Preface to "Set Optimization and Applications - The State of the Art"
Excerpt: In 1998, a special issue of the journal Mathematical Methods of Operations Research was published, edited by Guang-Ya Chen and Johannes Jahn. It was devoted solely to optimization problems with set-valued objective functions. Since then, major breakthroughs have been made including new “set relations,” new solution concepts for set optimization problems and a new framework for a set-valued convex analysis. The area has been pushed further by the discovery of its relevance for financial mathematics: risk evaluation in markets with “frictions” such as transaction costs or illiquidity effects is best done using set-valued functions. It turns out that results such as the superhedging theorem of Y. Kabanov from 1999 are essentially set-valued duality results, and the dual variables in this superhedging theorem are precisely what the recent theory expects them to be.