Section 1

Problem 1:

Search the web for a history lesson!

Make a table consisting of a list of 15 optimization software or optimization packages.

The table must contain the name of the software packages, year, the company or individual who invented them, the computer language used to write or run them, and price (if they are not open source).

Do not be afraid to trace the history to go back as far as the 50s.

Problem 2:

Literature web search for another history lesson!

Make a table consisting of a list of 15 papers or manuscripts that you or people think important or influential in the field of optimization or numerical algorithms for optimization. The table must contain the names of the papers, authors, years, and journals where they are published. Do not be afraid to trace the history to go back as far as the 50s.                                                                                                                                                  Problem 3:                                                                        Do the Vector Basics with Mathematical Tutorial.                                                                                                                                                                       Problem 4:                                                                   Do the Lines and Planes with Mathematical Tutorial.                                                                                                                                                       

Problem 5:

Do the Solving simple unconstrained optimization problems with Mathematical Tutorial.

Mathematical Project

MTH474 students can choose any 3 of the 6 problems

Section 2

1. Use Lagrange multipliers to find the maximum and minimum values of the function?f(x, y, z) = 2x + 6y + 10z subject to the constraint x?^{2 ?}+  y?² +^?  z²? ^?=  35?.           
2. Use Lagrange multipliers to find the maximum and minimum values of the function f(x ,y ,z) = xy + yz ? subject to the constraints ? xy = 1 ? and ? y? ?^{2 ?}+  z?^{2 ?}=  1  
3. The plane x? + y + z = 1 cuts? the cylinder x? ?^{2 ?}+  y²? ^?=  1 in? an ellipse. Find the points on the ellipse that lie closest to and farthest from the origin. It might be helpful to plot the cylinder and the plane on the same plot with Mathematical?    
4. Find a point ?A ?on the curve defined by the parametric equation x(t)? = ?sin?(t), y(t) = ?cos?(t), z(t) = 1 − ?sin?(t)− ?cos?(t), ?where ?− π ?≤ ?t ?≤ ?π, ?and a point ?B ?on another curve defined by the parametric equation ?x(s) = 2?sin?(s)+ 2, y(s) = 2?cos?(s), z(s) = 0, ?where −? π ?≤ ?s ?≤ ?π,

Such that the distance between A? ?and B is minimum. With Mathematical?, plot both curves, both points ?A and ? ?B, and a line joining the two points on the same graph.?

5. The surface of a donut given by the following parametric equation
   10. = (2 +cos (v))?cos?(u)

1. 25. = (2 +cos (v))?sin?(u) z = ?sin?(v)

where ?0 ?≤ ?u ?≤ ?2π ?and ?0 ?≤ ?v ?≤ ?2π ?intersects with the plane x? + y + 2z = 3.? Find the point

(or points) on the curve of intersection that lie closest to the point P with coordinate

(2,3,0)?. Plot the donut, the plane, the curve of intersection, and point(s) closest to P on the curve of intersection on the same plot.

6. Stellarator is a twisted toroidal apparatus for producing controlled fusion reactions in hot plasma. It is a cousin of Tokamak, which is an experimental donut-shaped machine designed to harness the energy of fusion. However, unlike Tokamak, a stellarator has cross section that looks like a bean shape.

 

Suppose the parametric equation of the boundary of the stellarator at one cross section is given by

r(θ) = cos?  (? θ)³? ^?+  sin?      (? θ)³?   ^?,            where ?− π/2 ≤? θ?      ≤? π?    /2,

Use whatever method of your own choice to find the location of two points on the boundary of the bean as shown in the picture. Compute the distance between the two points.

 

Section 3

Problem 1:

Implement the stopping criteria provided on page 6 in the week 3

Lecture slide in the MATLAB code your code for tolerance values ε=10?random_descentsearch−2 and ε=10−3.          . Test?

You are allowed to program in a language other than MATLAB.

Problem 2:

Test your code in Problem 1 for finding a local minimizer for:

• A function involving 3 variables
• A function involving 4 variables.
• A function involving 5 variables.

You can choose your own functions. For functions with 4 and 5 variables, you do not need to show the plots/animations.

You are allowed to program in a language other than MATLAB.

Problem 3:

In the MATLAB code αk at each step.random_descentsearch?          , we are using a?    fixed step value

Given a direction vector pk, implement a strategy in the code to find the largest αk such that f(xk+1)

 You are allowed to program in a language other than MATLAB.

Problem 4:

Modify the MATLAB code random_descentsearch? such that the domain of the objective function f(x,y) is restricted on a disk (x−0.5)2+(y−0.5)2≤0.25

Only first 8 problems need to be solved. You are allowed to program in a language other than MATLAB.