The origins of quasi-concavity: a development between mathematics and economics

Guerraggio, Angelo; Molho, Elena

doi:10.1016/j.hm.2003.07.001

Cited by 34 publications

(19 citation statements)

References 13 publications

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“…Moreover, in many practical problems, there are certain proportions in which the objectives should be improved to achieve the most intensive synergy effect. Here, we propose the idea of the most promising direction of simultaneous improvement of objectives which agrees with the well-known assumption of concavity of the utility function (Guerraggio and Molho, 2004), implying that this function grows faster in certain directions of simultaneous decrease of objective function values (starting e.g. from the nadir objective vector).…”

Section: Preference Handling Techniquesupporting

confidence: 56%

A new preference handling technique for interactive multiobjective optimization without trading-off

et al. 2015

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Luque, M. (2015). A new preference handling technique for interactive multiobjective optimization without trading-off. Journal of Global Optimization, 63 (4), 633-652. doi:10.1007/s10898-015- AbstractBecause the purpose of multiobjective optimization methods is to optimize conflicting objectives simultaneously, they mainly focus on Pareto optimal solutions, where improvement with respect to some objective is only possible by allowing some other objective(s) to impair. Bringing this idea into practice requires the decision maker to think in terms of trading-off, which may limit the ability of effective problem solving. We outline some drawbacks of this and exploit another idea emphasizing the possibility of simultaneous improvement of all objectives. Based on this idea, we propose a technique for handling decision maker's preferences, which eliminates the necessity to think in terms of trade-offs. We incorporate this technique into an interactive trade-off-free method for multiobjective optimization. We call the resulting method NAUTILUS 2, which is also suitable for negotiation support. We demonstrate the applicability of the new method with an example problem.

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Section: Preference Handling Techniquesupporting

confidence: 56%

A new preference handling technique for interactive multiobjective optimization without trading-off

et al. 2015

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“…A natural type of application, that comes immediately in mind, is the description of the family of indifference curves for a utility function in the field of Economics. In this respect it is worthwhile mentioning that the interest for the notion of quasi-convexity just came from the analysis of the family of level curves in economic problems (see e.g., [13,25]). We can notice that, in the exchangeable case considered here, quasi-convexity implies Schur-convexity and, at least in the study of survival functions, the latter is equivalent to the following condition on the ageing function B: B( u, v) B(u, v) for every ∈ I, 0 v u 1.…”

Section: Discussionmentioning

confidence: 99%

Semi-copulas, capacities and families of level sets

Durante

Spizzichino

2010

Fuzzy Sets and Systems

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“…For an historic introduction, see e.g. [4,5]; more recently quasi convex functions were used in [6,7]. Let 0 , 1 smooth bounded open convex sets, strictly convex, 0 & 1 .…”

Section: Isoclines Related To Quasi Convex Functionsmentioning

confidence: 99%

On isocline lines for functions and convex stratifications of two variables

Longinetti

Manselli

Venturi

2010

Applicable Analysis

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Let the isoclines of a function u be the level lines of the function ¼ arg(Du). Formulas for the curvature and the length of isocline lines in terms of the curvatures k, j of the level curves and of the steepest descent lines of u are given. The special case when all isoclines are straight lines is studied: in this case the steepest descent lines bend proportionally to the level lines; the support function of the level lines is linear function on the isoclines parameterized by the level values, possibly changing them. This characterization gives a new proof of a property of the developable surfaces found in [A. Fialkow, Geometric characterization of invariant partial differential equations, Amer. J. Math. 59(4) (1937), pp. 833-844]. When u is in the class of quasi convex functions, the L p norm of the length function I of the isoclines has minimizers with isoclines straight lines; the same occurs for other functionals on u depending on k, j. For a strictly regular quasi convex function isoclines may have arbitrarily large length and arbitrarily large L 1 norm of I .

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The origins of quasi-concavity: a development between mathematics and economics

Cited by 34 publications

References 13 publications

A new preference handling technique for interactive multiobjective optimization without trading-off

A new preference handling technique for interactive multiobjective optimization without trading-off

Semi-copulas, capacities and families of level sets

On isocline lines for functions and convex stratifications of two variables

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