APM462: Homework 3
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APM462: Homework 3
Due online: Sat Feb 18 (before 9pm) on Crowdmark.
(1) Recall Q.5 on HW2 where f (x) = x . Qx + x . (x + a) + 1990, Q is positive semidefinite 3 x 3 symmetric matrix and x, a e R3 . For the functions gb(x, y, z) in part 5(b) which are strictly convex, find first and second order conditions that the maximum of f (x, y, z) on the set Ω := {(x, y, z) e R3 | gb(x, y, z) < 2046} must satisfy. Remark: gb(x, y, z) is strictly convex when its hessian is positive definite. Note that when gb is strictly convex its sublevel sets are ellipsoids and hence are compact.
(2) Consider the minimization problem
subject to: h(x1 , x2 , y1 , y2 ) = x1 _ 2 > 0
g(x1 , x2 , y1 , y2 ) = y 1(2) + y2(2) _ 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 mini- mizer?
(d) Using the 2nd order conditions, what can you say about the candidate(s) you found in part (b) ?
Remark: it is easier to solve this problem if you first simplify it to an equivalent problem in two variables only.
(3) Let f (x) be a C2 function which depends on of x only (and not on y). Consider the problem:
minimize: f (x)
subject to: g1 (x, y) = x _ 1 < 0
g2 (x, y) = _x < 0
h(x, y) = y = 0
(a) Which feasible points are regular?
(b) Use Kuhn-Tucker conditions to find the candidate(s) for mini- mizer.
(c) Are the 2nd order conditions satisfied at the point(s) you found in part (b)? Explain.
(4) Let L > 0 be the length of a piece of wire. Suppose you use the wire to make a circle of radius x and a square of side y < . The following problem represents maximizing the total area of the circle and square:
subject to 2πx + 4y _ L = 0
0 < x < L, 0 < y < L
(a) Solve this problem as a optimization problem with inequlity constraints (use the Kuhn-Tucker 1st order necessary conditions to find candidates for maximizers and then check which 2nd
order conditions are satisfied).
(b) Convert this problem to a first year calculus problem of one variable by using the equation to solve for y in term of x and then solve it.
(5) Consider the problem:
minimize: f (x, y) = xy2
subject to: g(x, y) = x + y < 1
h(x, y) = x2 + y2 _ 2 = 0
(a) Which feasible points are regular?
(b) Use Kuhn-Tucker conditions to find the candidate(s) for mini- mizer.
(c) Are the 2nd order conditions satisfied at the point(s) you found in part (b)? Explain.
(d) Find the minimizer for this problem. Is it a global minimum?
(6) Fix αi e R and consider the minimization problem
n
minimize f (x) = _ log(αi + xi)
i=1
subject to x1 , . . . , xn > 0
x1 + . . . + xn = 1.
(a) Using the 1st order conditions show that xi has the form:
xi = max{0, _ αi}
for some λ e R.
(b) Using the equality constraint x1 + . . . + xn = 1, argue that λ is unique, hence that the solution to the mimization problem is unique. (You are not asked to solve for λ .)
Hiit: part b) does not require any tools from lectures, just a bit of independent thinking.
(7) Let’s have a second look at Q.6 HW1. Consider the function f which measures the distance squared in R2 from a point (a, b) to the parabola H := {(x, y) e R2 | y = x2 }:
f ((a, b)) = (北neH |(x _ a, y _ b)|2 .
(a) Write the first order conditions for a minimizer (x0 , y0 ).
(b) Find the cubic equation x0 must satisfy (no need to solve this equation). Note: this equation is a cubic equation with coeffi- cients that depend on a and b only.
(c) Suppose x0 is a solution to the cubic equation. Find conditions on x0 that guarantee that (x0 , x0(2)) is a local minimizer. That is find conditions on x0 which guarantee that the Hessian is posi- tive defininte on the tangent space at (x0 , x0(2)).
This last exercise is for practice only and is not to be turned in. It 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) Let f : Rn → R be a C1 function. Recall that for any point x0 e Rn , the gradient Vf (x0 ) e Rn . Recall also that the graph of f is the surface M := {(x, f (x)) e Rn x R | x e Rn } in Rn x R. Now, given a point p := (x0 , f (x0 )) e M on the graph of f , find a formula for the tangent space Tp M in terms of the gradient Vf (x0 ). Hint: think of the surface M as the zero set g(x, z) = f (x) _ z = 0.
2023-02-21