1 Unconstrained optimization

Apply optim() to solve the following unconstrained optimization problems:

\(\min_x f(x)=x^4.\)
\(\max_x \left ( 2\sin{x} -\frac{x^2}{10} \right ).\)
\(\max_{x,y} \left (2xy+2x-x^2-2y^2 \right ).\)

2 Linear Programming (LP)

Solve the following LP problem: \[ \max_{x_1, x_2, x_3,x_4} \left (x_1 +2x_2 +3x_3+4x_4+5 \right )\] subject to: \[ \left\{ \begin{array}{rl} 4x_1 + 3x_2 + 2x_3+ x_4 & \leq 10 \\ x_1 -x_3 +2x_4 & = 2 \\ x_1 + x_2 + x_3 +x_4 & \geq 1 \\ x_1\geq0, x_3\geq0, x_4 & \geq0 \end{array} \right . . \]

Apply lpSolveAPI and Rsolnp and compare the results.

3 Mixed Integer Linear Programming (MILP)

Apply lpSolveAPI to solve the following MILP problem: \[ \min_{x_1, x_2} ~{4x_1 +6x_2}\] subject to: \[ \left\{ \begin{align} 2x_1 + 2x_2 & \geq 5 \\ x_1 -x_2 & \leq 1 \\ x_1, x_2 &\geq 0 \\ x_1, x_2 & \in \text{ integers} \end{align} \right. . \]

4 Quadratic Programming (QP)

Solve the following QP problem: \[ \min_{x_1,x_2} ~{2x_1^2+x_2^2+x_1x_2+x_1+x_2}\] subject to: \[ \left\{ \begin{array}{rl} x_1 +x_2 & = 1 \\ x_1, x_2 &\geq 0 \end{array} \right. . \]

Apply quadprog to solve the QP
Use Rsolnp to solve the QP
Determine the Lagrange multiplier
Apply numDeriv to solve this Lagrange multiplier optimization manually
Compare the three versions of the results above.

5 Complex non-linear optimization

Solve the following nonlinear problem: \[ \min_{x_1,x_2} \left ( 100(x_2-x_1^2)^2+(1-x_1)^2 \right )\] subject to \(x_1,~x_2\geq 0.\)

6 Data Denoising

Use the example shown in [Chapter 21]http://www.socr.umich.edu/people/dinov/courses/DSPA_notes/21_FunctionOptimization.html). Try to change the noise level and replicate the denoising process and report your findings.

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Data Science and Predictive Analytics (UMich HS650)

Assessment: 21. Function Optimization