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TSM, M1 Finance 2019/20 Introduction to Optimisation for finance and insurance

Problem Set 2 : Mulitvariate Functions and their extrema -1-

Exercise 1. 1. Are the following functions affine ? If so determine a vector and a constant to represent the function. 1 f (x, y, z) = x + 2y + 3z + 4, 3

g(x, y) = x2 + y + 2,

h(x1 , . . . , xn ) = 1x1 + 2x2 + 3x3 + . . . + nxn 2. Determine the representation of the following quadratic forms using a symmetric matrix f (x, y) = x2 − 2y 2 ,

g (x, y) = x2 + 2xy + y 2 ,

h(x, y, z) = −x2 − 2y 2 − z 2

Do the functions attain a minimum or a maximum ? If so in which point ? 3. Determine the representation of a quadratic form for arbitrary a, b, c ∈ R. f (x, y) = ax2 + bxy + cy 2 Exercise 2. For each of the following subsets of R2 — Give a graphical representation. — Say whether the subset is open, closed or neither of both (without justification). — Determine wether the subset is bounded or not. Please justify your answer. — Is the subset compact ? — Say whether the subset is convex or not (without rigorous proof). U1 = [1, 2] × [2, 6],

U2 = R+ × (1, 2),

U4 = {(x, y) ∈ R2 ; 2x2 + 2y 2 = 4}, U6 = {(x1 , x2 ) ∈ R2 : 4x1 + x2 = 2}, U8 = {(x, y) ∈ R2 ; x + y < 1},

U3 = [−1, 1] × (1, 2)

U5 = {(x, y ) ∈ R2 ; (x − 2)2 ≤ 1 − (y − 1)2 }, U7 = {(x1 , x2 ) ∈ R2 : 3x1 + x2 ≥ 0},

U9 = {(x, y) ∈ R2 ; −1 ≤ x ≤ 0, y = 2},

U10 = {(x1 , x2 ) ∈ R2 : x1 + 3x2 ≤ 12, x1 ≥ 0, x2 ≥ 0}, U11 = {(x, y) ∈ R2 ; x2 − 2xy + y 2 = 0}, U13 = {(x, y) ∈ R2 ; x2 ≥ 1, y 2 ≥ 4}

U12 = {(x, y) ∈ R2 ; x2 − y ≤ 0, x + y ≤ 1},...