Information density and its exponential form, known as lift, play a central role in information privacy leakage measures. $α$-lift is the power-mean of lift, which is tunable between the worst-case measure max-lift ($α=\infty$) and more relaxed versions ($α<\infty$). This paper investigates the optimization problem of the privacy-utility tradeoff (PUT) where $α$-lift and mutual information are privacy and utility measures, respectively. Due to the nonlinear nature of $α$-lift for $α<\infty$, finding the optimal solution is challenging. Therefore, we propose a heuristic algorithm to estimate the optimal utility for each value of $α$, inspired by the optimal solution for $α=\infty$ and the convexity of $α$-lift with respect to the lift, which we prove. The numerical results show the efficacy of the algorithm and indicate the effective range of $α$ and privacy budget $\varepsilon$ with good PUT performance.