doi.org/10.1007/s10898-019-00820-y

Download citation

Keywords

  • Stochastic programming
  • Integer recourse
  • Generalized Benders decomposition
  • Branch and bound