# Linear programming over an infinite horizon by J.J.M. Evers By J.J.M. Evers

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Additional info for Linear programming over an infinite horizon

Sample text

For an LP-system (P- or D-directed; f(t) .... }; t .... co; p(t) .... ~, t .... 9 may be resumed as follows: If the LP-system is pO-feasible then the system (3. 10. 1) has a solution. 2) w > has a solution. 3) Z ~ has a sGlution. 4) w ~ 0 , 48 has a solution. 6) w > 0 have a solution. 6). 11 Theorem. 2) , has a solution for every a. > y. Proof. First consider the case that the LP-system is P-directed. 1) for some y > 0, hER:, then z If also 49 satisfies 'V -y Az. < f. This implies that for every a > y the following inequality holds: -(a-y)Az < (~-I)}.

L. This, however, I = 0 and for which a. L. j = 0 is impossible in connection with the definition of P-directedness. Sufficient: assume that the LP-system is not P-directed, then there to or f~ (t) < 0 for some t ::_ 1. This implies that there exists a (x,y) E Rn such that for some t > is a b .. < 0 such that a. LJ and row-index i: L.. 1.. 1.. (t) 1. -a. (t) 1.. 1. 3) . From this it appears that it is P ossible to choose a z E Rm. 2). 3 Theorem. The LP-system is then and only then D-directed if for every p(t),t~l: m+m+n .

5. 2) Proof. First consider the case that the LP-system is P-directed, so that A:= I . 1) for some y > 0, K ~ I, also satisfies -Ax(t-I) < ytf(t)+z(t), t > K. Since the sequence {8(t)};_1 is monotonous non-increasing, this implies that -(8(t-I)-8(t»Ax(t-I)~(8(t-I)-8(t»(y t f(t)+z(t»,t ~ K. 1) imply: G(t)(Bx(t)-Ax(t-I)+y(t»= G(t)(ytf(t)+z(t», t > K. 2). Now we consider the case that the LP-system is not P-directed. 8) K. 9) it appears that there exists a ~ K. 2). 6 Proposition. 6. I) K. ) f(t)+-Ax(K-I) -- (B--A)x+y a.