This paper proposes an efficient (semi-closed-form) zero-forcing (ZF) precoder design for the weighted sum-rate maximization problem under per-antenna power constraint (PAPC). Existing approaches for this problem are based on either interior-point methods that do not favor-ably scale with the problem size or subgradient methods that are widely known to converge slowly. To address these shortcomings, our proposed method is derived from three elements: minimax duality, alternating optimization (AO), and successive convex approximation (SCA). Specifically,the minimax duality is invoked to transform the considered problem into anequivalent minimax problem, for which we then recruit AO and SCA to finda saddle point, which enables us to take advantages of closed-form expressions and hence achieve fast convergence rate. Moreover, the complexity of the proposed method scales linearly with the number of users, compared to cubically for the standard interior-point methods. We provide an analytical proof for the convergence of the proposed method and numerical results to demonstrate its superior performance over existing approaches.Our proposed method offers a powerful tool to characterize the achievable rate region of ZF schemes under PAPC for massive multiple-input multiple-output