Mathematics 524 Final Name:

1. (40 points) Define or state precisely the following: (a) C is a convex subset of R^{n}.

(b) x is a convex combination of x_{1}, x_{2}, …, x_{k}.

(c) The Arithmetic-Geometric mean inequality.

(d) Let the real valued function f(x) be defined on the convex set C. Then define precisely what it means for f(x) to be a convex function.

(e) State what it means for a vector v to be a decent direction for f at x.

(f) Let g = [g_{1} ; ⋮ ; g_{n}] be a function g : R^{n} → R^{n}. Then define ∇g.

(g) Let g : R^{n} → R^{n}. Then state the chain rule for \frac{d}{dt}g(x + tv).

2. (15 points) Find the global and local maximizers and minimizers of f = (x^{2} − x + 1)e^{−x}.

3. (15 points) Classify the following matrices as to positive or negative semidefinite or indefinite. Explain briefly why your answer is correct.

(a) [\begin{matrix}1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{matrix}].

(b) [\begin{matrix}−8 & 0 & 0 \\ 0 & −4 & 0 \\ 0 & 0 & −2 \end{matrix}].

(c) [\begin{matrix}−8 & 2 & 5 \\ 2 & 4 & 3 \\ 0 & 0 & −2 \end{matrix}].

4. (15 points) State the basic geometric program and its dual. Then explain how they are related by use of the arithmetic-geometric mean inequality. Finally in some cases the dual problem will be messier than the original. When and why is this the case?

5. (20 points) Find all the critical points of f(x, y) = x^{3} + y^{3} − 3x − 12y + 20 and classify as to local maximums, minimums, or saddle points.

6. (20 points) Determine if the folowing functions are convex, concave or neither. Explain breifly why your answer is correct.

(a) f(x) = −2x^{2} − 4x + 7.

(b) f(x) = e^{−x^{2}}.

(c) f(x, y) = 4x^{2} + 2xy + y^{2} − 4y + 7.

(d) f(x, y) = 4e^{3x−y} + 5e^{x^{2}+y^{2}}.

7. (20 points) Maximize f(x, y, z) = xy^{2}z subject to x + y + z^{2} = k where k is a constant and x, y, z > 0.

8. (15 points) Find the minimum and the minimizer of the function f(x, y) = x + 2y + \frac{3}{xy} on the set x, y > 0.

9. (15 points) (a) What is Newton’s method for finding the minimum of a function of several variables?

(b) Explain, informally, the reason that Newton’s method works.

(c) What, if any, are the drawbacks of Newton’s method?

10. (15 points)

(a) What is the method of steepest decent for finding the minimum of a function of several variables?

(b) Explain, informally, the reason that the method of steepest decent works.

(c) What, if any, are the drawbacks of the method of steepest decent?

11. (10 points) 10 free points

Have a nice Holiday Break.