To address the conflict between variables and multipliers caused by the difficulty of handling inequality-constrained Karush-Kuhn-Tucker (KKT) conditions in the solution of optimal power flow (OPF) problems using the holomorphic embedding method (HEM), a novel HEM-based OPF solution framework is proposed. In this framework, inequality multipliers are parameterized in squared form to prevent path failure caused by negative multiplier values, and the KKT variables are partitioned into power-flow physical variables and multipliers. The direct current OPF solution is employed as the preset initial values for a subset of variables, while the preset initial values of reactive-power are derived from reactive power balance relations. Furthermore, an adaptive penalty-based initialization strategy adjusts inequality multipliers to guide analytical continuation toward the solution. Compared with the central-trajectory interior-point method on a 5-bus and the IEEE 6/9/14/30-bus systems, the proposed HEM approach can converge to the OPF solution with only a small number of recursive iterations. The results show that the proposed method improves the accuracy and feasibility of OPF computation, offering an analytical approach for OPF problems under inequality constraints.