A general approach is developed to solvability theorems involving a broad class of functions, here called H-convex functions and inf-H-convex functions. The concept of Minkowski duality is exploited to provide dual characterizations for certain infinite inequality systems. The results not only cover the recently developed solvability results involving DSL functions, concave functions and difference of sublinear and convex functions but also include a new dual characterization for systems with completely difference convex functions. Detailed examples are provided to illustrate the broad nature of the results. Applications to global optimization are also given.
|
342
|
Lattice Theory
– Birkhoff
- 1940
|
|
195
|
Convex Analysis and Minimization Algorithms I
– Hiriart-Urruty, Lemarecal
- 1993
|
|
100
|
Global optimization
– Horst, Tuy
- 1990
|
|
36
|
Rockafellar: Convex Analysis
– T
- 1970
|
|
8
|
Minkowski duality and its applications
– Kutateladze, Rubinov
- 1972
|
|
6
|
On the Class of H-Convex Functions
– ABASOV, RUBINOV
- 1994
|
|
6
|
Mathematical Theory of Economic Dynamic and Equilibria
– Makarov, M
- 1977
|
|
4
|
Subdifferentials of Some Classes of Nonsmooth Functions
– Abasov, M
- 1996
|
|
3
|
On conjugate convex functions, Canadian
– Fenchel
- 1949
|
|
3
|
Generalizations of Slater’s constraint qualification for infinite convex programs
– Jeyakumar, Wolkowicz
- 1992
|
|
3
|
On quasiconvex duality
– Penot, Volle
- 1990
|
|
3
|
The space of star-shaped sets and its application in nonsmooth optimization
– Rubinov, Yagubov
- 1986
|
|
2
|
Testing necessary and sufficient conditions for global optimality in the problem of maximizing a convex quadratic function over a convex polyhedron
– Hirriart-Urruty, Lemarechal
- 1990
|
|
2
|
Nonlinear extensions of Farkas' Lemma with applications to global optimization and least squares
– Jeyakumar, Glover
- 1995
|
|
2
|
Rockafellar: Conjugate Duality and Optimization
– T
- 1974
|
|
2
|
Kutateladze: Subdifferentials. Theory and applications
– Kusraev, S
- 1992
|
|
2
|
Quasiconvex duality theory by generalized conjugation methods
– Martinez-Legaz
- 1988
|
|
2
|
Singer: Dualities between complete lattices, Optimization 21
– Martinez-Legaz, I
- 1990
|
|
1
|
Abasov: Normal sets and monotone functions and their applications
– M
- 1993
|
|
1
|
An extension of duality-stability results to nonconvex optimization problems
– Balder
- 1977
|
|
1
|
About differentiability of order one of quasiconvex functions on IR n
– Crouzeix
- 1982
|
|
1
|
Glover: A generalized Farkas lemma with applications to quasidifferentiable programming, Zeitschrift fur
– M
- 1982
|
|
1
|
On quasidifferentiable functions and nondifferentiable programming
– Glover
- 1990
|
|
1
|
Tuan: Complete characterizations of global optimality of DC minimization problems, to appear
– Glover, Jeyakumar, et al.
- 1995
|
|
1
|
Oettli: A Farkas lemma for difference sublinear systems and quasidifferentiable programming
– Glover, Jeyakumar, et al.
- 1994
|
|
1
|
Oettli: Solvability theorems for classes of difference convex functions
– Glover, Jeyakumar, et al.
- 1994
|
|
1
|
Craven: Solvability theorems involving inf-convex functions
– Glover, Rubinov, et al.
- 1995
|
|
1
|
Results of Farkas type
– Gwinner
- 1987
|
|
1
|
On systems of convex inequalities
– Ha
- 1979
|
|
1
|
From convex to nonconvex minimization: Necessary and sufficient conditions for global optimality
– Hiriart-Urruty
- 1989
|
|
1
|
Ishizuka: Farkas' theorem of nonconvex type and its application to a min-max problem
– unknown authors
- 1988
|
|
1
|
Optimality conditions for quasidifferentiable programs with application to twolevel optimization
– Ishizuka
- 1988
|
|
1
|
Glover: A new Farkas lemma and global convex maximization
– Jeyakumar, M
- 1993
|
|
1
|
Martinez-Legaz: Quasiconvex duality theory by generalized conjugation methods
– E
- 1988
|
|
1
|
Martinez-Legaz: General conjugation and related topics, in Generalized Convexity and Fractional Programming with Economic Applications
– E
- 1990
|
|
1
|
Moreau: Inf-convolution des fonctions numerique sur un espace vectorial
– J
- 1963
|
|
1
|
Convex analysis without linearity
– Rolewicz
- 1994
|
|
1
|
Pallaschke: Foundations of Optimization
– Rolewicz, D
- 1994
|
|
1
|
Rubinov: Superlinear Multivalued Mappings and their Applications to
– M
- 1980
|
|
1
|
Rubinov and B. Shimshek: Conjugate quasiconvex nonnegative functions
– M
- 1995
|
|
1
|
Shveidel: Support sets of linear functionals and separation of star-shaped sets, unpublished manuscript
– P
- 1988
|
|
1
|
Tikhomirov: Collection of optimization problems
– Alexeev, Galeev, et al.
- 1984
|
|
1
|
Borwein: Adjoint process duality
– M
- 1983
|
|
1
|
Crouzeix: A duality framework in quasiconvex programming, in S.Schaible, W.T.Ziemba (eds): Generalized concavity in optimization and
– P
- 1981
|
|
1
|
Kusraev: Vector duality and its application
– G
- 1985
|
|
1
|
Generalized conjugation and related topics, Generalized convexity and fractional programming with economic applications
– Martinez-Legaz
- 1990
|
|
1
|
Rockafellar: Convex algebra and duality in dynamic models of production, in
– T
- 1974
|
|
1
|
Shimshek: Conjugate quasiconvex nonegative functions, Working Paper 9/94
– Rubinovi, B
- 1994
|
|
1
|
Yagubov: Of the operator of polarity
– Rubinov, A
- 1985
|
|
1
|
Conjugation Operators, in G.Hammer and D.Pallaschke (eds): Selected topics in operation research and mathematical economics
– Singer
- 1984
|
