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Qualitative robustness of set-valued value-at-risk
No full textRisk measures are defined as functionals of the portfolio loss distribution, thus implicitly assuming the knowledge of such a distribution. However, in practical applications, the need for estimation arises and with it the need to study the effects of mis-specification errors, as well as estimation errors on the final conclusion. In this paper we focus on the qualitative robustness of a sequence of estimators for set-valued risk measures. These properties are studied in detail for two well-known examples of set-valued risk measures: the value-at-risk and the maximum average value-at-risk. Our results illustrate, in particular, that estimation of set-valued value-at-risk can be given in terms of random sets. Moreover, we observe that historical set-valued value-at-risk, while failing to be sub-additive, leads to a more robust procedure than alternatives such as the maximum likelihood average value at-risk
Insights on the Theory of Robust Games
Full text linkA robust game is a distribution-free model to handle ambiguity generated by a bounded set of possible realizations of the values of players' payoff functions. The players are worst-case optimizers and a solution, called robust-optimization equilibrium, is guaranteed by standard regularity conditions. The paper investigates the sensitivity to the level of uncertainty of this equilibrium focusing on robust games with no private information. Specifically, we prove that a robust-optimization equilibrium is an epsilon-Nash equilibrium of the nominal counterpart game, where epsilon measures the extra profit that a player would obtain by reducing his level of uncertainty. Moreover, given an epsilon-Nash equilibrium of a nominal game, we prove that it is always possible to introduce uncertainty such that the epsilon-Nash equilibrium is a robust-optimization equilibrium. These theoretical insights increase our understanding on how uncertainty impacts on the solutions of a robust game. Solutions that can be extremely sensitive to the level of uncertainty as the worst-case approach introduces non-linearity in the payoff functions. An example shows that a robust Cournot duopoly model can admit multiple and asymmetric robust-optimization equilibria despite only a symmetric Nash equilibrium exists for the nominal counterpart game
Extended Well-Posedness of Quasiconvex Vector Optimization Problems.
No full textThe notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems
Existence of solutions of Minty type scalar and vector variational inequalities
No full textScalar and vector variational inequalities involving bifunctions are considered. Necessary and sufficient conditions for existence of solutions are proposed, based on the finite intersection
property of a suitable set-valued function. In the case of a properly quasi-monotone bifunction the relation to the KKM property is established. The result of John [19] concerning the
characterization of properly quasi-monotone functions in terms of existence of solutions of variational inequalities (scalar case) is extended to bifunctions both in the scalar and vector
cases. Existence of solutions for scalar equilibrium problems with bifunctions consider Bianchi and Pini [3], and their results admit a reformulation for variational inequalities as a special class of equilibrium problems. The present paper also generalizes this result, even in the scalar case (here the bifunctions are not assumed quasi-convex). As for the vector case, it should
be stressed that two type of variational inequalities are studied, and respectively the quasimonotonicity is understood in two different ways. Finally, as a particular case variational
inequalities of differentiable type are discussed
Some remarks on the Minty vector variational principle
No full textAbstractIn scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [F. Giannessi, On Minty variational principle, in: New Trends in Mathematical Programming, Kluwer Academic, Dordrecht, 1997, pp. 93–99] and subsequently in [X.M. Yang, X.Q. Yang, K.L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004) 193–201]. In these papers, in the particular case of a differentiable objective function f taking values in Rm and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions
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