  We propose the minimally nonlinear irreversible heat engine as a new general
theoretical model to study the efficiency at the maximum power $\eta^*$ of heat
engines operating between the hot heat reservoir at the temperature $T_h$ and
the cold one at $T_c$ ($T_c \le T_h $). Our model is based on the extended
Onsager relations with a new nonlinear term meaning the power dissipation. In
this model, we show that $\eta^*$ is bounded from the upper side by a function
of the Carnot efficiency $\eta_C\equiv 1-T_c/T_h$ as $\eta^*\le
\eta_C/(2-\eta_C)$. We demonstrate the validity of our theory by showing that
the low-dissipation Carnot engine can easily be described by our theory.
