The talk will introduce the concepts of multilevel optimization and motivate them in the context of problems arising from the discretization of infinite dimensional applications. It will be shown how optimization methods can accomodate a number of useful (and classical) ideas from the multigrid community, and thereby produce substantial efficiency improvements compared to existing large-scale minimization techniques. Two different classes of multilevel methods will be discussed: trust-region and linesearch algorithms. The first class will be described in the context of a multilevel generalization of the well-known trust-region-Newton method. The second will focus on limited-memory quasi-Newton algorithms. Preliminary numerical results will be presented which indicate that both types of multilevel algorithms may be practically very advantageous.
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