Mid term exam ”Optimization”.

20 october 2014 Teacher : Philippe Bich. Delay : 2H, no documents. Every exit is definitive.

Exercise 1 x on E = [0, +∞[. Consider the function f (x) = x+1 1) Prove that this function is well defined on E and bounded on E. 2) We consider the problem (P ) sup f (x). x∈E

a) What is the objective function ? What is the set of feasible points ? b) Compute the value of (P). c) Does there exists a maximizing sequence ? If yes, give it. d) Is there a solution of (P) ? if yes, compute it. If no, prove it. Exercise 2 P Let A = {x = (x1 , ..., xn ) ∈ IRn : ni=1 xi ≤ 0} and B = {(a, a, ..., a) ∈ IRn : a ∈ IR}. a) Prove that A and B are convex. Represent graphically A and B when n = 2. b) Prove that A ∪ B is not convex. Exercise 3 a) Recall the definition of f : IR → IR upper semicontinuous and g : IR → IR lower semicontinuous. b) Consider the function f : IR → IR defined by f (x) = 1 if x < 0 and f (x) = −1 if x ≥ 0. Is it upper semicontinuous ? lower semicontinuous ? (prove any affirmation). c) Consider the function f : IR → IR defined by f (x) = 1 if x is an integer (possibly negative) and f (x) = 0 otherwise. Is it upper semicontinuous ? lower semicontinuous ? (prove any affirmation). Exercise 4 a) Recall the definition of f : IR → IR quasi-concave and quasi-convexe. b) Consider the function f : IR → IR defined by f (x) = 0 if x < 0, f (x) = 2 − x if x ∈ [0, 1] and f (x) = 1 if x > 1. Is it quasiconcave ? quasiconvexe ? concave ? convexe ?

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Exercise 5 a) Is S = {(x, y, z) ∈ IR3 : x2 − y 2 = 0} an hypersurface of IR3 ? If yes, why ? in this case, what is the normal to S at (0, 0, 0) ? b) Is T = {(x, y, z) ∈ IR3 : x2 + 2y 2 + z 2 = 1} an hypersurface of IR3 ? If yes, why ? in this case, what is the normal to S at (1, 0, 0) ? c) Find the maximum of the function x + y + z on T .

Exercise 6 Consider a continuous function f : S → IR where S = {x ∈ IRn : kxk < 1} the open unit ball. We assume that for every x ∈ S, f (x) ≥

1 . 1 − kxk

We consider the problem (Q) inf f (x). x∈S

a) Prove that Q has a (finite) value. b) For every integer k, let Sk = {x ∈ IRn : kxk ≤ 1 − k1 }. Prove that Sk is compact. We consider for every n the problem (Qk ) inf f (x). x∈Sk

Prove that Qk has a solution. c) Prove that for k large enough, the solution of Qk is a solution of Q.

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