From scalar to vector optimization

Ivan Ginchev; Angelo Guerraggio; Matteo Rocca

Applications of Mathematics (2006)

  • Volume: 51, Issue: 1, page 5-36
  • ISSN: 0862-7940

Abstract

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Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem φ ( x ) → min , x ∈ ℝ m , are given. These conditions work with arbitrary functions φ ℝ m → ℝ ¯ , but they show inconsistency with the classical derivatives. This is a base to pose the question whether the formulated optimality conditions remain true when the “inconsistent” Hadamard derivatives are replaced with the “consistent” Dini derivatives. It is shown that the answer is affirmative if φ is of class 𝒞 1 , 1 (i.e., differentiable with locally Lipschitz derivative). Further, considering 𝒞 1 , 1 functions, the discussion is raised to unconstrained vector optimization problems. Using the so called “oriented distance” from a point to a set, we generalize to an arbitrary ordering cone some second-order necessary conditions and sufficient conditions given by Liu, Neittaanmäki, Křížek for a polyhedral cone. Furthermore, we show that the conditions obtained are sufficient not only for efficiency but also for strict efficiency.

How to cite

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Ginchev, Ivan, Guerraggio, Angelo, and Rocca, Matteo. "From scalar to vector optimization." Applications of Mathematics 51.1 (2006): 5-36. <http://eudml.org/doc/33241>.

@article{Ginchev2006,
abstract = {Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem $\phi (x)\rightarrow \min $, $x\in \mathbb \{R\}^m$, are given. These conditions work with arbitrary functions $\phi \:\mathbb \{R\}^m \rightarrow \overline\{\mathbb \{R\}\}$, but they show inconsistency with the classical derivatives. This is a base to pose the question whether the formulated optimality conditions remain true when the “inconsistent” Hadamard derivatives are replaced with the “consistent” Dini derivatives. It is shown that the answer is affirmative if $\phi $ is of class $\{\mathcal \{C\}\}^\{1,1\}$ (i.e., differentiable with locally Lipschitz derivative). Further, considering $\{\mathcal \{C\}\}^\{1,1\}$ functions, the discussion is raised to unconstrained vector optimization problems. Using the so called “oriented distance” from a point to a set, we generalize to an arbitrary ordering cone some second-order necessary conditions and sufficient conditions given by Liu, Neittaanmäki, Křížek for a polyhedral cone. Furthermore, we show that the conditions obtained are sufficient not only for efficiency but also for strict efficiency.},
author = {Ginchev, Ivan, Guerraggio, Angelo, Rocca, Matteo},
journal = {Applications of Mathematics},
keywords = {scalar and vector optimization; $\{\mathcal \{C\}\}^\{1,1\}$ functions; Hadamard and Dini derivatives; second-order optimality conditions; Lagrange multipliers; scalar and vector optimization; functions; Hadamard and Dini derivatives},
language = {eng},
number = {1},
pages = {5-36},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {From scalar to vector optimization},
url = {http://eudml.org/doc/33241},
volume = {51},
year = {2006},
}

TY - JOUR
AU - Ginchev, Ivan
AU - Guerraggio, Angelo
AU - Rocca, Matteo
TI - From scalar to vector optimization
JO - Applications of Mathematics
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 1
SP - 5
EP - 36
AB - Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem $\phi (x)\rightarrow \min $, $x\in \mathbb {R}^m$, are given. These conditions work with arbitrary functions $\phi \:\mathbb {R}^m \rightarrow \overline{\mathbb {R}}$, but they show inconsistency with the classical derivatives. This is a base to pose the question whether the formulated optimality conditions remain true when the “inconsistent” Hadamard derivatives are replaced with the “consistent” Dini derivatives. It is shown that the answer is affirmative if $\phi $ is of class ${\mathcal {C}}^{1,1}$ (i.e., differentiable with locally Lipschitz derivative). Further, considering ${\mathcal {C}}^{1,1}$ functions, the discussion is raised to unconstrained vector optimization problems. Using the so called “oriented distance” from a point to a set, we generalize to an arbitrary ordering cone some second-order necessary conditions and sufficient conditions given by Liu, Neittaanmäki, Křížek for a polyhedral cone. Furthermore, we show that the conditions obtained are sufficient not only for efficiency but also for strict efficiency.
LA - eng
KW - scalar and vector optimization; ${\mathcal {C}}^{1,1}$ functions; Hadamard and Dini derivatives; second-order optimality conditions; Lagrange multipliers; scalar and vector optimization; functions; Hadamard and Dini derivatives
UR - http://eudml.org/doc/33241
ER -

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Citations in EuDML Documents

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  1. Marie Dvorská, Karel Pastor, Necessary conditions for vector optimization in infinite dimension
  2. Marie Dvorská, Vector Optimization Results for ℓ -Stable Data
  3. Dušan Bednařík, Karel Pastor, A characterization of C 1 , 1 functions via lower directional derivatives
  4. Dušan Bednařík, Karel Pastor, Second-order sufficient condition for ℓ ˜ -stable functions
  5. Ivan Ginchev, Angelo Guerraggio, Matteo Rocca, Locally Lipschitz vector optimization with inequality and equality constraints
  6. Dušan Bednařík, Karel Pastor, Decrease of property in vector optimization
  7. Karel Pastor, Derivatives of Hadamard type in scalar constrained optimization

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