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On unbounded and binary parameters in multi-parametric programming: applications to mixed-integer bilevel optimization and duality theory

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Abstract

In multi-parametric programming an optimization problem is solved as a function of certain parameters, where the parameters are commonly considered to be bounded and continuous. In this paper, we use the case of strictly convex multi-parametric quadratic programming (mp-QP) problems with affine constraints to investigate problems where these conditions are not met. Based on the combinatorial solution approach for mp-QP problems featuring bounded and continuous parameters, we show that (i) for unbounded parameters, it is possible to obtain the multi-parametric solution if there exists one realization of the parameters for which the optimization problem can be solved and (ii) for binary parameters, we present the equivalent mixed-integer formulations for the application of the combinatorial algorithm. These advances are combined into a new, generalized version of the combinatorial algorithm for mp-QP problems, which enables the solution of problems featuring both unbounded and binary parameters. This novel approach is applied to mixed-integer bilevel optimization problems and the parametric solution of the dual of a convex problem.

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Notes

  1. Note that binary parameters are inherently bounded.

  2. This was the reason for the solution reported in Remark 4.

  3. Within our numerical studies, we successfully utilized \(M = 10^5\).

  4. The solutions of the 8 MILP problems correspond to critical regions \(CR1-CR8\) of Table 4.

  5. The solutions of the 2 MINLP problems correspond to critical regions \(CR1_n\) and \(CR2_n\) of Table 5.

References

  1. Acevedo, J., Pistikopoulos, E.N.: A multiparametric programming approach for linear process engineering problems under uncertainty. Ind. Eng. Chem. Res. 36(3), 717–728 (1997)

    Article  Google Scholar 

  2. Baes, M., Diehl, M., Necoara, I.: Every continuous nonlinear control system can be obtained by parametric convex programming. IEEE Trans. Autom. Control 53(8), 1963–1967 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bard, J.F., Moore, J.T.: An algorithm for the discrete bilevel programming problem. Nav. Res. Logist. 39(3), 419–435 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  6. Borrelli, F., Baotić, M., Bemporad, A., Morari, M.: Dynamic programming for constrained optimal control of discrete-time linear hybrid systems. Automatica 41(10), 1709–1721 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dempe, S.: Discrete bilevel optimization problems. Universitaet Leipzig, Wirtschaftswissenschaftliche Fakultaet, Institut fuer Informatik (1996)

  8. Dempe, S., Kalashnikov, V., Rios-Mercado, R.Z.: Discrete bilevel programming: application to a natural gas cash-out problem. Eur. J. Oper. Res. 166(2), 469–488 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. DeNegre S.T., Ralphs T.K.: A branch-and-cut algorithm for integer bilevel linear programs. In: Chinneck, J.W., Kristjansson, B., Saltzman, M.J. (eds.) Operations Research and Cyber-Infrastructure, pp 65–78. Springer, Boston (2009)

  10. Diangelakis, N.A., Manthanwar, A.M., Pistikopoulos, E.N.: A framework for design and control optimisation: application on a CHP system. In: Proceedings of the 8th International Conference on Foundations of Computer-Aided Process Design, Computer Aided Chemical Engineering, vol. 34, pp 765–770. Elsevier (2014)

  11. Domínguez, L.F., Pistikopoulos, E.N.: Global optimization of integer and mixed-integer bi-level programming problems via multi-parametric programming. In: 10th International Symposium on Process Systems Engineering: Part A, Computer Aided Chemical Engineering, vol. 27, pp 177–182. Elsevier (2009)

  12. Dua, V.: Mixed integer polynomial programming. Comput. Chem. Eng. 72, 387–394 (2015)

    Article  Google Scholar 

  13. Dua, V., Bozinis, N.A., Pistikopoulos, E.N.: A multiparametric programming approach for mixed-integer quadratic engineering problems. Comput. Chem. Eng. 26(4–5), 715–733 (2002)

