The increasing penetration of renewable energy sources (RESs) brings great challenges to the frequency security of power systems. The traditional frequency-constrained unit commitment (FCUC) analyzes frequency by simplifying the average system frequency and ignoring numerous induction machines (IMs) in load, which may underestimate the risk and increase the operational cost. In this paper, we consider a multi-area frequency response (MAFR) model to capture the frequency dynamics in the unit scheduling problem, in which regional frequency security and the inertia of IM load are modeled with high-dimension differential algebraic equations. A multi-area FCUC (MFCUC) is formulated as mixed-integer nonlinear programming (MINLP) on the basis of the MAFR model. Then, we develop a multi-direction decomposition algorithm to solve the MFCUC efficiently. The original MINLP is decomposed into a master problem and subproblems. The subproblems check the nonlinear frequency dynamics and generate linear optimization cuts for the master problem to improve the frequency security in its optimal solution. Case studies on the modified IEEE 39-bus system and IEEE 118-bus system show a great reduction in operational costs. Moreover, simulation results verify the ability of the proposed MAFR model to reflect regional frequency security and the available inertia of IMs in unit scheduling.