Abstract
In mathematical optimization, the Lagrangian approach is a general method to find an optimal solution of a finite (infinite) dimensional constrained continuous optimization problem. This method has been introduced by the Italian mathematician Joseph-Louis Lagrange in 1755 in a series of letters to Euler. This approach became known under the name The Principle of Lagrange and was also applied much later to integer optimization problems. The basic idea behind this method is to replace a constrained optimization problem by a sequence of easier solvable optimization problems having fewer constraints and penalizing the deletion of some of the original constraints by replacing the original objective function. To select the best penalization, the so-called Lagrangian dual function needs to be optimized and a possible algorithm to do so is given by the so-called subgradient method. This method is discussed in detail at the end of this chapter. The Lagrangian approach led to the introduction of dual optimization problems and penalization methods in nonlinear programming and recently to the development of interior point methods and the identification of polynomial solvable classes of continuous optimization problems. Also it had its impact on how to construct algorithms to generate approximate solutions of integer optimization problems. In this chapter, we discuss in the first part the main ideas behind this approach for any type of finite dimensional optimization problem. In the remaining parts of this chapter we focus in more detail on how this approach is used in continuous optimization problems and show its full impact on the so-called K-convex continuous optimization problems. Also we consider its application within linear integer programming problems and show how it is used to solve these type of problems. To illustrate its application to the well-known integer programming problems, we consider in the final section its application to some classical vehicle routing and location models. As such this chapter should be regarded as an introduction to duality theory and the Lagrangian approach for less mathematically oriented readers proving at the same time most of the results using the simplest possible proofs.
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References
Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (1990). Linear programming and network flows. New York: John Wiley and Sons.
Bazaraa, M. S., Sherali, H. D., & Shetty, C. M. (1993). Nonlinear programming: theory and algorithms. New York: John Wiley and Sons.
Beasley, J., Dowsland, K., Glover, F., Laguna, M., Peterson, C., Reeves, C. R., & Soderberg, B. (1993). Modern heuristic techniques for Combinatorial Problems. New York: John Wiley and Sons.
Ben-Tal, A., & Nemirovski, A. (2001), Lectures on modern convex optimization: analysis, algorithms and engineering applications. Philadelphia: SIAM.
Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge: Cambridge University Press.
Choquet, G. (1976). Lectures on analysis volume II: representation theory. London: W.A. Benjamin.
Chvatal, V. (1983). Linear programming. New York: Macmillan Company.
Correa, R., & Lemaréchal, C. (1993). Convergence of some algorithms for convex minimization. Mathematical Programming, 62(1), 261–275.
Daskin, M. S. (2011). Network and discrete location: models, algorithms, and applications. New York: John Wiley and Sons.
Erlenkotter, D. (1978). A dual-based procedure for uncapacitated facility location. Operations Research, 26(6), 992–1009.
Everett III, H. (1963). Generalized Lagrange multiplier method for solving problems of optimum allocation of resources. Operations Research, 11(3), 399–417.
Fiacco, A. V., & McCormick, G. P. (1969). Nonlinear programming: sequential unconstrained minimization techniques. New York: Wiley.
Francis, R. L., McGinnis, L. F., & White, J. A. (1992). Facility Layout and Location: an analytical approach. Pearson College Division.
Frenk, J. B. G., & Kassay, G. (1999). On classes of generalized convex functions, Gordan–Farkas type theorems, and Lagrangian duality. Journal of Optimization Theory and Applications, 102(2), 315–343.
Frenk, J. B. G., & Kassay, G. (2005). Introduction to convex and quasiconvex analysis. In Handbook of generalized convexity and generalized monotonicity (pp. 3–87). New York: Springer.
Frenk, J. B. G., & Kassay, G. (2006). The level set method of Joo and its use in minimax theory. Mathematical Programming, 105(1), 145–155.
Frenk, J. B. G., & Kassay, G. (2007). Lagrangian duality and cone convexlike functions. Journal of Optimization Theory and Applications, 134(2), 207–222.
Frenk, J. B. G., & Kassay, G. (2008). On noncooperative games, minimax theorems and equilibrium problems. In A. Chinchuluum, P. M. Pardalos, A. Migdalas & L. Pitsoulis (Eds.) On Noncooperative Games, Minimax Theorems and Equilibrium Problems. (pp. 211–250). New York: Springer.
Frenk, J. B. G., Kas, P., & Kassay, G. (2007). On linear programming duality and necessary and sufficient conditions in minimax theory. Journal of Optimization Theory and Applications, 132, 423–439.
Garey, M. R., & Johnson, D. S. (1979). Computers and intractability, a guide to the theory of NP completeness. San Francisco: W.F Freeman.
Giannessi, F. (1984). Theorems of the alternative and optimality conditions. Journal of Optimization Theory and Applications, 42(3), 331–365.
Hiriart-Urruty, J. B., & Lemaréchal, C. (2013). Convex analysis and minimization algorithms I: fundamentals. New York: Springer Verlag.
Lagrange, J. L. (1806). Leçons sur le calcul des fonctions. Paris: Courcier.
Martello, S. (1990). Knapsack problems: algorithms and computer implementations. New York: John Wiley and Sons.
Nemhauser, G. L., & Wolsey, L. A. (1988). Integer and combinatorial optimization. New York: John Wiley and Sons.
Neumann, J. V. (1928). Zur theorie der gesellschaftsspiele. Mathematische Annalen, 100(1), 295–320.
Nesterov, Y., & Nenirovski, A. (2001). Interior point polynomial algorithms in convex programming. Philadelphia: SİAM.
Nocedal, J., & Wright, S. (2006). Numerical optimization. New York: Springer Verlag.
Rockafellar, R. (1970). Convex analysis. Princeton: Princeton University Press.
Saigal, R. (1997). Linear programming: a modern integrated analysis. Journal of the Operational Research Society, 48(7), 762–762.
Schrijver, A. (1998). Theory of Linear and Integer Programming. New York: John Wiley and Sons.
Shor, N. Z. (2012). Minimization methods for non-differentiable functions. New York: Springer Verlag.
Sion. (1958). On general minimax theorems. Pacific Journal of Mathematics (8), 171–178.
Wolkowicz, H. (1981). Some applications of optimization in matrix theory. Linear Algebra and Its Applications, 40, 101–118.
Wolsey, L. A. (1998). Integer programming. New York: John Wiley and Sons.
Yuan, G. X. (1999). KKM theory and applications to nonlinear analysis. New York: Marcel Dekker.
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The authors like to thank the comments of anonymous referees, which greatly improved the previous version of this chapter.
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Frenk, J.B.G., Javadi, S. (2022). On the Principle of Lagrange in Optimization Theory and Its Application in Transportation and Location Problems. In: Topcu, Y.I., Önsel Ekici, Ş., Kabak, Ö., Aktas, E., Özaydın, Ö. (eds) New Perspectives in Operations Research and Management Science. International Series in Operations Research & Management Science, vol 326. Springer, Cham. https://doi.org/10.1007/978-3-030-91851-4_2
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