Criteria for efficiency in vector optimization |
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Authors: | Francisco Guerra Vázquez Hubertus Th Jongen Vladimir Shikhman Maxim Ivanov Todorov |
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Institution: | (1) Department of Physics and Mathematics, UDLA, 72820 San Andrés Cholula, Puebla, Mexico;(2) Department of Mathematics – C, RWTH Aachen University, 52056 Aachen, Germany |
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Abstract: | We consider unconstrained finite dimensional multi-criteria optimization problems, where the objective functions are continuously
differentiable. Motivated by previous work of Brosowski and da Silva (1994), we suggest a number of tests (TEST 1–4) to detect,
whether a certain point is a locally (weakly) efficient solution for the underlying vector optimization problem or not. Our
aim is to show: the points, at which none of the TESTs 1–4 can be applied, form a nowhere dense set in the state space. TESTs
1 and 2 are exactly those proposed by Brosowski and da Silva. TEST 3 deals with a local constant behavior of at least one
of the objective functions. TEST 4 includes some conditions on the gradients of objective functions satisfied locally around
the point of interest. It is formulated as a Conjecture. It is proven under additional assumptions on the objective functions,
such as linear independence of the gradients, convexity or directional monotonicity.
This work was partially supported by grant 55681 of the CONACyT. |
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Keywords: | Vector optimization Efficiency criteria Density |
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