As a variant of the Area Under the ROC Curve (AUC), the partial AUC (PAUC) focuses on a specific range of false positive rate (FPR) and/or true positive rate (TPR) in the ROC curve. It is a pivotal evaluation metric in real-world scenarios with both class imbalance and decision constraints. However, selecting instances within these constrained intervals during its calculation is NP-hard, and thus typically requires approximation techniques for practical resolution. Despite the progress made in PAUC optimization over the last few years, most existing methods still suffer from uncontrollable approximation errors or a limited scalability when optimizing the approximate PAUC objectives. In this paper, we close the approximation gap of PAUC optimization by presenting two simple instance-wise minimax reformulations: one with an asymptotically vanishing gap, the other with the unbiasedness at the cost of more variables. Our key idea is to first establish an equivalent instance-wise problem to lower the time complexity, simplify the complicated sample selection procedure by threshold learning, and then apply different smoothing techniques. Equipped with an efficient solver, the resulting algorithms enjoy a linear per-iteration computational complexity w.r.t. the sample size and a convergence rate of $O(ε^{-1/3})$ for typical one-way and two-way PAUCs. Moreover, we provide a tight generalization bound of our minimax reformulations. The result explicitly demonstrates the impact of the TPR/FPR constraints $α$/$β$ on the generalization and exhibits a sharp order of $\tilde{O}(α^{-1}\n_+^{-1} + β^{-1}\n_-^{-1})$. Finally, extensive experiments on several benchmark datasets validate the strength of our proposed methods.