Problem $1.$ Consider a linear programming problem:

minimize$\_($x$\_(1),$x$\_(2))(-$x$\_(1)-$x$\_(2))$

subject to:

x$\_(1)+2$x$\_(2)<=5,$

$2$x$\_(1)+$x$\_(2)<=4,$

x$\_(1)>=0,$x$\_(2)>=0.$

Solve this problem graphically similarly to the way it was done in the notebook. To do this, draw a

feasible set, the objective function, and determine the optimal value and optimal solution.

Write a Lagrangian function for this problem $($remember that you need a Lagrangian multiplier for

each constraint, including x$\_(1)>=0,$x$\_(2)>=0).$ Write a system of KKT conditions for the problem and

solve this system $($hint: graphical solution from the part $(1)$ and complementary slackness conditions

will help you determine which Lagrangian multipliers are equal to zero$).$

Write a logarithmic barrier problem for the linear programming problem above and show that this is a

convex optimization problem in its feasible region. One of the ways to do it is to look at a logarithmic

barrier for each of the constraints and show that the corresponding Hessian is symmetric positive

semi$-$definite in the interior of the feasible region. Then the sum of convex functions is convex.

I understand that the question does have instructions, but I am not sure on how to apply those instructions.

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