  We study the non-equilibrium thermodynamics of a heat engine operating
between two finite-sized reservoirs with well-defined temperatures. Within the
linear response regime, it is found that the uniform temperature of the two
reservoirs at final time $\tau$ is bounded from below by the entropy production
$\sigma_{\mathrm{min}}\propto1/\tau$. We discover a general power-efficiency
trade-off depending on the ratio of heat capacities ($\gamma$) of the
reservoirs for the engine. And a universal efficiency at maximum average power
of the engine for arbitrary $\gamma$ is obtained. For practical purposes, the
operation protocol of an ideal gas heat engine to achieve the optimal
performance associated with $\sigma_{\mathrm{min}}$ is demonstrated. Our
findings can be used to develop an general optimization scenario for
thermodynamic cycles with finite-sized reservoirs in real-world circumstances.
