Solution concepts in vector optimization: a fresh look at an old story

Abstract. Extending the approach of Jouini, Meddeb, and Touzi [Finance Stoch., 8 (2004) 1. Introduction. The concept of set-valued coherent measures of risk has been introduced recently by Jouini, Meddeb, and Touzi [16]. The basic question is how to evaluate the (financial) risk of a multivariate random outcome in terms of more than one reference instrument, for example if the regulator accepts deposits in more than one currency. This is of particular importance if transaction costs have to be paid for each transaction between assets including the reference instruments.This question became unexpectedly topical as the European Central Bank decreed [7] that temporarily, until the end of 2009, "the list of assets eligible as collateral in Eurosystem credit operations will be expanded" by "marketable debt instruments denominated in other currencies than the euro, namely the US dollar, the British pound and the Japanese yen, and issued in the euro area. These instruments will be subject to a uniform haircut add-on of 8%." See also [8].The following exposition is based on the model used in [16]; in particular, we assume the presence of proportional transaction costs modeled via a closed convex cone which goes back to [17]. A special feature of this model is that the risk of a d-dimensional random variable is evaluated in terms of m reference instruments with 1 ≤ m ≤ d. Usually m d will be true, but the case m = d is not excluded (and also, of course, neither m = 1, d > 1, nor m = d = 1).We shall make a few generalizations compared to [16]. First, we do not assume a "substitutability condition"; i.e., we do not assume that everything could be transferred into one distinguished currency. Second, we consider a general subspace M of R d as the collection

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“…See also Remark 2.5 below. For more information about lower semicontinuity of set-valued functions, compare [14].…”

Section: Definition Of Risk Measures Bymentioning

confidence: 99%

Duality for Set-Valued Measures of Risk

Hamel¹,

Heyde²

2010

SIAM J. Finan. Math.

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119

142

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“…Conjugate maps and Farkas-type results are crucial in any duality theory, and this is why they constitute the main tool and the main objective, respectively, of our research. The state of the art in vector optimization is described, e.g., in [3], [16], [18], [23], and references therein.…”

Section: Introductionmentioning

confidence: 99%

Farkas-Type Results for Vector-Valued Functions with Applications

Dinh

Goberna

López

et al. 2017

J Optim Theory Appl

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The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space de…ned by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.

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“…A function f : R k → G(R K , C) is called convex if s ∈ (0, 1) and x, y ∈ R k imply Again, if g is concave, the set x∈R k g(x) is convex and the convex hull in the formula for the sup x∈R k g(x) can be dropped. Finally, a solution concept for set optimization problems is given which is due to Heyde, Löhne [22].…”

Section: Discussionmentioning

confidence: 99%

A set optimization approach to zero-sum matrix games with multi-dimensional payoffs

Hamel

Löhne

2018

Math Meth Oper Res

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A new solution concept for two-player zero-sum matrix games with multidimensional payoff is introduced. It is based on extensions of the vector order in R K to order relations in the power set of R K , so-called set relations, and strictly motivated by the interpretation of the payoff as multi-dimensional loss for one and gain for the other player. The new concept provides coherent worst case estimates, i.e. minimax and maximin strategies, for games with multi-dimensional payoffs. It is shown that-in contrast to games with one-dimensional payoffs-minimax/maximin strategies are independent from Shapley's notion of equilibrium strategies for such games. Therefore, the two concepts are combined into new equilibrium notions for which existence theorems are given. By means of examples, relationships of the new concepts to existing ones such as Shapley and vector equilibria, vector minimax/maximin solutions and Pareto optimal security strategies are clarified. A algorithm for computing optimal strategies is presented.

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Solution concepts in vector optimization: a fresh look at an old story

Cited by 52 publications

References 19 publications

Duality for Set-Valued Measures of Risk

Duality for Set-Valued Measures of Risk

Farkas-Type Results for Vector-Valued Functions with Applications

A set optimization approach to zero-sum matrix games with multi-dimensional payoffs

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