I am working with the following optimization problem: $$ \min_{\Delta} \boldsymbol{\theta}^T\boldsymbol{\Delta} \\ \text{Such that:} ~~~0 \leq \mu_i + \Delta_i \leq 1 ~~\forall~~ i\in\{1,2,\ldots, n\} \\ \text{and}~~ \boldsymbol{\Delta}^T\boldsymbol{\Delta} \leq \rho $$ where $\boldsymbol{\theta}\in \mathbb{R}^n$ and $\mu_i$ is a scalar between 0 and 1.
I understand that the KKT conditions can help to give an optimality condition for optimization problems. I wanted to see if that could be applied here. I tried to write the KKT conditions, and I obtained the following: $$ \boldsymbol{\theta} + 2\lambda \boldsymbol{\Delta} + \boldsymbol{\alpha} - \boldsymbol{\beta}=\mathbf{0} \\ \alpha_i (\Delta_i + \mu_i - 1) = 0 ~~\forall~~ i \in \{1,2,\ldots,n\}\\ \beta_i (-\Delta_i - \mu_i) = 0 ~~\forall~~ i \in \{1,2,\ldots,n\}\\ \lambda(\boldsymbol{\Delta}^T\boldsymbol{\Delta} - \rho) = 0 $$ My understanding is that if I can find a $\boldsymbol{\Delta}$ that satisfies these equations for some $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ then that $\boldsymbol{\Delta}$ must be an optimal solution. My problem is that I am not sure how to proceed at this point. Can someone help push me in the right direction?