Solving a non-linear program $\min x + y^p$

Question

Can anyone tell me how I can solve the following NLP for fixed p > 1:

$$ \min x + y^p \\ st \ x+y=1 \\ \ x,y \ge 0 $$

Thanks!

I tried using KKT theorem, but it seems this program has no Slater points. Perhaps I am overthinking this?

Answer 1

The question is equivalent to

$$\min x+(1-x)^p$$

subject to $$0 \le x \le 1.$$

Let $$f(x)=x+(1-x)^p$$

Notice that when $p=1$, $f(x)=1$, hence the optimal solution set is just $[0,1]$.

Now, we focus on $p>1$,

$$f'(x)=1-p(1-x)^{p-1}$$

$$f'(0)=1-p<0$$

and $$f'(1)=1>0$$

The solution is in the interior, try to equate $f'(x)$ to $0$ and solve for x.

Solve for $$1-p(1-x)^{p-1}=0$$

I will leave that as an exercise.