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.
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This work was partially supported by grant 55681 of the CONACyT.
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Vázquez, F.G., Jongen, H.T., Shikhman, V. et al. Criteria for efficiency in vector optimization. Math Meth Oper Res 70, 35–46 (2009). https://doi.org/10.1007/s00186-008-0230-0
- Vector optimization
- Efficiency criteria
Mathematics Subject Classification (2000)