By Dorea C. E.
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Additional info for (A, B)-Invariant Polyhedral Sets of Linear Discrete-Time Systems
5. Exercises 17 c. Use part b to show that P is infeasible. d. Show that y(t) = (2t, 3t, t)T is D-feasible for all t ≥ 0. e. Use part d to prove that D is unbounded. 15 Repeat part a N times, each with a diﬀerent LOP of your own design. a. Let P be a LOP in standard max form and let D be its dual LOP. Write D in standard max form as D and let Q be its dual. Finally, write Q in standard max form as Q. Compare P and Q. b. Prove a statement about the relationship between P and Q in general. 16 Consider the problem: Max.
2. Recall that this row is merely the row of coeﬃcients that represents the equality 22x3 + 33x4 + 11x5 = −22 . Because every variable is nonnegative, the left side of the equation is nonnegative, which contradicts the equality. 1 is infeasible; that is, there are no values for its variables that satisfy both its problem and nonnegativity constraints. 2 might then distrust this conclusion. Therefore, it would be beneﬁcial to produce a certiﬁcate of this result. 38), we should be able to produce the exact linear combination necessary.
2 See Appendix A. pivot operation basic coeﬃcient 36 Chapter 2. 9 Max. t. 10. 9. It says that z = (720 − 4x3 − 2x4 )/30, which means that if x3 or x4 takes on any value other than zero, then z < 720/30 = 24. Therefore z cannot be increased! Hence z ∗ = 24 and x∗ = (15, 9 | 0, 0, 6)T . These are the workings of the Simplex Algorithm (Tableau Environment) in the simplest case (Phase II): given a feasible tableau, ﬁnd a parameter whose increase from zero will increase z. That is, ﬁnd a negative entry in the objective row (not including the rightmost, which can be negative at times), and pivot in the same column as that entry.
(A, B)-Invariant Polyhedral Sets of Linear Discrete-Time Systems by Dorea C. E.