    Article  Google Scholar 

  14. Faísca, N., Saraiva, P., Rustem, B., Pistikopoulos, E.: A multi-parametric programming approach for multilevel hierarchical and decentralised optimisation problems. Comput. Manag. Sci. 6(4), 377–397 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Faísca, N.P., Dua, V., Rustem, B., Saraiva, P.M., Pistikopoulos, E.N.: Parametric global optimisation for bilevel programming. J. Glob. Optim. 38(4), 609–623 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Feller, C., Johansen, T.A., Olaru, S.: An improved algorithm for combinatorial multi-parametric quadratic programming. Automatica 49(5), 1370–1376 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Fiacco, A.V.: Sensitivity analysis for nonlinear programming using penalty methods. Math. Program. 10(1), 287–311 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  18. Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Mathematics in Science and Engineering vol v, vol. 165. Academic Press, New York (1983)

    MATH  Google Scholar 

  19. Floudas, C.A.: Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications. Topics in Chemical Engineering. Oxford University Press, New York (1995)

    MATH  Google Scholar 

  20. Gumus, Z.H., Floudas, C.A.: Global optimization of mixed-integer bilevel programming problems. Comput. Manag. Sci. 2, 181–212 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gupta, A., Bhartiya, S., Nataraj, P.: A novel approach to multiparametric quadratic programming. Automatica 47(9), 2112–2117 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hempel, A.B., Goulart, P.J., Lygeros, J.: Inverse Parametric Optimization With an Application to Hybrid System Control. IEEE Trans. Autom. Control 60(4), 1064–1069 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  23. Herceg, M., Jones, C.N., Kvasnica, M., Morari, M.: Enumeration-based approach to solving parametric linear complementarity problems. Automatica 62, 243–248 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  24. Jones, C.N., Barić, M., Morari, M.: Multiparametric linear programming with applications to control. Eur. J. Control 13(2–3), 152–170 (2007)

    Article  MATH  Google Scholar 

  25. Kassa, A.M., Kassa, S.M.: A branch-and-bound multi-parametric programming approach for non-convex multilevel optimization with polyhedral constraints. J. Glob. Optim. 64, 1–20 (2015)

    MATH  MathSciNet  Google Scholar 

  26. Kleniati, P.M., Adjiman, C.: Branch-and-sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. part i: theoretical development. J. Glob. Optim. 60(3), 425–458 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kleniati, P.M., Adjiman, C.S.: A generalization of the branch-and-sandwich algorithm: From continuous to mixed-integer nonlinear bilevel problems. Comput. Chem. Eng. 72, 373–386 (2015)

    Article  Google Scholar 

  28. Koppe, M., Queyranne, M., Ryan, C.T.: Parametric integer programming algorithm for bilevel mixed integer programs. J. Optim. Theory Appl. 146(1), 137–150 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  29. Li, H.C., Wang, Y.P.: Exponential distribution-based genetic algorithm for solving mixed-integer bilevel programming problems. J. Syst. Eng. Electron. 19(6), 1157–1164 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Mitsos, A.: Global solution of nonlinear mixed-integer bilevel programs. J. Glob. Optim. 47(4), 557–582 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  31. Mitsos, A., Lemonidis, P., Barton, P.: Global solution of bilevel programs with a nonconvex inner program. J. Glob. Optim. 42(4), 475–513 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Moore, J.T., Bard, J.F.: The mixed integer linear bilevel programming problem. Oper. Res. 38(5), 911–921 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  33. Nguyen, N.A., Olaru, S., Rodriguez-Ayerbe, P., Hovd, M., Necoara, I.: Inverse Parametric Convex Programming Problems Via Convex Liftings. In: World Congress, IFAC Proceedings Volumes, IFAC, pp. 2489–2494. Elsevier (2014)

  34. Nishizaki, I., Sakawa, M.: Computational methods through genetic algorithms for obtaining stackelberg solutions to two-level integer programming problems. Cybern. Syst. 36(6), 565–579 (2005)

    Article  MATH  Google Scholar 

  35. Oberdieck, R., Pistikopoulos, E.N.: Explicit hybrid model-predictive control: the exact solution. Automatica 58, 152–159 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  36. Oberdieck, R., Pistikopoulos, E.N.: Multi-objective optimization with convex quadratic cost functions: a multi-parametric programming approach. Comput. Chem. Eng. 85, 36–39 (2016)

