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Second-order convex functions were introduced by Mond (Opsearch 11(2-3):90-99, ) in order to deal with second-order duality. Then that notion was generalized again and again, using more and more parameters introduced using several quantifiers. In the present paper, we show that most of these notions have quite simple intrinsic characterizations. This paper can be viewed as a continuation of our paper (Zălinescu in An Ştiinţ Univ Al I Cuza Iaşi Secţ I a Mat 35(3):213-220, ) in which we characterized generalized bonvex functions. [ABSTRACT FROM AUTHOR]

The support function is strongly related to the cost function in economics. Because the differentiability of the cost function is an important property, being related to the celebrated Shephard's lemma, our propose is to study the differentiability of the support function of a nonempty closed and convex subset of a finite dimensional normed space. So we provide characterizations of the differentiability of the support function on three subsets of its domain. [ABSTRACT FROM AUTHOR]

The main goal of this note is to give a counterexample to the Triality Theorem in Gao and Ruan (Math Methods Oper Res 67:479-491, ). This is done first by considering a more general optimization problem with the aim to encompass several examples from Gao and Ruan (Math Methods Oper Res 67:479-491, ) and other papers by Gao and his collaborators (see f.i. Gao Duality principles in nonconvex systems. Theory, methods and applications. Kluwer, Dordrecht, ; Gao and Sherali Advances in applied mathematics and global optimization. Springer, Berlin, ). We perform a thorough analysis of the general optimization problem in terms of local extrema while presenting several counterexamples. [ABSTRACT FROM AUTHOR]

In their paper 'Duality of linear conic problems' Shapiro and Nemirovski considered two possible properties (A) and (B) for dual linear conic problems (P) and (D). The property (A) is 'If either (P) or (D) is feasible, then there is no duality gap between (P) and (D)', while property (B) is 'If both (P) and (D) are feasible, then there is no duality gap between (P) and (D) and the optimal values val(P) and val(D) are finite'. They showed that (A) holds if and only if the cone K is polyhedral, and gave some partial results related to (B). Later Shapiro conjectured that (B) holds if and only if all the nontrivial faces of the cone K are polyhedral. In this note we mainly prove that both the 'if' and 'only if' parts of this conjecture are not true by providing examples of closed convex cone in $${\mathbb{R}^{4}}$$ for which the corresponding implications are not valid. Moreover, we give alternative proofs for the results related to (B) established by Shapiro and Nemirovski. [ABSTRACT FROM AUTHOR]

In this short note we give two counterexamples to the Triality Theorem concerning the optimization problem presented in Ruan et al., Comput. Optim. Appl. (). In order to understand the problem first we introduce the framework of this paper and quote the theorem we have in view. Then, while trying to verify this theorem we encounter several difficulties and note their nature. We end by revealing our counterexamples. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show that the results on triality and tri-duality in Gao (J Glob Optim 17:127-160, 2000; J Glob Optim 29:377-399, 2004; J Glob Optim 35:131-143, 2006; Encyclopedia of optimization, 2nd edn. Springer, New York, pp 822-828, 2009) and Gao et al. (J Glob Optim 45:473-497, 2009) are false. To prove this we provide simple counterexamples. [ABSTRACT FROM AUTHOR]

We use representations of maximal monotone operators for studying recession (or asymptotic) operators associated to maximal monotone operators. Such a concept is useful for dealing with unboundedness. [ABSTRACT FROM AUTHOR]

Several generalizations of the Hahn–Banach extension theorem to K-convex multifunctions were stated recently in the literature. In this note we provide an easy direct proof for the multifunction version of the Hahn–Banach–Kantorovich theorem and show that in a quite general situation it can be obtained from existing results. Then we derive the Yang extension theorem using a similar proof as well as a stronger version of it using a classical separation theorem. Moreover, we give counterexamples to several extension theorems stated in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study conditioning problems for convex and nonconvex functions defined on normed linear spaces. We extend the notion of upper Lipschitzness for multivalued functions introduced by Robinson, and show that this concept ensures local conditioning in the nonconvex case via an abstract subdifferential; in the convex case, we obtain complete characterizations of global conditioning in terms of an extension of the upper-Lipschitz property. [ABSTRACT FROM AUTHOR]

In this paper there is stated a result on sets in ordered linear spaces which can be used to show that some properties of the sets are inherited by their convex hulls under suitable conditions. As applications one gives a characterization of weakly efficient points and a duality result for nonconvex vector optimization problems. [ABSTRACT FROM AUTHOR]

In this paper we give a criterion for d-stability of convex programs and a perturbation result which subsume and generalize some recent results in semi-infinite programming and limiting Lagrangians. [ABSTRACT FROM AUTHOR]

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