# 1 Unconstrained optimization

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

1. $$\min_x f(x)=x^4.$$

2. $$\max_x \left ( 2\sin{x} -\frac{x^2}{10} \right ).$$

3. $$\max_{x,y} \left (2xy+2x-x^2-2y^2 \right ).$$

# 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 13. Try to change the noise level and replicate the denoising process and report your findings.

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