2. For r =0 the problem max (x -r)? has two solutions, x =±1. Forr #0, there is only one solutioni f*(r) is not differentiable at r = 0. (a) Conid

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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Can you help with question 2?

i = 1,..., k
()
(1),x=x
The interpretation of the right-hand side of this formula is analogous to the interpretation
of formula (3.3.15).
PROBLEMS FOR SECTION 3.7
SM 1. (a) Solve the nonlinear programming problem (a and b are constants)
maximize
95x 'p52+ 6+ x 0} ɔɔqns _ə -- k.? - 1_2 – 001
(b) Let f*(a, b) be the (optimal) value function. Compute the partial derivatives of ƒ* with
respect to a and b, and relate them to the Lagrange multipliers.
(c) Put b = 0, and show that F*(a) = f*(a,0) is concave in a.
2. For r = 0 the problem
max (x – r)²
has two solutions, x =±1. For r #0, there is only one solution. Show that the value function
f*(r) is not differentiable at r = 0.
SM 3. (a) Consider the problem
max(min) x² + y? subject to r < 2x² + 4y² < s?
where 0 <r < s. Solve the maximization problem and verify (5) in this case.
(b) Reformulate the minimization problem as a maximization problem, solve it, and verify (5)
in this case.
(c) Can you give a geometric interpretation of the problem and its solution?
HARDER PROBLEMS
SM 4. Prove that f*(r) defined in (4) is concave if f is concave and g1,..., &m are convex in (x, r).
(This generalizes Theorem 3.7.1.)
021S QF202 Recitat..R
6.ArimaX.Regression..r
n13.pdf
Re
Transcribed Image Text:i = 1,..., k () (1),x=x The interpretation of the right-hand side of this formula is analogous to the interpretation of formula (3.3.15). PROBLEMS FOR SECTION 3.7 SM 1. (a) Solve the nonlinear programming problem (a and b are constants) maximize 95x 'p52+ 6+ x 0} ɔɔqns _ə -- k.? - 1_2 – 001 (b) Let f*(a, b) be the (optimal) value function. Compute the partial derivatives of ƒ* with respect to a and b, and relate them to the Lagrange multipliers. (c) Put b = 0, and show that F*(a) = f*(a,0) is concave in a. 2. For r = 0 the problem max (x – r)² has two solutions, x =±1. For r #0, there is only one solution. Show that the value function f*(r) is not differentiable at r = 0. SM 3. (a) Consider the problem max(min) x² + y? subject to r < 2x² + 4y² < s? where 0 <r < s. Solve the maximization problem and verify (5) in this case. (b) Reformulate the minimization problem as a maximization problem, solve it, and verify (5) in this case. (c) Can you give a geometric interpretation of the problem and its solution? HARDER PROBLEMS SM 4. Prove that f*(r) defined in (4) is concave if f is concave and g1,..., &m are convex in (x, r). (This generalizes Theorem 3.7.1.) 021S QF202 Recitat..R 6.ArimaX.Regression..r n13.pdf Re
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