Elements of Structural Optimization pp 61-98 | Cite as

# Linear Mathematical Programming

## 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 form. 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 the decision making process. 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 the *linear* and the *nonlinear programming* (commonly designated as LP and NLP, respectively).

## Keywords

Design Variable Linear Programming Problem Collapse Mechanism Collapse Load Basic Feasible Solution## Preview

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## References

- [1]Charnes, A. and Greenberg, H. J., “Plastic collapse and linear programming,” Bull. Am. Math. Soc., 57, 480, 1951.Google Scholar
- [2]Calladine, C.R., Engineering Plasticity. Pergamon Press, 1969.Google Scholar
- [3]Cohn, M.Z., Ghosh, S.K. and Parimi, S.R., “Unified approach to theory of plastic structures,” Journal of the EM Division, ASCE 5, pp. 1133–1158, 1972.Google Scholar
- [4]Neal, B. G., The plastic methods of structural analysis, 3rd edition, Chapman and Hall Ltd., London, 1977.Google Scholar
- [5]Zeman, P. and Irvine, H. M., Plastic design: an imposed hinge—rotation approach, Allen and Unwin, Boston, 1986.Google Scholar
- [6]Massonet, C.E. and Save, M.A., Plastic Analysis and Design, Beams and Frames, Vol. 1. Blaisdell Publishing Co., 1965.Google Scholar
- [7]Lin, T.Y. and Burns, N.H., Design of Prestressed Concrete Structures, 3rd ed. John Wiley and Sons, New York, 1981.Google Scholar
- [8]Parme, R.L. and Paris, G.H., “Designing for continuity in prestressed concrete structures,” Am. Concr. Inst. J., 48, pp. 54–64, 1951.Google Scholar
- [9]Morris, D., “Prestressed concrete design by linear programming,” J. Struct. D.v., ASCE 3, pp. 439–452, 1978.Google Scholar
- [10]Kirsch, U., “Optimum design of prestressed beams,” J. Computers and Structures 2, pp. 573–583, 1972.CrossRefGoogle Scholar
- [11]Luenberger, D., Introduction to Linear and Nonlinear Programming, Addison—Wesley, Reading, Mass., 1973.zbMATHGoogle Scholar
- [12]Majid, K.I., Nonlinear Structures, London, Butterworths, 1972.Google Scholar
- [13]Dantzig, G., Linear Programming and Extensions, Princeton University Press, Princeton, NJ, 1963.zbMATHGoogle Scholar
- [14]Karmarkar, N., “A new polynomial—time algorithm for linear programming,” Combinatorica, 4, pp. 373–395, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
- [15]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.MathSciNetzbMATHCrossRefGoogle Scholar
- [16]Rinaldi, G., “A projective method for linear programming with box—type constraints,” Algorithmica, 1, pp. 517–527, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
- [17]Strang, G., “Karmarkar’s algorithm and its place in applied mathematics,” The Mathematical Intelligence, 9, 2, pp. 4–10, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
- [18]Vanderbei, R. F., Meketon, M. S., and Freedman, B. A., “A modification of Karmarkar’s linear programming algorithm,” Algorithmica, 1, pp. 395–407, 1986.MathSciNetzbMATHCrossRefGoogle Scholar