Learning curves
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LAGRANGE MULTIPLIERS The multiplicative, linear-combination constants that appear in the Lagrangian of a mathematical programming problem. They are generally dual variables if the dual exists, so-called "shadow prices" in linear programming, giving the rate of change of the optimal value with constraint changes, under appropriate conditions. See Lagrangian function; Nonlinear programming.
LAGRANGIAN DECOMPOSITION See Combinatorial and integer optimization.
LAGRANGIAN FUNCTION The general mathematical-programming problem of minimizing f(X) subject to a set of constraints {g;(x) ::; b;) has associated with it a Lagrangian function defined as L(x, A) = f(x) + L;A;[g;(x) bi)], where the components A; of the nonnegative vector A are called Lagrange multipliers. For a primal linear-programming problem, the Lagrange multipliers can be interpreted as the variables of the corresponding dual problem. See Nonlinear programming.
LAGRANGIAN RELAXATION An integer programming decomposition method. See Combinatorial and integer optimization.
LANCHESTER ATTRITION The concept of an explicit mathematical relationship between opposing military forces and casualty rates. The two classical "laws" are the linear law, which gives the casualty rate (derivative of force size with
respect to time) of one side as a negative constant multiplied by the product of the two sides' force sizes, and the square law, which gives the casualty rate of one side as a negative constant multiplied by the opposing side's force size. See Battle modeling; Homogeneous Lanchester equations; Lanchester's equations.
LANCHESTER'S EQUATIONS Joseph H. Engel Bethesda, Maryland HISTORICAL BACKGROUND: Lanchester's equations are named for the Englishman, F. W. Lanchester, who formulated and presented them in 1914 in a series of articles contributed to the British journal, Engineering, which then were printed in toto in Lanchester (1916). More recent presentation of these results appeared in the 1946 Operations Evaluation Group Report No. 54, Methods of Operations Research by Philip M. Morse and George E. Kimball, which was published commercially by John Wiley and Sons (Morse and Kimball, 1951). In addition, a reprint of the original1916 Lanchester work, "Mathematics in Warfare," appeared in The World of Mathematics, Volume 4, prepared by James R. Newman and published by Simon and Schuster in 1956. The significance of these equations is that they represented possibly the first mathematical analysis of forces in combat, and served as the guiding light (for the U.S.A. and its allies) behind the development, during and after World War II, of all two sided combat models, simulations and other methods of calculating combat losses during a battle. It appears that M. Osipov developed and published comparable equations in a Tsarist Russian military journal in 1915, perhaps independent ofLanchester's results. A translation of his work into English, prepared by Robert L. Helmbold and Allen S. Rehm, was printed in September 1991 by the U.S. Army Concepts Analysis Agency. Lanch
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