    Article  Google Scholar 

  37. Oberdieck, R., Wittmann-Hohlbein, M., Pistikopoulos, E.N.: A branch and bound method for the solution of multiparametric mixed integer linear programming problems. J. Glob. Optim. 59(2–3), 527–543 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  38. Pistikopoulos, E.N., Diangelakis, N.A., Manthanwar, A.M.: Towards the integration of process design, control and scheduling: Are we getting closer? In: 12th International Symposium on Process Systems Engineering and 25th European Symposium on Computer Aided Process Engineering, Computer Aided Chemical Engineering, vol. 37, pp 41–48, Elsevier (2015a)

  39. Pistikopoulos, E.N., Diangelakis, N.A., Oberdieck, R., Papathanasiou, M.M., Nascu, I., Sun, M.: PAROC—an integrated framework and software platform for the optimization and advanced model-based control of process systems. Chem. Eng. Sci. 136, 115–138 (2015b)

    Article  Google Scholar 

  40. Rivotti, P., Pistikopoulos, E.N.: A dynamic programming based approach for explicit model predictive control of hybrid systems. Comput. Chem. Eng. 72, 126–144 (2014)

    Article  Google Scholar 

  41. Romanko, O., Ghaffari-Hadigheh, A., Terlaky, T.: Multiobjective optimization via parametric optimization: models, algorithms, and applications. In: Modeling and Optimization: Theory and Applications, Springer Proceedings in Mathematics and Statistics, vol. 21, pp 77–119 . Springer, New York (2012)

  42. Saharidis, G.K., Ierapetritou, M.G.: Resolution method for mixed integer bi-level linear problems based on decomposition technique. J. Glob. Optim. 44(1), 29–51 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  43. Sahin, K.H., Ciric, A.R.: A dual temperature simulated annealing approach for solving bilevel programming problems. Comput. Chem. Eng. 23(1), 11–25 (1998)

    Article  Google Scholar 

  44. Sakizlis, V., Perkins, J.D., Pistikopoulos, E.N.: Recent advances in optimization-based simultaneous process and control design. Comput. Chem. Eng. 28(10), 2069–2086 (2004)

    Article  Google Scholar 

  45. Tang, Y., Richard J.P., Smith, J.: A class of algorithms for mixed-integer bilevel min-max optimization. J. Glob. Optim. (2015). doi:10.1007/s10898-015-0274-7

  46. Tsoukalas, A., Rustem, B., Pistikopoulos, E.: A global optimization algorithm for generalized semi-infinite, continuous minimax with coupled constraints and bi-level problems. J. Glob. Optim. 44(2), 235–250 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  47. Wen, U.P., Huang, A.D.: A simple tabu search method to solve the mixed-integer linear bilevel programming problem. Eur. J. Oper. Res. 88(3), 563–571 (1996)

    Article  MATH  Google Scholar 

  48. Wiesemann, W., Tsoukalas, A., Kleniati, P.M., Rustem, B.: Pessimistic bilevel optimization. SIAM J. Optimi. 23(1), 353–380 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  49. Xu, P., Wang, L.Z.: An exact algorithm for the bilevel mixed integer linear programming problem under three simplifying assumptions. Comput. Oper. Res. 41, 309–318 (2014)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, London, UK

    Richard Oberdieck, Nikolaos A. Diangelakis & Styliani Avraamidou

  2. Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, TX, USA

    Richard Oberdieck, Nikolaos A. Diangelakis, Styliani Avraamidou & Efstratios N. Pistikopoulos

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  1. Richard Oberdieck
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  2. Nikolaos A. Diangelakis
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  3. Styliani Avraamidou
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  4. Efstratios N. Pistikopoulos
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Correspondence to Efstratios N. Pistikopoulos.

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Financial support from EPSRC (EP/M027856/1), Texas A&M University and Texas A&M Energy Institute is gratefully acknowledged.

Appendices

Appendix 1: The parametric solution of LP and QP problems based on their active set

Given problem (1) and an optimal active set, the task is to obtain the optimization variables and the Lagragian multipliers as a function of the parameters, i.e. \(x\left( \theta \right) \) and \(\lambda \left( \theta \right) \). Some of these equations have been derived in [37], while the others follow immediately.

Remark 8

These general equations can be derived directly by solving the Karush-Kuhn-Tucker conditions parametrically, given a candidate active set.

