## Farkas-type results for max-functions and applications (2004)

Citations: | 3 - 2 self |

### BibTeX

@TECHREPORT{Bot04farkas-typeresults,

author = {R. I. Bot and G. Wanka},

title = {Farkas-type results for max-functions and applications},

institution = {},

year = {2004}

}

### OpenURL

### Abstract

Abstract. We present some Farkas-type results for inequality systems involving finitely many convex constraints as well as convex max-functions. Therefore we use the dual of a minmax optimization problem. The main theorem and its consequences allows us to establish, as particular instances, some set containment characterizations and to rediscover two famous theorems of the alternative. Keywords: duality, Farkas-type results, minmax programming, set containment, theorems of the alternative AMS subject classification: 49N15, 90C25, 90C46 1.

### Citations

3621 | Convex Analysis - Rockafellar - 1970 |

463 |
Convex analysis and minimization algorithms I
- Hiriart-Urruty, Lemaréchal
- 1993
(Show Context)
Citation Context ... i = 1, ..., p. We have f ∗ i (pi ) = h ∗ i (pi ) + ai, for i = 1, ..., p. On the other hand, the conjugate of hi, i = 1, ..., p can be calculated by using the Moore-Penrose pseudo-inverse U − i (see =-=[5]-=-, [6]) h ∗ i (p i ) = Relation (ii) becomes � 1 2 (pi − Ai) T U − i (pi − Ai), if p i ∈ Ai + ImUi, +∞, otherwise. (ii) ∀i = 1, ..., p there exist p i ∈ R n and q i ∈ R m , q i ≧ 0 such that p i ∈ Ai +... |

211 |
Nonlinear Programming
- Mangasarian
- 1969
(Show Context)
Citation Context ... Tucker, Stiemke, Gordan and Slater, can be obtained from the results we mentioned above. For a detailed presentation of theorems of the alternative we invite the reader to consult Mangasarian’s book =-=[8]-=-. Throughout this section the set X will be the whole space R n and all the functions involved will be affine. THEOREM 5. (Gale’s theorem for linear inequalities) Let A ∈ R k×n and c ∈ R k be given. T... |

18 |
Strong duality for generalized convex optimization problems
- Bot, Kassay, et al.
(Show Context)
Citation Context ...convex constraints. This approach bases on the theory of conjugate duality for convex optimization problems, namely by using the so-called Fenchel and Fenchel-Lagrange duality concepts (see also [1], =-=[2]-=-, [10], [12]). Moreover the authors show how these new Farkas-type results generalize some of the results obtained by Jeyakumar in [7]. The aim of the present paper is to extend the results obtained i... |

14 |
Characterizing set containments involving infinite convex constraints and reverse-convex constraints
- Jeyakumar
(Show Context)
Citation Context ...ful in the determination of knowledge-based classifiers, the most famous example being here the so-called support vector machines classifiers. Motivated by the paper [9], Jeyakumar has established in =-=[7]-=- dual characterizations for the containment of a closed convex set, defined by infinitely many convex constraints, in an arbitrary polyhedral set, in a reverse convex set and in another convex set, re... |

14 |
On the relations between different dual problems in convex mathematical programming
- Wanka, Bot¸
- 2002
(Show Context)
Citation Context ...traints. This approach bases on the theory of conjugate duality for convex optimization problems, namely by using the so-called Fenchel and Fenchel-Lagrange duality concepts (see also [1], [2], [10], =-=[12]-=-). Moreover the authors show how these new Farkas-type results generalize some of the results obtained by Jeyakumar in [7]. The aim of the present paper is to extend the results obtained in [3] by con... |

5 | Fenchel-Lagrange versus geometric duality in convex optimization
- Bot¸, Grad, et al.
- 2006
(Show Context)
Citation Context ...r of convex constraints. This approach bases on the theory of conjugate duality for convex optimization problems, namely by using the so-called Fenchel and Fenchel-Lagrange duality concepts (see also =-=[1]-=-, [2], [10], [12]). Moreover the authors show how these new Farkas-type results generalize some of the results obtained by Jeyakumar in [7]. The aim of the present paper is to extend the results obtai... |

5 | Set containment characterization
- Mangasarian
(Show Context)
Citation Context ... the alternative. Keywords: duality, Farkas-type results, minmax programming, set containment, theorems of the alternative AMS subject classification: 49N15, 90C25, 90C46 1. Introduction In the paper =-=[9]-=-, Mangasarian introduced a new approach in order to give dual characterizations for different set containment problems. He succeeded to characterize the containment of a polyhedral set in another poly... |

4 |
Einführung in die Nichtlineare
- Elster, Reinhardt, et al.
- 1977
(Show Context)
Citation Context ...1 ∈ ri dom(fi) i=1 � � � X × R such that ⎧ ⎪⎨ gj(x ⎪⎩ ′ ) ≤ 0, j ∈ L, gj(x ′ ) < 0, j ∈ N, fi(x ′ � ) − max i=1,...,k {fi(x ′ � )} + 1 < 0, i = 1, ..., k. Under the present hypotheses, Theorem 5.7 in =-=[4]-=- states the existence of ¯q 1 ∈ Rk , ¯q 1 ≧ 0, k� ¯q i=1 1 i = 1 and ¯q2 ∈ Rm , ¯q 2 ≧ 0 such that strong duality for the Lagrange dual holds, i. e. v(P ′ ) = max q1≧0,q2≧0, k� q i=1 1 i =1 inf x∈Rn �... |

2 |
R.: Duality for Minmax Programs
- Scott, Jefferson
- 1984
(Show Context)
Citation Context ... v(P ′ ) represent the optimal objective values of the problems (P ) and (P ′ ), respectively. We formulate (P ′ ), which is also a convex optimization problem, in the following way (see for instance =-=[11]-=- and [1]) (P ′ ) inf x,a a, s.t. x ∈ X, g(x) ≦ 0, a ∈ R, fi(x) − a ≤ 0, i = 1, ..., k. Proposition 1 states the equality between the optimal objective values of the problems (P ) and (P ′ ). PROPOSITI... |

1 |
Farkas type results with conjugate functions
- Bot¸, Wanka
(Show Context)
Citation Context ...lyhedral set, in a reverse convex set and in another convex set, respectively. The characterizations are given in terms of epigraphs of conjugate functions. Recently, Bot¸ and Wanka have presented in =-=[3]-=- some new Farkas-type results for inequality systems involving a finite as well as an infinite number of convex constraints. This approach bases on the theory of conjugate duality for convex optimizat... |