Approximation and Optimization: Proceedings of the by Juan A. Gomez-Fernandez, Francisco Guerra-Vazquez, Miguel A.

By Juan A. Gomez-Fernandez, Francisco Guerra-Vazquez, Miguel A. Jimenez-Pozo, Guillermo Lopez-Lagomasino

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Extra resources for Approximation and Optimization: Proceedings of the International Seminar, held in Havana, Cuba, January 12-16, 1987

Sample text

IYi 1) 0 X yi I k matrix defined as follows if i = j , otherwise. ;y E Uy - {y*}} = infa>o{a I *Iy E C} = maxi{l(Iy)il} = maxi{IYi - yil-1IYil}, and 1I·lIy becomes the weighted Tchebycheff norm. 6, to construct the norm II . 3. But two instances of such cones are of special interest. 2. l, :f Yi, i = 1, ... , k, i = 1, ... , k. 3, for a given Kg (obviously Kp and Kp,y are polyhedral) there exist positive p and p' such that Kp C intKg U {O}, Kp',y C intKg U {O}. Observe also if and y* . represents the cones K p and K p,y (for if = 0) .

On the other hand, max min YEKp n B y'EK n B Ily - y'IIE = d' > 0, and d' decreases strictly monotonically with p tending to zero . Hence, there exists p> 0, such that Kp ~ intO U {O}. 8 Let J( -properly to ensure J( be polyhedral. An efficient element f} E Z is efficient if and only if there exists p > 0 sufficiently small J(p ~ K(f}). Proof. 3 for some j semi-strictly at f}. 2, ({jj} ~ + Kj) # 0), the jo separate Z and jj n Z = {jj} , hence CHAPTER 3. CONES, EFFICIENCY, ... 36 J(j 2:: jo is the required ~ k(fj) .

In general, efficiency in va problems can be interpreted in terms of separation with respect to a certain subset of nk containing {O} , in a similar manner as has been defined for cones. 4 can be easily extended to set separability. Thus, cone separation is only a special case of conceivable more general separability techniques. Such a general technique has been developed by Wierzbicki (Wierzbicki (1976,1980,1986)) and has been presented in terms of so called scalarizing functionals. There is no difficulty to interpret scalarization functionals in terms of set separability with a larger class of sets than just cones.

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