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Isolated minimizers, proper efficiency and stability for C0,1 constrained vector optimization problems

Author

Listed:
  • Ginchev Ivan

    (Department of MathematicsVarna, Bulgaria)

  • Guerraggio Angelo

    (Department of Economics, University of Insubria, Italy)

  • Rocca Matteo

    (Department of Economics, University of Insubria, Italy)

Abstract

In this paper we consider the vector optimization problem minC f(x), g(x) 2 -K, where f : Rn ! Rm and g : Rn Rp are C0,1 functions and C Rm and K Rp are closed convex cones. We give several notions of solutions (efficiency concepts), among them the notion of a properly efficient point (p-minimizer) of order k and the notion of an isolated minimizer of order k. We show that each isolated minimizer of order k > = 1 is a p-minimizer of order k. The possible reversal of this statement in the case k = 1 is the main subject of the investigation. For this study we apply some first order necessary and sufficient conditions in terms of Dini derivatives. We show that the given optimality conditions are important to solve the posed problem, and a satisfactory solution leads to two approaches toward efficiency concepts, called respectively sense I and sense II concepts. Relations between sense I and sense II isolated minimizers and p-minimizers are obtained. In particular, we are concerned in the stability properties of the p-minimizers and the isolated minimizers. By stability, we mean that they still remain the same type of solutions under small perturbations of the problem data. We show that the p-minimizers are stable under perturbations of the cones, while the isolated minimizers are stable under perturbations both of the cones and the functions in the data set. Further, we show that the sense I concepts are stable under perturbations of the objective data, while the sense II concepts are stable under perturbations both of the objective and the constraints.

Suggested Citation

  • Ginchev Ivan & Guerraggio Angelo & Rocca Matteo, 2004. "Isolated minimizers, proper efficiency and stability for C0,1 constrained vector optimization problems," Economics and Quantitative Methods qf0404, Department of Economics, University of Insubria.
    • Handle: RePEc:ins:quaeco:qf0404
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    References listed on IDEAS

    as
    1. J. B. Hiriart-Urruty, 1979. "Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces," Mathematics of Operations Research, INFORMS, vol. 4(1), pages 79-97, February.
    2. E. Miglierina & E. Molho, 2002. "Scalarization and Stability in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 114(3), pages 657-670, September.
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