Abstract
In a discrete programming problems the feasible solution set is a multi-connected and non-convex domain. Due to this fact the direct application of well elaborated methods of continuous optimum structural design is not possible.
In this chapter the dual approach in a discrete structural optimization problems is presented The constraints of the problem are scaled by means of Lagrangean multipliers, added to the cost function and relaxing from the problem, giving an unconstrained problem. To obtain the best lower bound (in the case of minimization) the dual of the relaxed problem, with the Lagrangean multipliers as variables must be solved. The dual function is a continuous, concave but not differentiable everywhere. We treat the dual problem as a nondifferentiable steepest ascent problem using in a solution a generalization of the gradient — the subgradient.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Bauer, (1992), Algorithms of nondifferentiable optimization in discrete optimum structural design, ZAMM, 72, T563 - T566.
J. Bauer, (1995), Discrete variable optimization of nonhomogeneous circular and annular plastic plates, In: N. Olhoff and G.I.N. Rozvany eds., First World Congress of Structural and Multidisciplinary Optimization, Pergamon, 933–938.
M. L. Fischer, W. D. Northup and J. F. Shapiro, (1975), Using duality to solve discrete optimization problem: theory and computational experience, In: M. L. Balinski and Philip Wolfe eds., Math. Programming Study, Vol. 3–Nondifferentiable Optimization, North-Holland, 56–94.
Renpu Ge and Changbin Huang, (1989), A continuous approach to nonlinear integer programming, Appl. Math. Comput., 34,1, 39–60
A. M. Geoffrion, (1974), Lagrangean relaxation for integer programming, In: M.L.Balinski ed.,Math. Programming Study, Vol. 2 Approaches to Integer Programming, North-Holland, 82–114
J. A. Hoogeveen and S. L. van de Velde, (1995), Lagrangian bounds by use of slack variables: applications to machine scheduling problems, Mathematical Programming, 70, 2, 173–190
O. Jonsson and T. Larsson, (1990), Lagrange’an relaxation and subgradient optimization applied to optimal design with discrete sizing, Engineering Optimization, 16, 221–233.
D. O. Lamblin, G. Guerlement and M. A. Save, (1985), Solutions de dimensionnement plastique de volume minimal de plaques circulaires pleines et sandwiches en presence de constraintes technologiques, J. de Mec., teor. et appl., 4, 4. 433–461.
M. M. Makela and P. Neittaanmaki, (1992), Nonsmooth Optimization, World Scientific, Singapore.
G. L. Nemhauser and L. A. Wolsey, (1988), Integer and Combinatorial Optimization, Wiley, New York.
B. T. Poljak, (1983), Introduction to Optimization (in Russian), Nauka, Moskva
U. T. Ringertz, (1988), Discrete-continuous structural optimization, In: G.I. N. Rozvany, B.1. Karihaloo eds., Structural Optimization, Kluwer, 257–264.
L. A. Schmit and C. Fleury, (1980), Discrete-continuous variable structural synthesis using dual methods, AIAA J., 18, 1515–1524.
A. Sepulveda and J. H. Cassis, (1986), An efficient algorithm for the optimum design of trusses with discrete variables, Int. J. Num. Meth. Engng., 23, 1111–1130.
J. F. Shapiro, (1979), A survey of Lagrange’an techniques for discrete optimization, Annals of Discrete Mathematics. North-Holland, 5, 113–138.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag Wien
About this chapter
Cite this chapter
Bauer, J. (1997). Dual Methods in Discrete Structural Optimization. In: Gutkowski, W. (eds) Discrete Structural Optimization. International Centre for Mechanical Sciences, vol 373. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2754-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-7091-2754-4_5
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-82901-1
Online ISBN: 978-3-7091-2754-4
eBook Packages: Springer Book Archive