MATH 589B due March 19, 2023 Algorithms of Applied Mathematics II

Section 001, Spring 2023

Midterm

1. Minimize

���1(���,���,���) = exp(13��� +21��� −34���) +exp(−21��� −34��� +55���) + ?exp(2��� +���) +exp(−2��� −���)?/1000

Solution: Use only gradient method I get

��� = 0.001239572480721367, ��� = 0.002031358808021369, ��� = −0.003770604145803503

and

��� (���,���,���) = 1.9465798782509387.

I tried backtracking, but the code will stuck in the loop. Also, I tried newton’s method, but keep having the singular matrix issue.

2. Let ���(���) = 45 + ���1��� + ���2���2 + ���3���3. Minimize

1

2?2

Plot 1/���(���) as a function of ���, over the range 0 ≤ ��� ≤ 1, for the optimal values of ���1, ���2, and ���3.

���2(���1,���2,���3) =

d���

?d��� d���

−���

∫

0

3.Maximize���3(���,���,���)=���2+���2+���2subjectto���4+���4+���4+10���2+16���2 =154. Solution: Tried newton;s methods, stuck in backtracking. Used gradient ascent,

4. Maximize

34 ���4(���1,���1,���2,���2,���3,���3,���4,���4) = ∑︁ ∑︁ h(������ −������)2 + (������ −������)2i

subjectto������2+������2 ≤1,where1≤���≤4.

���=1 ���=���+1

5. An elastic ring between two vertical sheets of glass (so it does not fall, and ��� = 0) is standing on a table (��� ≥ 0). Minimize

20

���5(���1,���2,...,���20,���1,���2,...,���20)=∑︁h 1 ������+?������−1−2������+������+1?2+?������−1−2������+������+1?2i 150

���=1

subject to (������ − ������−1)2 + (������ − ������−1)2 = (1/20)2 for all 1 ≤ ��� ≤ 20 equality constraints, and ������ ≥ 0 for all 1 ≤ ��� ≤ 20 inequality constraints. The manipulations with indices are done modulo 20, so (���0, ���0) and (���21, ���21) are identified with (���20, ���20) and (���1, ���1), respectively. Plot the solution by drawing points (������, ������) and connecting consecutive points by segments on the ������-plane.

Minimizing or maximizing means finding the position where the optimum is achieved, also report the optimal value. You can use built-in elementary functions, linear algebra and ODE functions/solvers.