Text over the overbrace beyond "scope"

You can achieve directly what you want withmathtools' \mathclap. I also provide another variant that uses amsmaths \substack macro that gives you a multilined over-brace text (note also the \hfill to right-align the first line).

I also added a pair of braces {} between the + and the column separater & to get the right spacing around the plus sign. (This would be also necessary in \substack with the plus signs at the start of the lines, but since we're in script-style no horizontal space is added anyway.)

I get overfull boxes, by the way, so maybe you should split the equation onto different lines (see below).

Code

\documentclass[pdftex,a4paper]{scrartcl}
\usepackage[latin1]{inputenc}
\usepackage[left=2.5cm,top=2.5cm]{geometry}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{bbm}

\begin{document}
\begin{align*}
\min \biggl(
    \omega_1 \cdot \sum_{i \in \mathcal{F}  } \sum_{c    \in \mathcal{C}    }
                 \sum_{e \in \mathcal{E}_c} \sum_{\tau \in \mathcal{S}^i_e}
                 \mathbf{w}^i_{c,e,\tau} \cdot x_{i,c,e,\tau} + {} &
  \omega_2 \cdot  \sum_{t \in \mathcal{T}} \sum_{c \in \mathcal{C}}
                 \overbrace{\mathbbm{1}_{\{ u_{t, c} \geq 0.8 \cdot \kappa^{\text{cap}}_c \}} (t,c) }^
                   {\mathclap{ \eta_{0.7} \mathbbm{1}_{\{ u_{t, c} \geq 0.7 \cdot \kappa^{\text{cap}}_c\}} (t,c)
                             + \eta_{0.8} \mathbbm{1}_{\{ u_{t, c} \geq 0.8 \cdot \kappa^{\text{cap}}_c\}} (t,c)
                             + \eta_{0.9} \mathbbm{1}_{\{ u_{t, c} \geq 0.9 \cdot \kappa^{\text{cap}}_c\}} (t,c)}
                   }
                 + \omega_3 \cdot \frac{1}{2}\sum\limits_{t \in \mathcal{T}} \sum_{c \in \mathcal{C}} n_{t,c}
     \biggr) \\
  & \text{with} \quad \omega_i \in \left[0,1\right], \sum\nolimits_{i=1}^3\omega_i = 1
\end{align*}

\begin{align*}
\min \biggl(
    \omega_1 \cdot \sum_{i \in \mathcal{F}  } \sum_{c    \in \mathcal{C}    }
                 \sum_{e \in \mathcal{E}_c} \sum_{\tau \in \mathcal{S}^i_e}
                 \mathbf{w}^i_{c,e,\tau} \cdot x_{i,c,e,\tau} + {} &
  \omega_2 \cdot  \sum_{t \in \mathcal{T}} \sum_{c \in \mathcal{C}}
                 \overbrace{\mathbbm{1}_{\{ u_{t, c} \geq 0.8 \cdot \kappa^{\text{cap}}_c \}} (t,c) }^
                   {\mathclap{\substack{ \hfill\eta_{0.7} \mathbbm{1}_{\{ u_{t, c} \geq 0.7 \cdot \kappa^{\text{cap}}_c\}} (t,c) \\
                             + \eta_{0.8} \mathbbm{1}_{\{ u_{t, c} \geq 0.8 \cdot \kappa^{\text{cap}}_c\}} (t,c) \\
                             + \eta_{0.9} \mathbbm{1}_{\{ u_{t, c} \geq 0.9 \cdot \kappa^{\text{cap}}_c\}} (t,c)}
                   }}
                 + \omega_3 \cdot \frac{1}{2}\sum\limits_{t \in \mathcal{T}} \sum_{c \in \mathcal{C}} n_{t,c}
     \biggr) \\
  & \text{with} \quad \omega_i \in \left[0,1\right], \sum\nolimits_{i=1}^3\omega_i = 1
\end{align*}
\end{document}

