Exam 3

Exercise 3. Consider the following multi-objective problem:

$$\left\lbrace \begin{array}{ll} \mathrm{minimize} & x_2 \ ,-x_1 -2x_2 \\ s\ldotp t & x_1 \ge 0\\ & x_2 \ge 0\\ & x_1 +x_2 \le 5 \end{array}\right.$$

3.1 Is it a convex problem? Why?

We need to analyze the convexity of all the objective function and the feasible region:

• Objective Function: $f_1 \left(x\right)=x_2$ and $f_2 \left(x\right)={-x}_1 -2x_2$ .The objective function $f_1 \left(x\right)$ is a linear function, as same as $f_2 \left(x\right)$, which it means both of the objective functions are convex.

• Constraints: $g_1 \left(x\right)=x_1 \ge 0$ , $g_2 \left(x\right)=x_2 \ge 0$ and $g_3 \left(x\right)=x_1 +x_2 \le 5$ . The constraints functions $g_1 \left(x\right)$ and $g_2 \left(x\right)$ are both linear inequalities and the feasible region belong to the non-negative Castersian plane, and it means is convex set.
  The constraint function $g_3 \left(x\right);$ is also linear inequality constraint and the region is defined by $x_1 +x_2 =5$ , $x_1 =0$ and $x_2 =0$ makes a triangle set that is also convex. Since all the objective functions and the feasible region are convex, we can confirm that this multi-objective optimization problem is convex.

3.2 Do minima exist? Why?

3.3 Is the point (0, 2) a weak minimum? Why?

3.4 Find all the minima by using the scalarization method.

3.5 Find all the weak minima by using the scalarization method.

3.6 Find the ideal point.

3.7 Apply the goal method with L1

3.8 Apply the goal method with L2

3.9 Apply the goal method with L-infinity