By Hiroshi Konno (auth.), Prof. Nicolas Hadjisavvas, Prof. Juan Enrique Martínez-Legaz, Prof. Jean-Paul Penot (eds.)

A number of generalizations of convex features were brought in components equivalent to mathematical programming, economics, administration technological know-how, engineering, stochastics and technologies, for instance. Such capabilities defend a number of houses of convex capabilities and provides upward push to versions that are extra adaptable to real-world occasions than convex types. equally, generalizations of monotone maps were studied lately. A starting to be literature of this interdisciplinary box has seemed, and plenty of foreign conferences are totally committed or contain clusters on generalized convexity and generalized monotonicity. the current ebook incorporates a choice of refereed papers offered on the sixth foreign Symposium on Generalized Convexity/Monotonicity, and goals to study the most recent advancements within the box.

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InSection4, we showtheresultsof our computationalexperiments . ", bERn, and e is a vector of all ones whose dimensionis given by context. , r; = { y E Rn I- e ::;y ::;e }. 50 Y. Yajima et al. It is well knownthattheproblemcan behomogenizedby introducinga new variableYn+! in the following way: (Q P)' T ISubjectto Ma~imize x Qx xE F n+l Q= where x = (YT Yn+! , [Pb] bT 0 . Let us introducean equivalentformulationwith matrix variables. Let Sn+! : FCut n+! = Conv{X E Sn+! IX = xx T , X E Fn+d. , and (A, B) denotestheinnerproductof symmetricmatrices A and B.

Convex Functionsand their Applications If, on the otherhand, x x + k > h, then ,1k g(x) + k :S h, then trivially ,1k g(x) x+k-h = l: (-1)i(7)g(x i=O = 0; if x 45 < hand + k - i) (6. 10) = e a hLl x+k-h l: (-l)i(7) (e a(x+k-h-i)4 -1). i=O The last termin the above sum is zerobut we keep it, ifx otherwisewe drop it. It is well-knownthat for any odd,j +k- h is odd, < k . This implies thatif x + k - h is odd then ,1k g ( X ) and if x +k- > e a hLl X~\_1)iG)e(*+k-h-i)4, h is even,then In both cases wecombineeach term,correspondingto an eveni, with the next term.

Inequalitieson ExpectationsBased on the Knowledge of MultivariateMoments. In : StochasticInequalities(M . Z . ), Instituteof MathematicalStatistics,LectureNotes- Monograph Series 22, 309-331. 24. Prekopa, A. (1995). StochasticProgramming. Kluwer Academic Publishers, Dodrecht, Boston. 25. Prekopa,A . (1998). Bounds on Probabilitiesand ExpectationsUsing Multivariate Moments of Discrete Distributions. Studia ScientiarumMathematiearum Hungarica34, 349-378. 26. Prekopa,A . (1999). The Use of DiscreteMoment Bounds in ProbabilisticConstrainedStochasticProgrammingModels.

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