How to find a pareto optimal solution in a smart way (3 variables)

Question

$\max\left( { 3x }_{ 1 }+4{ x }_{ 2 }+2{ x }_{ 3 } \right) $

${ x }^2_{ 1 }+{ x }^2_{ 2 }+{ x }^2_{ 3 }\le 1 $

${ x }_{ i }\ge 0 $

I have to find a Pareto Optimal solution, but I can't solve this by making calculus (Lagrangian function). So Is there a smart way to do this?

I know the objective function is a plane, and the constraint is a sphere. But is very difficult to find where the plane intersect the sphere

Answer 1

Using the Cauchy-Schwarz inequality, we have $$\begin{align}(3x_1+4x_2+2x_3)^2&\leq (3^2+4^2+2^2)(x_1^2+x_2^2+x_3^2)=29(1) \\\Rightarrow 3x_1+4x_2+2x_3&\leq\sqrt{29}\end{align}$$

Answer 2

since we have maximization problem and the objective is linear, the second condition is idle:

The refined version of the problem :

$ \min-(3x_1 + 4x_2 +2x_3) $

$ S.j\ \ ||x_i ||_2 <=1 $

Answer:

Lagrange : $ -(3x_1 + 4x_2 +2x_3)-\lambda (x_1^2+x_2^2+x_3^2) $

take derivatives and ... But if you want a shortcut in $4$ options-based tests, check the the options one-by-one what option has norm $\leq 1$ and biggest value of objective function, simultaneously.

Answer 3

You can easily solve this using the Lagrange multiplier method but a smarter way to do this would be as follows.

The question seems to me like an optimisation problem regarding utilisation maximisation of a consumer with finite income. Consider a consumer which is faced to buy three different products with amounts ($x_1$, $x_2$, $x_3$) and prices (1, 1, 1). If the utility function is described as

f($x_1$, $x_2$, $x_3$)= 3$x_1$+4$x_2$+2$x_3$

and the income of the consumer is 1 unit, then the problem is identical to the one stated above.

The optimal solution is reached for

($x_1$, $x_2$, $x_3$)= (0, 1, 0)

since the marginal utility is constant for all variables and largest for $x_2$.

The way described here is only another way to say that the problem is convex ( even linear ) and the optimum is reached on the boundary.

I hope I' m not wrong and it has helped. For further research on this type of specific problem in economics you can start by this link and continue with any undergraduate microeconomics book, the whole theory is related to such problems.