Optimum beam design via stochastic programming. (English) Zbl 1201.90145

Summary: The purpose of the paper is to discuss the applicability of stochastic programming models and methods to civil engineering design problems. In cooperation with experts in civil engineering, the problem concerning an optimal design of beam dimensions has been chosen. The corresponding mathematical model involves an ODE-type constraint, uncertain parameter related to the material characteristics and multiple criteria. As a result, a multi-criteria stochastic nonlinear optimization model is obtained. It has been shown that two-stage stochastic programming offers a promising approach to solving similar problems. A computational scheme for this type of problems is proposed, including discretization methods for random elements and ODE constraint. An approximation is derived to implement the mathematical model and solve it in GAMS. The solution quality is determined by an interval estimate of the optimality gap computed by a Monte Carlo bounding technique. The parametric analysis of a multi-criteria model results in efficient frontier computation. Furthermore, a progressive hedging algorithm is implemented and tested for the selected problem in view of the future possibilities of parallel computing of large engineering problems. Finally, two discretization methods are compared by using GAMS and ANSYS.


90C15 Stochastic programming
65C05 Monte Carlo methods
90C29 Multi-objective and goal programming
49M27 Decomposition methods


Full Text: EuDML Link


[1] Betts, J. T.: Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, London 2001. · Zbl 0995.49017
[2] Branda, M., Dupačová, J.: Approximations and contamination bounds for probabilistic programs. SPEPS 13 (2008), 1-25.
[3] Conti, S., Held, H., Pach, M., Rumpf, M., Schultz, R.: Shape optimization under uncertainty - a stochastic programming perspective. SIAM J. Optim. 19 (2009), 1610-1632. · Zbl 1176.49045
[4] Haslinger, J., Mäkinen, R. A. E.: Introduction to Shape Optimization: Theory, Approximation, and Computation (Advances in Design and Control). SIAM, 2003. · Zbl 1020.74001
[5] Helgason, T., Wallace, S. W.: Approximate scenario solutions in the progressive hedging algorithm. Ann. Oper. Res. 31 (1991), 425-444. · Zbl 0739.90047
[6] Hořejší, J., Novák, O.: Statics of Structures (in Czech). SNTL, Praha 1972.
[7] Mathews, J. H., Fink, K. D.: Numerical Methods Using Matlab. Prentice Hall, New Jersey 2004.
[8] Mauder, T., Kavička, F., Štětina, J., Franěk, Z., Masarik, M.: A mathematical & stochastic modelling of the concasting of steel slabs. Proc. METAL Conference, Tanger 2009, pp. 41-48.
[9] Mak, W.-K., Morton, D. P., Wood, R. K.: Monte Carlo bounding techniques for determining solution quality in stochastic programs. Oper. Res. Lett. 24 (1999), 47-56. · Zbl 0956.90022
[10] Plšek, J., Štěpánek, P.: Optimisation of reinforced concrete cross-section. Engenharia Estudo e Pesquisa 1 (2007), 24-36.
[11] Plšek, J., Štěpánek, P., Popela, P.: Deterministic and reliability based structural optimization of concrete cross-section. J. Advanced Concrete Technology 1 (2007), 63-74.
[12] Rockafellar, R. T., Wets, R. J.-B.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16 (1991), 1-29. · Zbl 0729.90067
[13] Ruszczynski, A., Shapiro, A.: Handbooks in Operations Research and Management Science, Vol. 10: Stochastic Programming. Elsevier, Amsterdam 2003.
[14] Steuer, R. E.: Multiple Criteria Optimization: Theory, Computation and Application. John Wiley, New York 1986. · Zbl 0742.90068
[15] Šarlej, M., Petr, P., Hájek, J., Stehlík, P.: Computational support in experimental burner design optimisation. Appl. Thermal Engrg. 16 (2007), 2727-2732.
[16] Wets, R. J.-B.: The aggregation principle in scenario analysis and stochastic optimization. Algorithms and Model Formulations in Mathematical Programming (S. Wallace, Springer-Verlag, Berlin 1989, pp. 91-113.
[17] Žampachová, E., Popela, P.: The selected PDE constrained stochastic programming problem. Proc. Risk, Quality and Reliability Conference, Ostrava 2007, pp. 233-237.
[18] Žampachová, E.: Approximate solution of PDE constrained stochastic optimization problems (in Czech). Proc. FME Junior Conference, Brno 2009, pp. 16-23.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.