3. Augmented Lagrangian Methods for the Solution of Variational Problems
Society for Industrial and Applied Mathematics eBooks · 1989
1. Introduction and synopsis. 1.1. Introduction. Duality principles play an important role in mechanics, physics, mathematical economy, and other branches of science and engineering; their role in mechanics for proving existence results for some particular classes of nonlinear problems has already been illustrated in Chapter 2, § 2. Another field in which such principles play a pivotal role is mathematical programming, i.e., the science (and sometimes the art) of minimizing or maximizing functions over sets of various kinds. Actually, some optimization techniques are founded on the application of these duality principles, most of them using those vectors called multipliers (Lagrange multipliers, John-Kuhn-Tucker multipliers, etc.). Such multiplier methods are discussed in Arrow, Hurwicz, and Uzawa [1958] (motivated by economical equilibria) and in Glowinski, Lions, and Tremolières [1976], [1981] (motivated by nonlinear mechanics). Unfortunately, these multiplier methods (at least the original one) converge linearly at best. Therefore, the resulting algorithms may be slow and, to improve the speed of convergence, one may think of conjugate-gradient variants of the original algorithms.