Final exam ”Optimization” QEM. 2014. 10H30-12H30 02 december QEM. Delay : 2H, no documents, no computers, no electronic devices, no cellphone ! Exercise 1 On the following map, you have cities (named by letters), and roads between 2 cities. Each number on a road gives the distance between the two extreme cities of the road. Two cities being given, you have several paths between these cities : for example, to go from A to B, you can take the direct path AB (this path measures 2 kilometers) or take AC then CB (this path measures kilometers 4+3=7 kilometers). For every city named Z, V (Z) represents the distance of the shortest path from Z to R. For example V (Q) = 2, V (M ) = 3,... Represent the diagram. Explain carefully, using some dynamic programmation principle you will explain, how you can compute V (N ), V (O), and V (J). More quickly, compute the other numbers V (M ) for every M (explain the order of your computation), and represent these numbers on your diagram. Give the shortest path from A to R (draw it on your diagram) and its length.
Exercise 2 Let 0 < β < 1. Consider the following optimization problem (Q) :
+∞ X
sup
k0 =1,∀n∈N,0≤cn ≤kn ,kn+1 =kn −cn n=0
βn
cn . 1 + cn
This is the maximization of an intertemporal utility under some constraints. The variable k0 (equal to 1) can be seen as an initial capital, cn ∈ IR is consumption at time n, kn ∈ IR capital at time n. You should choose consumption cn at each time so that it is between 0 and kn , and the capital at time n + 1 is equal to kn − cn . 1) Prove that the real sequence of capitals (kn )n∈IN is decreasing. Prove the terms of the real sequence (cn )n∈IN are all beween 0 and 1. 2) Let X be the set of real sequences whose all terms are between 0 and 1. We consider on X a distance (we admit it is a distance) d((xn )n∈IN , (yn )n∈IN ) =
+∞ X
β n | xn − yn |
n=0
2)a) Prove that this infinite sum is well defined. Recall the definition of a distance. If you consider a sequence (x(k))k∈IN of X (thus, for each k, x(k) = (xn (k))n∈IN is itself a real sequence), recall without proof what it means that x(k) converges to some real sequence x when k → ∞. 3) a) Recall the the definition of ”X is a compact for the distance d”. 1
b) Now prove that X is a compact for the distance d. c) Recall a simple criterium on K that guarantees that a subset K of X is compact. d) Now prove that the set of feasible consumptions plans, i.e. C = {(cn )n∈IN : k0 = 1, ∀n ∈ N, 0 ≤ cn ≤ kn , kn+1 = kn − cn } is a compact for the distance d. 4) The function Φ we want to maximize can be written as follows : for every sequence (cn )n∈IN ∈ C, +∞ X cn . Φ((cn )n∈IN ) = βn 1 + cn n=0 a) Recall the the definition of ”Φ is continuous from C to IR”, where we consider on C the distance d and on IR the standard distance defined by | . | . b) Prove that Φ is continuous. 5) Prove that the optimization problem (Q) has a solution.
Exercise 3 We have a thread of length 1 to tie a box from top to bottom along the two perpendicular directions. The aim of the problem is to find the maximum volume that such a box can contain. 1) Prove that it can be mathematically written : (P ) max f (x, y, z) = xyz under the constraint x ≥ 0, y ≥ 0, z ≥ 0, 2x + 2y + 4z ≤ 1. 2) 3) 4) 5)
Prove there exists a solution, now denoted (x, y, z), of (P). Write the Kuhn and tucker equations (call µ1 , µ2 , µ3 and µ4 the multiplicators). Prove that xyz 6= 0, and that µ1 = µ2 = µ3 = 0. Prove that the following system called (S) is true : x.(yz + 2µ4 ) = 0 y.(xz + 2µ4 ) = 0 z.(xy + 4µ4 ) = 0 (2x + 2y + 4z − 1)µ4 = 0
6) Express µ4 using only x, y and z. 7) Find x, y, z. Bonus An insect (ant) wants to go from point A to point B on a cube. The insect cannot enter into the interior of the cube (say the cube is metallic). Knowing that B and A are in the middle of two edges (see picture) explain, by proving it, what is the shortest path from A to B.
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