Output

No overfull boxes

\begin{multline*}
\min \biggl(
    \omega_1 \cdot \sum_{i \in \mathcal{F}  } \sum_{c    \in \mathcal{C}    }
                 \sum_{e \in \mathcal{E}_c} \sum_{\tau \in \mathcal{S}^i_e}
                 \mathbf{w}^i_{c,e,\tau} \cdot x_{i,c,e,\tau} \\ {} +
  \omega_2 \cdot  \sum_{t \in \mathcal{T}} \sum_{c \in \mathcal{C}}
                 \overbrace{\mathbbm{1}_{\{ u_{t, c} \geq 0.8 \cdot \kappa^{\text{cap}}_c \}} (t,c) }^
                   {\mathclap{ \eta_{0.7} \mathbbm{1}_{\{ u_{t, c} \geq 0.7 \cdot \kappa^{\text{cap}}_c\}} (t,c)
                             + \eta_{0.8} \mathbbm{1}_{\{ u_{t, c} \geq 0.8 \cdot \kappa^{\text{cap}}_c\}} (t,c)
                             + \eta_{0.9} \mathbbm{1}_{\{ u_{t, c} \geq 0.9 \cdot \kappa^{\text{cap}}_c\}} (t,c)}
                   }
                 + \omega_3 \cdot \frac{1}{2}\sum\limits_{t \in \mathcal{T}} \sum_{c \in \mathcal{C}} n_{t,c}
     \biggr) \\
  \text{with} \quad \omega_i \in \left[0,1\right], \sum\nolimits_{i=1}^3\omega_i = 1
\end{multline*}

Here are some additional suggestions regarding your current typesetting:

1. align's alignment should use a & <sym> style rather than <sym> &. Taking yours as an example, do not use ... + & ... but instead use ... & + .... Note the difference:

   

   \documentclass{article}
   \usepackage{amsmath}% http://ctan.org/pkg/amsmath
   \begin{document}
   \begin{align*}
     f(x) = ax^2 + bx +& c \\ % Wrong use of &
            c &+ bx + ax^2 = g(x) % Correct use of &
   \end{align*}
   \end{document}
   

2. For large \overbrace descriptions, it may be better to use \overbrace{<stuff>}^{<sym>} and then define <sym> somewhere else. For example,

   

   \documentclass{article}
   \usepackage{amsmath,bbm}% http://ctan.org/pkg/{amsmath,bbm}
   \begin{document}
   \begin{multline*}
     \min \biggl(\omega_1 \cdot \sum\limits_{i\in\mathcal{F}}\sum\limits_{c\in\mathcal{C}} 
     \sum\limits_{e\in\mathcal{E}_c}\sum\limits_{\tau\in\mathcal{S}^i_{e}}
     \mathbf{w}^i_{c,e,\tau} \cdot x_{i,c,e,\tau} + \omega_2 \cdot  \sum\limits_{t \in 
     \mathcal{T}}\sum\limits_{c \in \mathcal{C}}\overbrace{\mathbbm{1}_{\{ u_{t, c} \geq 0.8
     \cdot \kappa^{\text{cap}}_c \}}(t,c)}^{\alpha} \\
     \hspace{5em}{} + \omega_3 \cdot \frac{1}{2}\sum\limits_{t \in \mathcal{T}}
     \sum\limits_{c \in \mathcal{C}}n_{t,c} \biggr)
   \end{multline*}
   \vspace*{-\baselineskip}
   \begin{align*}
     \text{where\quad} \phantom{\textstyle\sum_{i=1}^3}\omega_i & \in \left[0,1\right], \\
     \textstyle\sum_{i=1}^3\omega_i & = 1, \\
     \alpha & =\eta_{0.7}\mathbbm{1}_{\{ u_{t, c} \geq 0.7 \cdot \kappa^{\text{cap}}_c\}}(t,c) + 
       \eta_{0.8}\mathbbm{1}_{\{ u_{t, c} \geq 0.8 \cdot \kappa^{\text{cap}}_c\}}(t,c) \\
       & \phantom{{}={}} \eta_{0.9}\mathbbm{1}_{\{ u_{t, c} \geq 0.9 \cdot \kappa^{\text{cap}}_c\}}(t,c)
   \end{align*}
   \end{document}
   

3. Since you only have ω₁, ω₂ and ω₃, it may be better to write this out in your where clause, rather than using sum notation over 3 elements.