程序代写案例-APM462

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APM462: Homework 5
Comprehensive assignment for first term 1
Due: Sat Feb 27 (before 9pm) on Crowdmark.
(1) Recall that in lecture I proved that at a regular point p, TpM ⊆
Tp and then said that equality follows from the Implicit Function
Theorem. In this problem you are asked to prove Tp ⊆ TpM in a
special case.
Let f : Rn → R be a C1 function. Recall from HW4 that the
graph of f is the surface M := {(x, f(x)) ∈ Rn × R | x ∈ Rn} in
Rn × R.
(a) Let p := (x0, f(x0)) ∈M . Find the space Tp.
(b) Show that Tp ⊆ TpM . That is show that any vector in the
space Tp is a tangent vector to M at p. Note: unlike the case
on HW4 Q7, here you are not allowed to assume that Tp = TpM .
(2) Let f : R3 → R, and g : R2 → R1 be C1 functions. Use Lagrange
multipliers (1st order conditions) to show that first constrained op-
timization problem and the second unconstrained problem have the
same candidates for minimizers:
min f(x, y, z)
subject to h(x, y, z) = z − g(x, y) = 0
(x, y, z) ∈ R3,
and
min f(x, y, g(x, y))
(x, y) ∈ R2.
Note: the first problem involves the variables x, y, z while the sec-
ond one only x, y.
(3) This question explores a surprising relationship called “linear programming duality” be-
tween two related linear problems. If you took APM236 you would have seen duality
there. Here we will prove duality using the Kuhn-Tucker conditions.
Let A be an m × n matrix, b ∈ Rm, and c ∈ Rn. Consider the
following two optimization problems, the “primal problem”:
1Copyright c©2021 J. Korman. Sharing this material publically is not allowed without
permission of author.
1
2max cTx
subject to Ax ≤ b
x ≥ 0,
and the “dual problem”:
min bT p
subject to AT p ≥ c
p ≥ 0.
Let x∗ be an primal optimal solution and p∗ be an optimal dual
solution.
(a) Write the Kuhn-Tucker (1st order neccessary) conditions for x∗.
Hint: use Lagrange multipliers p∗ (for the Ax ≤ b constraints)
and µ (for the x ≥ 0 constraints).
(b) Show that the Lagrange mutipliers p∗ for the optimal primal x∗
satisfy the constraints of the dual problem.
(c) Use the comlementary slackness conditions for the primal prob-
lem to show that (p∗)TAx∗ = pT∗ b.
(d) Write the Kuhn-Tucker (1st order neccessary) conditions for p∗.
Hint: use Lagrange multipliers x∗ (for the AT p ≥ c constraints)
and ν (for the p ≥ 0 constraints).
(e) Show that the Lagrange mutipliers x∗ for the optimal dual p∗
satisfy the constraints of the primal problem.
(f) Use the comlementary slackness conditions for the dual problem
to show that cTx∗ = (p∗)TAx∗.
(4) Let Q be a 2× 2 positive definite symmetric matrix. Let f ∈ C2 be
an increasing, non-negative, and convex function on R1. Let F (s)
denote the area of the sublevel sets {xTQx ≤ f(s)2}:
F (s) = area {x ∈ R2 | xTQx ≤ f(s)2}.
Prove that F is convex.
(5) Let R > 0 and assume the following problem has a solution:
min f(x, y, z)
subject to x2 + y2 − z2 ≤ R2.
3Note that the Kuhn-Tucker conditions (*) are:
∇f(x, y, z) + µ
 2x2y
−2z
 =
00
0

µ(x2 + y2 − z2 −R2) = 0, µ ≥ 0.
(a) Write the Kuhn-Tucker conditions for the following problem
using Lagrange multipliers µ1 and µ2:
min f(r

R2 + z2 cos θ, r

R2 + z2 sin θ, z)
subject to r − 1 ≤ 0
−r ≤ 0
z, θ ∈ R1
(b) Show that the conditions you found in part (a) imply the Kuhn-
Tucker conditions (*). Start by expressing µ in terms of µ1 and
µ2. There should be three cases to consider: when r = 0,
0 < r < 1, and r = 1.
(6) Consider the following optimization problem where f, h are C1 func-
tions on Rn:
min f(x)
x ∈ Rn
subject to h(x) = 0.
Suppose p is a regular point of the constraint h. Prove that TpM =
Tp. Hint: HW4 Q.7 and the implicit function theorem.

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