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\title{Final exam. \\
MATH 5770-001 and 6640-001 and ME EN 6025-001 \\
Introduction to Optimization}
\author{Due Day: Friday, December 13, 2019}
\date{}
\maketitle

\begin{enumerate}
\item 
Using Lagrange multipliers method,
 solve the problem
 $$
 \min_{x_1 , ..., x_n} \sum_{i=1}^n a_i^2 x_i \quad \mbox{subject to }~ \sum_{i=1}^n \frac{c}{ x_i -b}= 1
 $$
 assuming that $x_i > a$. Here, $a$, $b$, $c$ are real parameters. 
\item
Using Lagrange Multiplyers method,
 solve the problem
 $$
 \min_{x_1 , x_2} \left(x_1-1 \right)^2 + \left(x_2-t  \right)^2, \quad \mbox{subject to }~ |x_1| + |x_2| \leq 2
 $$
 Here $t$ is a real parameter. Find the solutions for all values of $t$. Sketch the graph of the feasible region and illustrate the found solutions. 
  
\item Convert the following problem to the standard LP form
$$
\min_{x_1 , ..., x_4} x_1+ x_2 + 3 x_3 + x_4
$$
subject to
\begin{eqnarray*}
x_1+ x_2 =1\\
x_1 + 3 x_3 \leq 1 \\
x_2 + x_4 \leq 10 \\
1\leq x_2 \leq 3\\
-1\leq x_3 \leq 1
\end{eqnarray*}
\newpage
\item Solve 
$$\min_{x\in R_n} ( x -x_0)^T(x-x_0) \quad \mbox{subject to }~ Ax=b
$$
Here, $ x\in R_n,~b\in R_m, ~ n>m$. Rank $A=m$ (in other words, all constraints are linearly independent.

Find the formulas for the vector Lagrange multiplyer, and the optimal value $x_*$ of $x$.
\item Consider the problem
\begin{equation}
\min_{x_1, x_2} {e^{x_1 + x_2}} \quad \mbox{subject to }~ x_1^2 + x_2^2 \leq 1
\label{1}
\end{equation}
Write down the KKT conditions.          t

Formulate the problem using logarithmic barrier function [formula (19.4) in the textbok] for accounting for the constraint. Write down the KKT conditions for the obtained Interior-Point problem. Compute the minimum, using different values of the barrier parameter $\mu= 1, 01, 0.001$. Comment.

\item Use quadratic penalty to solve problem (\ref{1}). Compute the minimum, using different values of the penalty parameter $\mu= 0.1, 1, 10, 30$. Comment.

\item Use augmented Lagrangian method to numerically solve problem % (\ref{1}), 
 $$
 \min_{x_1, x_2, x_3} (x_1-3)^2 + 2 x_2 x_3 + x_3^2 
 \quad \mbox{subject to }~ x_1^2 + x_2^2+ x_3^2 = 4
 $$
Starting at the point $x^{(0)}= [2, 0, 0]$. Perform 8 iterations. Comment. 



\end{enumerate}


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