# 代写接单-MATH 589B due March 19, 2023 Algorithms of Applied Mathematics II Section 001, Spring 2023

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.

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