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9) ai , x ≤ 0 for i = 1, 2, . . , m, c, x > 0, x ∈ E. Proof. 9) has no solution. 8) has a solution by using induction on the number of elements m. The result is clear for m = 0. Suppose then that the result holds in any Euclidean space and for any set of m − 1 elements and any element c. Deﬁne a0 = −c. 9) shows there are scalars i λ0 , λ1 , . . , λm ≥ 0 in R, not all zero, satisfying λ0 c = m 1 λi a . If λ0 > 0 the proof is complete, so suppose λ0 = 0 and without loss of generality λm > 0.

7). Exercises and commentary Versions of the Lagrangian necessary conditions above appeared in [161] and [99]: for a survey, see [140]. The approach here is analogous to [72]. The Slater condition ﬁrst appeared in [152]. 1. 3). 2. 3) to solve the following problems. 2 The value function 57 (a) ⎧ ⎪ ⎪ ⎪ ⎨ inf x21 + x22 − 6x1 − 2x2 + 10 subject to 2x1 + x2 − 2 ≤ 0, ⎪ x2 − 1 ≤ 0, ⎪ ⎪ ⎩ x ∈ R2 . (b) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ inf −2x1 + x2 subject to x21 − x2 ≤ 0, x2 − 4 ≤ 0, x ∈ R2 . (c) ⎧ ⎪ ⎪ ⎪ ⎨ inf x1 + (2/x2 ) subject to −x2 + 1/2 ≤ 0, ⎪ −x1 + x22 ≤ 0, ⎪ ⎪ ⎩ x ∈ {(x1 , x2 ) | x2 > 0}.

Prove the following functions x ∈ R → f (x) are convex and calculate ∂f : (a) |x|; (b) δR+ ; √ (c) − x if x ≥ 0, and +∞ otherwise; (d) 0 if x < 0, 1 if x = 0, and +∞ otherwise. 6. 6 (Subgradients and directional derivatives). 7. 7. 8. (Subgradients of norm) Calculate ∂ · . 9. (Subgradients of maximum eigenvalue) Prove ∂λ1 (0) = {Y ∈ Sn+ | tr Y = 1}. 10. 2, Exercise 9 (Schur-convexity)). 48 Fenchel duality ∗ 11. Deﬁne a function f : Rn → R by f (x1 , x2 , . . , xn ) = maxj {xj }, let x¯ = 0 and d = (1, 1, .