Remark 9

Note that it is necessary to explicitly calculate the inverse of the matrices involving \(A_p\) at each iteration, as the active set changes. However, due to the relatively small sizes normally associated with multi-parametric programming problems, this operation is computationally tractable.

1.1 Parametric solution for LP problems

The parametric solution of a LP problem for a given active set are given by:

$$\begin{aligned} x\left( \theta \right) = A_p^{-1}\left( F_p\theta + b_p\right) , \end{aligned}$$
(24)

where the subscript p denotes the active constraints. Note that the Lagragian multipliers \(\lambda \) are not a function of \(\theta \) in LP problems.

1.2 Parametric solution of QP problems

The parametric solution of a QP problem for a given active set are given by:

$$\begin{aligned} x\left( \theta \right)= & {} -Q^{-1}\left( A_p\lambda \left( \theta \right) + H^T\theta + c\right) \end{aligned}$$
(25a)
$$\begin{aligned} \lambda \left( \theta \right)= & {} -\left( A_pQ^{-1}A_p\right) ^{-1}\left( A_pQ^{-1}H^T\theta + F_p\theta + b_p + A_pQ^{-1}c\right) \end{aligned}$$
(25b)

where the subscript p denotes the active constraints.

Appendix 2: The complete solution to the bilevel problem in 5.1.1

Whithin our approach, in order to solve the complete bilevel optimization problem the objective function of the upper level needs to be minimized within every critical region, i.e. the problem(s) in Eq. (26) need to be solved. The solutions from this problems that provide the best objective function in the upper level is the solution of the overall bilevel problem.

$$\begin{aligned} \begin{array}{ll} \underset{x,y}{\text {minimize}} &{} F(x,y) \\ \text {subject to} &{} G(x,y) \le 0 \\ &{} x \in CR_i, \quad \forall i\in \mathscr {I} \\ &{} y = p(x) \end{array} \end{aligned}$$
(26)

where \(\mathscr {I}\) is the finite set of critical regions identified from the parametric solution of the lower level problem and p(x) the parametric solution expressed as a function of the upper level problem optimization variables. Based on the nature of p(x) the problem of Eq. (26) can be either a (MI)LP or (MI)NLP problem. In our case, we can choose to either solve 8 MILP problems or 2 MINLP problems to global optimality. Note that the nonlinearity occurs only in the form of bilinear terms between binary and continuous terms. Table 6 presents the solutions of the 8 MILP problemsFootnote 4 and Table 7 presents the solutions of the 2 MINLP problems.Footnote 5

Table 6 Solution of the bilevel MILP-MILP at every linear critical region
Full size table

Columns Obj.L and Obj.U of the Table 6 correspond to the values of the objective functions for the lower and upper problem, respectively. Between critical region CR1 and CR2, the solution that minimizes Obj.U is kept. Similarly for the pairs \(CR3-CR4\), \(CR5-CR6\) and \(CR7-CR8\). The pairs correspond to the complete solution of the mp-MILP lower problem for a common value set of binary variable parameters. The overall solution is the one that corresponds to CR1. The value of the binary variables \(y_1\) and \(y_2\) is determined in the lower level together with the parametric expressions for \(x_1\) and \(x_2\) as a function of \(x_3\) and \(x_4\). Therefore, optimality of the lower objective function is guaranteed. The values of optimization variables of the upper problem determine the overall solution.

By considering the reduced parametric solution of the lower level mp-MILP with binary parameters of Table 5 the solutions corresponding to the overall problem are presented in Table 7.

Table 7 Solution of the bilevel MILP-MILP at every non-linear critical region
Full size table

In order to verify the solution, the binary variables were fixed to the values reported in CR1 and \(CR1_n\). The MILP-MILP bilevel optimization problem was transformed into a LP-LP problem. By formulating the full set of the KKT conditions for the latter we arrived at the same result for the continuous variables. This approach cannot be applied to the MILP-MILP bilevel problem as the derivation of KKT conditions for such a system is impossible.

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Oberdieck, R., Diangelakis, N.A., Avraamidou, S. et al. On unbounded and binary parameters in multi-parametric programming: applications to mixed-integer bilevel optimization and duality theory. J Glob Optim 69, 587–606 (2017). https://doi.org/10.1007/s10898-016-0463-z

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