Optimization using KKT

Question

Optimization of: $x_{1}^{2}+x_{2}^{2} + x_{3}^{2}$, given that $x_{1}-x_{2}\leq0$, $x_{2}-x_{3}\leq0$, $(x_{1} - x_{2})^{2} + (x_{2} - x_{3})^{2} = \delta$ and $x_{1} + x_{2} + x_{3} = 0$. I tried using KKT condtion but not getting the solution.

Answer 1

Hint : Use the substitution $$x= x_2 - x_1 \quad y = x_3- x_2 $$ then using the constraint $x_1 + x_2 + x_3 = 0$ find $x_i$ in terms of $x, y$ and plug in back into objective function to get a two variable function, call it $f(x,y)$ So you need to solve following simple problem

\begin{aligned} & {\text{Minmize}} & & f(x,y) \\ & \text{subject to} & & x^2 + y^2 = \delta \\ & && x \geq0 ,~y\geq0 \end{aligned}

Now apply KKT!

or you could rewrite problem in one variable via $y = \sqrt{\delta - x^2}$ therefore problem becomes maximizing the one variable function, $g(x)=f(x,\sqrt{\delta - x^2})$ over the interval $[0, \sqrt{\delta}]$.