By J.J.M. Evers
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Tat L (-) f(T+I-t) t=O p is Consequently a vector gERm exists such that for everyP-feasible solution x and every 1 - (B--A)x a ~ T~I T P g x(T) has a solution. 5) by (~,~)', one can derive that ~,~ T< P u'g ~,; > v'x(T) = ) has a solution for every P-feasible x and every T~I. Since v > 0, this implies that every P-feasible solution is p-dominated. 18 Remark. 2) > p. 11. 2) has a solution for (l = p' The next two theorems give a summary of the latter three propositions for the primal and dual system.
2 Proposition. 2. 1 ) Proof. If «x,y),(u,v» of periods is a feasible solution, then for every pair L~K;I:
= 7T K,L L L (B'u(t)-A'u(t+I)-v(t»'x(t) t=K L L (u (t) t=K I (Bx (t) - Ax (t - 1 ) ) -v (t) , x (t» + + u(K)'Ax(K-I)-u(L+I)'Ax(L)
Proof. (a) Let (x,y) E In:m be a P-feasible solution for some x(O). 4) 1 t Y > - E (-) y(t) = T t= 1 P has a solution for every T > I. 7) g(T) E C, T>I. '" The supposition f(t) .... f, '" g (T) .... f, t .... 8) 00 From the definition of C it follows ( 8:174) that this set is '" closed. 8) imply f E C. 8. I) has a solution. 9) I, m for some y E int(R+). 9) it may be derived that (g(T)-~y» E C, T>I. 6 it may be derived that the system '\, '" (B-7TA)x+y '"x 1 T_K(7TAx(K-I)+ ~ T ~ t=K f(t» 0 has a solution for every T>K.