# (Multi-objective optimization) Find efficient points.

The problem is as follows:

\begin{array}{ll}\tag{P}\label{prob} \text{maximize} & f_1(\mathbf{x}) = -x_1+x_2\\ \text{maximize} & f_2(\mathbf{x}) = 10x_1\\ \text{minimize} & f_3(\mathbf{x}) = 2x_1-3x_2\\ \text{subject to} & -x_1+x_2 \le 2\\ & x_1 \le 7\\ & x_1+x_2 \le 8\\ & x_1, x_2 \ge 0 \end{array}

Find sets of efficient points and properly efficient points.

I first tried to solve the problem using graphical method.

A figure above represents a feasible set created by four constraints of the multi-objective optimization problem \eqref{prob}.

According to the above (second) figure, an ideal solution set of $f_1$ is $$S_{f_1} = \{\mathbf{x}:\theta\,\mathbf{x}'+(1-\theta)\,\mathbf{x}'', ~\forall\theta\in[0,1], ~\mathbf{x}'=(0,2), ~\mathbf{x}''=(3,5)\},$$ an ideal solution set of $f_2$ is $$S_{f_2}=\{(3,5)\},$$ and an ideal solution set of $f_3$ is $$S_{f_3}=\{\mathbf{x}:\theta\,\mathbf{x}'+(1-\theta)\,\mathbf{x}'', ~\forall\theta\in[0,1], ~\mathbf{x}'=(7,0), ~\mathbf{x}''=(7,1)\},$$

In a figure above, only a black region is direction set from a black point, i.e., any points in the black region dominates the black pint.

According to the above figure, a set $E$ of efficient points is defined by $$\{\mathbf{x}:\theta\,\mathbf{x}'+(1-\theta)\,\mathbf{x}'', ~\forall\theta\in[0,1], ~\mathbf{x}'=(3,5), ~\mathbf{x}''=(7,1)\}.$$

Then, a set of properly efficient points is defined by $$\{\mathbf{x}:\theta\,\mathbf{x}'+(1-\theta)\,\mathbf{x}'', ~\forall\theta\in(0,1), ~\mathbf{x}'=(3,5), ~\mathbf{x}''=(7,1)\}.$$

Additionally, I want to know a method to find a properly efficient set using $P_\lambda$ problem.
• When a problem that maximizes $f_1(\mathbf{x}), \ldots, f_p(\mathbf{x})$ with constraints $\mathbf{x}\in\mathcal{S}$ (where $\mathcal{S}$ is a feasible space) is given, the $P_\lambda$ problem is to maximize $\sum_{i=1}^p \lambda_i f_i(\mathbf{x})$ such that $\mathbf{x}\in\mathcal{S}$, where $\sum_{i=1}^p \lambda_i = 1$ and $\lambda_1,\ldots,\lambda_p\ge0$.
• If $\mathcal{S}$ is a closed convex set and $f_i$'s are all concave function on $\mathcal{S}$, then $\mathbf{x}\in\mathcal{S}$ is properly efficient if and only if there exist $\mathbf{lambda}$ such that $\mathbf{x}$ is optimal solution of the $P_\lambda$ problem.
Is someone answer me the process of solving the above problem using the $P_\lambda$ problem explained above?