Set Optimization and Applications - The State of the Art: From Set Relations to Set-Valued Risk Measures (Springer Proceedings in Mathematics & Statistics Book 151)

This volume presents five surveys with extensive bibliographies and six original contributions on set optimization and its applications in mathematical finance and game theory. The topics range from more conventional approaches that look for minimal/maximal elements with respect to vector orders or set relations, to the new complete-lattice approach that comprises a coherent solution concept for set optimization problems, along with existence results, duality theorems, optimality conditions, variational inequalities and theoretical foundations for algorithms. Modern approaches to scalarization methods can be found as well as a fundamental contribution to conditional analysis. The theory is tailor-made for financial applications, in particular risk evaluation and [super-]hedging for market models with transaction costs, but it also provides a refreshing new perspective on vector optimization. There is no comparable volume on the market, making the book an invaluable resource for researchers working in vector optimization and multi-criteria decision-making, mathematical finance and economics as well as [set-valued] variational analysis. Front Matter....Pages i-xii Front Matter....Pages 1-1 A Comparison of Techniques for Dynamic Multivariate Risk Measures....Pages 3-41 Nonlinear Scalarizations of Set Optimization Problems with Set Orderings....Pages 43-63 Set Optimization—A Rather Short Introduction....Pages 65-141 A Survey of Set Optimization Problems with Set Solutions....Pages 143-158 Linear Vector Optimization and European Option Pricing Under Proportional Transaction Costs....Pages 159-176 Front Matter....Pages 177-177 Conditional Analysis on \(\mathbb {R}^d\) ....Pages 179-211 Set Optimization Meets Variational Inequalities....Pages 213-247 Estimates of Error Bounds for Some Sets of Efficient Solutions of a Set-Valued Optimization Problem....Pages 249-273 On Supremal and Maximal Sets with Respect to Random Partial Orders....Pages 275-291 Generalized Minimality in Set Optimization....Pages 293-311 On Characterization of Nash Equilibrium Strategy in Bi-Matrix Games with Set Payoffs....Pages 313-331