Exam 11

Exercise 11. Consider the following multi-objective problem:

11.1 Is it a convex problem? Why?

We have to analyze individualy the convexity of each Objective Function and the constraints.

• Objective Function: $f_1 \left(x\right)=x_1^2 +x_2^2 -16x_1 -12x_2$ and $f_2 \left(x\right)=-x_1 +2x_2$ . On the case of $f_1 \left(x\right)$ we can recognize a quadratic function: $f\left(x\right)=\frac{1}{2}x^T Q;x+q^T x+b$ . We need to check if $\nabla^2 f_1 \left(x\right)\ge 0$ ,the Hessian Matrix, is positive semi-definite to conclude if the function is convex or not. To check if the Hessian Matrix is positive semi-definite we need to check the eigen-values of Q are not-negative.

$$H_1 =\left\lbrack \begin{array}{cc} 4 & 0\\ 0 & 4 \end{array}\right\rbrack$$

With the second Objective function we can confirm is a linear function, which it means is convex. Constraints

The constraints functions and are both linear inequalities and the feasible region belong to the non-negative Castersian plane, and it means is convex set.
The constraint function is also linear inequality constraint and the region is defined by , and 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.

11.2 Do minima exist? Why?

11.3 Is the point (5, 5) a minimum? Why?

11.4 Is the point (10, 0) a weak minimum? Why?

11.5 Find a subset of minima by using the scalarization method.

11.6 Find the set of all weak minima by using the scalarization method.

11.7 Find the set of all minima.

11.8 Find the ideal point.