An optimal sensor placement is defined as a sensor configuration that achieves the minimum capital cost while observing prespecified performance criteria. Previous formulations of this problem have resulted in the definition of a mixed-integer nonlinear program (MINLP) with dimensions dependent on the value of the integer decision variables. The main contribution of this work is an equivalent reformulation of the design problem such that the dimension of the NLP is independent of all decision variables. Additionally, the traditional sensor-placement problem, based on static process conditions, is extended to linear dynamic processes. The final contribution is the exact conversion of the general NLP into a convex program through the use of linear matrix inequalities. The aggregation of these results show that the sensor-placement problem can be solved globally and efficiently using standard interior-point and branch-and-bound search algorithms.