Abstract
Mathematical programming is concerned with the extremization of a function f defined over an n-dimensional design space R n and bounded by a set S in the design space. The set S may be defined by equality or inequality constraints, and these constraints may assume linear or nonlinear forms. The function f together with the set S in the domain of f is called a mathematical program or a mathematical programming problem. This terminology is in common usage in the context of problems which arise in planning and scheduling which are generally studied under operations research, the branch of mathematics concerned with decision making processes. Mathematical programming problems may be classified into several different categories depending on the nature and form of the design variables, constraint functions, and the objective function. However, only two of these categories are of interest to us, namely linear and nonlinear programming problems (commonly designated as LP and NLP, respectively).
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References
Charnes, A. and Greenberg, H. J., “Plastic Collapse and Linear Programming,” Bull. Am. Math. Soc, 57, 480, 1951.
Calladine, C.R., Engineering Plasticity. Pergamon Press, 1969.
Cohn, M.Z., Ghosh, S.K. and Parimi, S.R., “Unified Approach to Theory of Plastic Structures,” Journal of the EM Division, 98(EM5), pp. 1133–1158, 1972.
Neal, B. G., The Plastic Methods of Structural Analysis, 3rd edition, Chapman and Hall Ltd., London, 1977.
Zeman, P. and Irvine, H. M., Plastic Design, An Imposed Hinge-Rotation Approach, Allen and Unwin, Boston, 1986.
Massonet, C.E. and Save, M.A., Plastic Analysis and Design, Beams and Frames, Vol. 1. Blaisdell Publishing Co., 1965.
Lin, T.Y. and Burns, N.H., Design of Prestressed Concrete Structures, 3rd ed. John Wiley and Sons, New York, 1981.
Parme, A.L. and Paris, G.H., “Designing for Continuity in Prestressed Concrete Structures,” J. Am. Concr. Inst., 23(1), pp. 45–64, 1951.
Morris, D., “Prestressed Concrete Design by Linear Programming,” J. Struct. Div., 104(ST3), pp. 439–452, 1978.
Kirsch, U., “Optimum Design of Prestressed Beams,” Computers and Structures 2, pp. 573–583, 1972.
Luenberger, D. G., Introduction to Linear and Nonlinear Programming, Addison-Wesley, Reading, Mass., 1973.
Majid, K.I., Nonlinear Structures, London, Butterworths, 1972.
Dantzig, G., Linear Programming and Extensions, Princeton University Press, Princeton, NJ, 1963.
Karmarkar, N., “A New Polynomial-Time Algorithm for Linear Programming,” Combinatorica, 4(4), pp. 373–395, 1984.
Todd, M. J. and Burrell, B. P., “An Extension of Karmarkar’s Algorithm for Linear Programming Using Dual Variables,” Algorithmica, 1, pp. 409–424, 1986.
Rinaldi, G., “A Projective Method for Linear Programming with Box-type Constraints,” Algorithmica, 1, pp. 517–527, 1986.
Strang, G., “Karmarkar’s Algorithm and its Place in Applied Mathematics,” The Mathematical Intelligencer, 9, 2, pp. 4–10, 1987.
Vanderbei, R. F., Meketon, M. S., and Freedman, B. A., “A Modification of Karmarkar’s Linear Programming Algorithm,” Algorithmica, 1, pp. 395–407, 1986.
Garfinkel, R. S., and Nemhauser, G. L., Integer Programming, John Wiley & Sons, Inc., New York, 1972.
Lawler, E. L., and Wood, D. E., “Branch-and-Bound Methods—A Survey,” Operations research, 14, pp. 699–719, 1966.
Tomlin, J. A., “Branch-and-Bound Methods for Integer and Non-convex Programming,” in Integer and Nonlinear Programming, J. Abadie (ed.), pp. 437–450, Elsevier Publishing Co., New York, 1970.
Land, A. H., and Doig, A. G., “An Automatic Method for Solving Discrete Programming Problems,” Econometrica, 28, pp. 497–520, 1960.
Johnson, E. L., and Powell, S., “Integer Programming Codes,” in Design and Implementation of Optimization Software, Greenberg, H. J. (ed.), pp. 225–240, 1978.
Schrage, L., Linear, Integer, and Quadratic Programming with LINDO, 4th Edition, The Scientific Press, Redwood City CA., 1989.
Kovács, L. B., Combinatorial Methods of Discrete Programming, Mathematical Methods of Operations Research Series, Vol. 2, Akadémiai Kiadó, Budapest, 1980.
Haftka, R. T., and Walsh, J. L., “Stacking-sequence Optimization for Buckling of Laminated Plates by Integer Programming,” AIAA J. (in press).
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© 1992 Springer Science+Business Media Dordrecht
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Haftka, R.T., Gürdal, Z. (1992). Linear Programming. In: Elements of Structural Optimization. Solid Mechanics And Its Applications, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2550-5_3
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DOI: https://doi.org/10.1007/978-94-011-2550-5_3
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