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How could one calculate the KKT conditions in the case of bound-constrained optimization?

In the general sense, given an objective function $J$ and design variables $x \in \mathcal{R}^n$, we consider the following optimization problem:

$\min\limits_{x\in \mathcal{R}^n} J(x)$

with the following bounds for $x_j$

$x_j -1 \le 0\\ -x_j \le 0$

The question is how we could express (theoretically) the KKT conditions.

boutsitron
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Oct 06 '22 at 14:14
  • Can you describe your optimization problem? I would guess that you are dealing with structural optimization for what you have written. Also, it looks like you are asking us to help you with your homework. If that's the case, please be explicit about it and show what you have tried. – nicoguaro Oct 06 '22 at 16:02
  • Excuse me for the short question, I tried to be as laconic and as general as possible, since it is a theoretical question and not my homework. That being said, @nicoguaro I am honestly sorry that I made you feel this way. Hopefully, now the question is more clearly formulated. I am not dealing with structural optimization but I don't think this is relevant to the question, which is more general. – boutsitron Oct 07 '22 at 15:17

1 Answers1

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For minimizing an objective with bound constraints

\begin{align*} \min_{x\in\mathbb{R}^{N}}J(x) \\ \text{such that } c_{i}(x) \geq 0,\,\,\,1\leq i\leq k \end{align*}

one can follow the Lagrangian formalism by first forming the Lagrangian functional

\begin{align*} \mathcal{L}(x,\lambda)=J(x)-\sum_{i=1}^{k}\lambda_{i}c_{i}(x) \end{align*}

The KKT conditions are composed of two equations, first is stationarity of the Lagrangian

\begin{align*} \nabla_{x}\mathcal{L}(x,\lambda)=\nabla_{x}J(x)-\sum_{i=1}^{k}\lambda_{i}\nabla_{x}c_{i}(x)=0. \end{align*}

The next equation is that a complementarity condition holds

\begin{align*} \lambda_{i}\geq 0,\,\,\,c_{i}(x)\geq0,\,\,\,\lambda_{i}c_{i}(x)=0,\,\,\,1\leq i\leq k. \end{align*}

For more details, I would suggest looking at the following text (https://link.springer.com/book/10.1007/978-0-387-40065-5).

Tucker
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