# Exercise 5
* Due: 11.2.2016 at noon
* Max points: 8

## General rules
* Send the source code or the written notes of your answers as an attachment to markus.hartikainen@jyu.fi before the due time
* You will get feedback about your answers a week from the due date
* From the second week on, the exercises will be given on the previous Wednesday's lecture

## Exercises

For exercises 1-2, we study optimization problem
$$
\begin{align}
\min \qquad & x_1^2+x_2^2 + x_3^3+(1-x_4)^2\\
\text{s.t.}\qquad &x_1^2+x_2^2-1=0\\
    &x_1^2+x_3^2-1=0\\
    &x\in\mathbb R^4
\end{align}
$$

1. **(2 points)** Use the SQP method to solve the above problem.
2. **(2 points)** Use the Lagrangian method to solve the above problem.
3. **(2 points)** Solve the problem
$$
\begin{align}
\min   \  & x_1^2 + x_2^2\\
\text{s.t. } & x_1 + x_2 \geq 1.
\end{align}$$ by using just the optimality conditions.
4. **(2 points)** Consider a problem
$$\begin{align}
\min   \  & f(x)\\
\text{s.t. } & h_k(x)=0, \text{ for all } k=1,\dots,K,
\end{align}
$$
where all the functions are twice differentiable. Show, that the gradient of the augmented Lagrangian function is zero in the minimizer $x^*$ of the above problem. In other words, show that $\nabla_xL_c(x^*,\lambda^*)=0$, where $\lambda^*\in R^n$ is the corresponding optimal Lagrange multiplier vector.




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ipynb
Jupyter Notebook
Exercise 5.ipynb
maeehart/TIES483
cce5c779aeb0ade5f959a2ed5cca982be5cf2316
CC-BY-3.0
2019-04-26T12:46:14.000Z
2021-11-23T03:38:59.000Z
Exercise 5.ipynb
maeehart/TIES483
cce5c779aeb0ade5f959a2ed5cca982be5cf2316
CC-BY-3.0
Exercise 5.ipynb
maeehart/TIES483
cce5c779aeb0ade5f959a2ed5cca982be5cf2316
CC-BY-3.0
2016-01-08T16:28:11.000Z
2021-04-10T05:18:10.000Z
Qwen/Qwen-72B
1. YES
2. YES