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i need details of these 2 math questions .

these questions are about optimization (basics of set-constrained and unconstrained optimization)


6.12.png6.32.png

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6.12 Consider the problem minimize f (ac) subject to a: 6 Q, where f : R2 a R is given by f(:c) = 5:32 with a: = [$1,32]T, and Q = {a3 = [m1,z:2]T :xf +132 2 1}. a. Does the point :6" = [0, 1]T satisfy the first-order necessary condition? 1). Does the point (3" = [0, 1]T satisfy the second-order necessary condition? 0. Is the point 9;" = [0, 1]T a local minimizer?

6.32 Prove the following generalization of the second-order sufficient condi- tion: Theorem: Let Q be a. convex subset of IR", f E (32 a. real—valued function on Q, and m" a point in 9. Suppose that there exists a E R, c > 0, such that for all feasible directions or! at :3" (d 75 0), the following hold: 1. dTVf(a:*) 2 o. 2. dTF(a:*)d 2 clldllz. Then, cc" is a strict local minimizer of f.

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6.12 Consider the problem minimize f (ac) subject to a: 6 Q, where f : R2 a R is given by f(:c) = 5:32 with a: = [$1,32]T, and Q = {a3 = [m1,z:2]T
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