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Second-Order Characterizations of Convex and Pseudoconvex Functions

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Veröffentlicht/Copyright: 9. Juni 2010
Journal of Applied Analysis
Aus der Zeitschrift Journal of Applied Analysis Band 9 Heft 2

Abstract

The present paper gives characterizations of radially u.s.c. convex and pseudoconvex functions f : X → ℝ defined on a convex subset X of a real linear space E in terms of first and second-order upper Dini-directional derivatives. Observing that the property f radially u.s.c. does not require a topological structure of E, we draw the possibility to state our results for arbitrary real linear spaces. For convex functions we extend a theorem of Huang, Ng [Math. Oper. Res. 22: 747–753, 1997]. For pseudoconvex functions we generalize results of Diewert, Avriel, Zang [J. Econom. Theory 25: 397–420, 1981] and Crouzeix [Generalized Convexity, Generalized Monotonicity: Recent Results: 237–256, Kluwer Academic Publisher, 1998]. While some known results on pseudoconvex functions are stated in global concepts (e.g. Komlosi [Math. Pro Programming 26: 232–237, 1983]), we succeeded in realizing the task to confine to local concepts only.

Key words and phrases.: Nonsmooth analysis; second-order Dini-directional derivatives; second-order characterizations of convex and pseudoconvex functions
Received: 2002-06-16
Revised: 2003-04-28
Published Online: 2010-06-09
Published in Print: 2003-December

© Heldermann Verlag

Artikel in diesem Heft

  1. Critical Cardinalities and Additivity Properties of Combinatorial Notions of Smallness
  2. Existence of Global Weak Solutions for Coupled Thermoelasticity with Barber's Heat Exchange Condition
  3. A Priori Solution Estimates for Nonlinear Vector Integral Equations
  4. On a Boundary Value Problem for a Nonlocal Elliptic Equation
  5. On Noncoercive Elliptic Problems with Discontinuities
  6. Some Nonlinear Problems in Hyperconvex Metric Spaces
  7. Linearly Invariant Families of Holomorphic Mappings in . Transition to Other Dimensions
  8. On Continuous Selection Problems for Multivalued Mappings with the Local Intersection Property in Hyperconvex Metric Spaces
  9. Second-Order Characterizations of Convex and Pseudoconvex Functions
  10. On Marczewski-Burstin Representations of Algebras and Ideals
  11. New and Generalized Convergence Conditions for the Newton-Kantorovich Method
https://doi.org/10.1515/JAA.2003.261
Schlagwörter für diesen Artikel
Nonsmooth analysis; second-order Dini-directional derivatives; second-order characterizations of convex and pseudoconvex functions

Artikel in diesem Heft

  1. Critical Cardinalities and Additivity Properties of Combinatorial Notions of Smallness
  2. Existence of Global Weak Solutions for Coupled Thermoelasticity with Barber's Heat Exchange Condition
  3. A Priori Solution Estimates for Nonlinear Vector Integral Equations
  4. On a Boundary Value Problem for a Nonlocal Elliptic Equation
  5. On Noncoercive Elliptic Problems with Discontinuities
  6. Some Nonlinear Problems in Hyperconvex Metric Spaces
  7. Linearly Invariant Families of Holomorphic Mappings in . Transition to Other Dimensions
  8. On Continuous Selection Problems for Multivalued Mappings with the Local Intersection Property in Hyperconvex Metric Spaces
  9. Second-Order Characterizations of Convex and Pseudoconvex Functions
  10. On Marczewski-Burstin Representations of Algebras and Ideals
  11. New and Generalized Convergence Conditions for the Newton-Kantorovich Method
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