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