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APM462 minimizing the function

创建日期：2024-06-11 16:23:12

所属分类：数学mathematics

标签： APM462

minimizing the function

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APM462: Homework 3

Due online: Sat Feb 17 (before 9pm) on Crowdmark.1
(1) Let’s minimizing f(x, y) = (y−5)2+2(x−3)2 on unit disk D := {(x, y) ∈ R2 | x2+y2 ≤ 1}.
(a) Use the fact that f is strictly convex to show that it has a unique minimizer on D.
(b) After showing no interior points of D satisfy the 1st order necessary conditions, con-
clude the unique minimizer is on the boundary ∂D. Write the Lagrage multiplier
equations the minimizer must satisfy. Note: you do not need to solve these equations.
(c) Use the 2nd order conditions to show that one of the candidates for minimizers (those
points satisfying the Lagrange multiplier equations) is a (strict) local minimizer.
(2) Consider the minimization problem
minimize: f(x) = |x|2
subject to: h(x) = x1 + · · ·+ xn = 1
Here x ∈ Rn and xi are the components of the vector x.
(a) Which feasible points are regular?
(b) Use the first order conditions to find the candidate(s) for a local minimizer.
(c) What is the tangent space at the candidate(s) for a local minimizer?
(d) Using the 2nd order conditions, what can you say about the candidate(s) you found in
part (b) ?
(3) Inscribe a rectangle of maximal area into the ellipse x
2
a2
+ y
2
b2
= 1. That is, solve the following
maximization problem where a, b > 0:
maximize: f(x, y) = xy
subject to: h(x, y) =
x2
a2
+
y2
b2
− 1 = 0.
(4) Solve the following problem of minimizing the function f(x, y) over the ellipse h(x, y) = 0.
Here the point (x0, y0) is outside the ellipse and a, b > 0:
minimize: f(x, y) =
(x− x0)2
a2
+
(y − y0)2
b2
subject to: h(x, y) =
x2
a2
+
y2
b2
− 1 = 0
(a) Which feasible points are regular?
(b) Use the first order conditions to find the candidate(s) for a local minimizer.
(c) What is the tangent space at the candidate(s) for a local minimizer?
(d) Using the 2nd order conditions, what can you say about the candidate(s) you found in
part (b) ?
(5) Let f(x) = 12(x− x0)TQ(x− x0) be a quadratic function where Q is positive definite n× n
symmetric matrix and |x0| > 1, that is the point x0 is outside the unit ball in Rn. Consider
the following problem:
minimize: f(x)
subject to: g(x) = |x|2 − 1 ≤ 0
1Copyright ©2024 J. Korman. Sharing or selling this material without permission of author is a copyright
violation.
1
2(a) Which feasible points are regular? Explain.
(b) Write the 1st-order Kuhn-Tucker conditions that a local minimizer x0 must satisfy.
You do not need to solve for the Lagrange multiplier µ. Express the minimizer in
terms of x0, Q, and µ.
(c) At which of the points x0 you found in part (b) are the 2
nd-order sufficient conditions
satisfied? Hint: recall that the Lagrange multiplier µ is non-negative.
(6) Consider the problem
min f = x21 + x
2
2 − 2x3
subject to x1 + 2x2 = 5
x1 + x2 − x3 ≤ 3
0 ≤ x3 ≤ 1
(a) Which feasible points are regular? Explain.
(b) Use the 1st-order Kuhn-Tucker conditions to find the candidate(s) for local min.
(c) Are the 2nd-order sufficient conditions satisfied at the point(s) you found in part (b)?
(7) Suppose A is an m× n matrix of rank n, b ∈ Rm, and c ∈ Rn. Consider the problem:
minimize: f(x) =
1
2
||Ax− b||2
subject to: ||x||2 ≤ 1.
Recall from HW2 that you already computed the gradient and Hessian of f(x) = 12 ||Ax−
b||2 and have shown that ATA is positive definite.
(a) Write the 1st-order Kuhn-Tucker conditions that a local minimizer x0 must satisfy.
You do not need to solve for the Lagrange multiplier µ. Express the minimizer in
terms of x0, A, and µ.
(b) At which of the points x0 you found in part (b) are the 2
nd-order sufficient conditions
satisfied? Hint: recall that the Lagrange multiplier µ is non-negative.
The last two exercises are for practice only and are not to be turned in. The first one is
about the Lagrange multiplier condition in the spcial case when the surface is a graph of a
function and the second problem is intended to help you understand the relations between
the gradient of a function, the graph of the function, and the tangent space to the graph.
(8) Suppose (x0, y0) is a minimizer for the following constrained minimization problem
minimize: F (x, y)
subject to: H(x, y) = y − h(x) = 0
(a) Convert this problem into an unconstrained minimization problem of the form
minimize: f(x)
x ∈ R
Hint: Use the constraint to write y = h(x).
(b) Find the 1st order necessary condition that a minimizer x0 of f(x) must satisfy.
(c) Use part (b) to show that ∇F (x0, y0) is a multiple of ∇H(x0, y0).
(9) Let f : Rn → R be a C1 function. Recall that for any point x0 ∈ Rn, the gradient ∇f(x0) ∈
Rn. Recall also that the graph of f is the surface M := {(x, f(x)) ∈ Rn × R | x ∈ Rn} in
Rn × R. Now, given a point p := (x0, f(x0)) ∈M on the graph of f , find a formula for the
3tangent space TpM in terms of the gradient ∇f(x0). Hint: think of the surface M as the
zero set g(x, z) = f(x)− z = 0.

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