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By Dr. Matthias Ehrgott (auth.)

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36)). 24 below ~ 0, j = 1, ... , Q with implies that for the ith such system there exist 2::~ 1 A~ = 1 such that for all x E X holds: A; #i #i #i #i Q ==? #i Q L A)/i(x) + M L A;IJ(x) ~ L A)fi(x*) + M L A)IJ(x*) j#i j=l ¢=:::? fi(x) +ML j=l A)IJ(x) ~ fi(x*) #i We have such an inequality for each i ito obtain +M L j#i A)fi(x*) #i = 1, ... 3 Proper Pareto Optimality and Proper Efficiency Q Q Lf;(x) i=l Q 43 Q + MLL,\;/j(x);::: Lf;(x*) + MLL,\;/j(x*) i=l i=l # i j=l # i for all x EX. We can now norm the values (1 + M · L # i ,\;),so that they sum up to one, and have-\;, i = 1, ...

If x is strictly Pareto optimal for (X, fp, ]RIP I) /id/ (JRIPI, ~) it is also strictly Pareto optimal for (X, f, IRQ)/id/(IRQ, ~). Stronger results in this direction can be obtained for convex functions. Rn -t IR are convex functions. This implies that all level sets are convex. 16 is then about intersections of convex sets. A fundamental theorem on such intersections is known in convex analysis: Helly's Theorem. 36 2. 18 (Helly 1923, [Hel23]). C 1 , •.. n be convex sets. + 1 holds # 0. j=l Equivalently stated, we can say if and only if there is a subset {i1, ...

4 .. 6. ~ -semicompactness. ~ -compactness. 4. Y C IRQ is called (y n Y is compact. 9. If Y is Proof. Let {(y"' y"'' E Y take IR2 -compact then Y -IR2 Y : y"' E Y, { ( y"' - IR2 -compact, if for ally E Y is 27 the section IR2 -semicompact. a E A} be an open cover of Y. For arbitrary IR2) c : y"' E Y, a E A, a -:f. a'} . 9 gives an open cover of (y"'' n Y, a compact set, since Y is compact. 9 contains a finite subcover of (y"'' n Y. This finite sub cover together with (y"'' c yields a -semicompactness.

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