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New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization

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Abstract

We present a new second-order directional derivative and study its properties. Using this derivative and the parabolic second-order derivative, we establish second-order necessary and sufficient optimality conditions for a general scalar optimization problem by means of the asymptotic and parabolic second-order tangent sets to the feasible set. For the sufficient conditions, the initial space must be finite dimensional. Then, these conditions are applied to a general vector optimization problem obtaining second-order optimality conditions that generalize the differentiable case. For this aim, we introduce a scalarization, and the relationships between the different types of solutions to the vector optimization problem and the scalarized problem are studied.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, E.T.S.I. Informática, Universidad de Valladolid, Edificio de Tecnologías de la Información y las Telecomunicaciones, Campus Miguel Delibes, s/n, 47011, Valladolid, Spain

    C. Gutiérrez

  2. Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, c/ Juan del Rosal 12, Ciudad Universitaria, 28040, Madrid, Spain

    B. Jiménez & V. Novo

Authors
  1. C. Gutiérrez
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  2. B. Jiménez
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  3. V. Novo
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Corresponding author

Correspondence to V. Novo.

Additional information

Communicated by X.Q. Yang.

This research was partially supported by the Ministerio de Educación y Ciencia (Spain), under projects MTM2006-02629 and Ingenio Mathematica (i-MATH) CSD2006-00032 (Consolider-Ingenio 2010), and by the Consejería de Educación de la Junta de Castilla y León (Spain), Project VA027B06.

The authors are grateful to the anonymous referees for valuable comments and suggestions.

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Gutiérrez, C., Jiménez, B. & Novo, V. New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization. J Optim Theory Appl 142, 85–106 (2009). https://doi.org/10.1007/s10957-009-9525-4

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