convex optimization (problem)

Question

Let us have the following task: $$\min f(x)$$ $$g_{i}(x)\le 0,\quad\forall i$$ $$h_{j}(x)= 0,\quad\forall j$$

What is the correct form of rewriting as a maximize problem? $$\max - f(x)$$ $$g_{i}(x)\le 0,\quad\forall i$$ $$h_{j}(x)= 0,\quad\forall j$$

or $$-\max - f(x)$$ $$g_{i}(x)\le 0,\quad\forall i$$ $$h_{j}(x)= 0,\quad\forall j$$ Thanks for your answer.

Answer 1

If $f$ is a convex function, you can write the maximization problem as:

$$ max -f(x) $$

Answer 2

$$\max f(x) = - \min - f(x)$$

If you have a black box algorithm that knows how to find minima, to find maximum:

1. Multiply the objective by $-1$.
2. Plug the modified problem into the black box
3. Multiply the resulting objective value by $-1$, to reverse the first step. No need to adjust the $x$ of the solution.