Nonsmooth Analysis and Parametric Optimization

In an optimization problem that depends on parameters, an important issue is the effect that perturbations of the parameters can have on solutions to the problem and their associated multipliers. Under quite broad conditions the possibly multi-valued mapping that gives these elements in terms of the parameters turns out to enjoy a property of " proto-differentiability. " Generalized derivatives can then be calculated by solving an auxiliary optimization problem with auxiliary parameters. This is constructed from the original problem by taking second-order epi-derivatives of an essential objective function. From an abstract point of view, a general optimization problem relative to elements x of a Banach space X can be seen in terms of minimizing an expression f (x) over all x ∈ X , where f is a function on X with values in IR = IR ∪ {±∞}. The effective domain dom f := x ∈ X f (x) < ∞ gives the " feasible " or " admissible " elements x. Under the assumption that f is lower semicontinuous and proper (the latter meaning that f (x) < ∞ for at least one x, but f (x) > −∞ for all x), a solution ¯ x to the problem must satisfy 0 ∈ ∂f (¯ x), where ∂f denotes subgradients in the sense of Clarke [1] (see also Rockafellar [2]). When f is convex, such subgradients coincide with the ones of convex analysis, and the condition 0 ∈ ∂f (¯ x) is not only necessary for optimality but sufficient. A substantial calculus, part of a broader subject called nonsmooth analysis, has been built up for determining the set ∂f (¯ x) in the case of particular structure of f. Dual elements such as Lagrange multipliers are often involved, and under convexity assumptions these typically solve a dual problem of optimization. It has long been known that in order to derive and interpret the dual elements appearing in optimality conditions, it is important to study optimization problems not in isolation but in parametrized form. Only recently, however, have the tools of analysis reached the stage where it is possible to analyze in a general and effective manner the dependence of