Graphical Convergence of Set-Valued Maps

Aubin, J.-P. (1987). Graphical Convergence of Set-Valued Maps. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-87-083

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Abstract

This is an introduction to graphical convergence of set-valued maps and the epigraphical convergence of extended real-valued functions.

It is now well established that maps and, more generally, set-valued maps should be regarded not only as maps from one space to another, but should be characterized in an intrinsic and symmetric way by their graphs.

When dealing with limits of maps, either single-valued or set-valued, it is quite advantageous to overcome the natural reluctance to handle convergence of subsets and to replace pointwise convergence by "graphical convergence": Instead of studying (more or less uniform) limits of the images, one consider the limits of their graphs.

One of the main reasons is that doing so is that a map and its inverse are treated on the same footing. This is quite important in approximation theory and numerical analysis.

The concepts of graphical convergence of set-valued maps are related to the concepts of epigraphical limits of functions, which had recently met an important success to overcome the failure of pointwise convergence in many problems of calculus of variations, optimization. stochastic programming. etc.

Finally, this report provides a first study of the Kuratowski upper and lower limits of tangent cones, which is needed to compute generalized derivatives and epi-derivatives of graphical and epigraphical limits of maps and functions.

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