This paper investigates the problem of optimally locating passive vibration isolators to minimize unwanted vibration caused by exogenous disturbance forces. The stiffness and damping parameters of the isolators are assumed to be known, leaving the isolator locations, which are nonlinearly related to system states, as unknown optimization variables. An approach for reformulating the nonlinear isolator placement problem as a linear time-invariant (LTI) feedback control problem, by linking fictitious control forces to fictitious measured outputs using a nonzero feedforward term, is proposed. Accordingly, the isolator locations show up within a static output feedback gain matrix which can be optimized, using methods from optimal control theory, to minimize the H2 and/or H∞ norms of transfer functions representing unwanted vibration. The proposed framework also allows well-established LTI control theories to be applied to the analyses of the optimal isolator placement problem and its results. The merits of the proposed approach are demonstrated using single and multivariable case studies related to isolator placement in precision manufacturing machines. However, the framework is applicable to optimal placement of passive isolators, suspensions, or dampers in automotive, aerospace, civil, and other applications.