A simple yet effective optimization technique is developed to solve nonlinear conjugate heat transfer. The proposed Nonlinear Optimization with Replacement Strategy (NORS) is a mutation of several existing optimization processes. With the improvements of 3D metal printing of turbine components, it is feasible to have film holes with unconventional diameters, as these holes are created while printing the component. This paper seeks to optimize each film hole diameter at the leading edge of a turbine vane to satisfy several optimum thermal design objectives with given design constraints. The design technique developed uses linear regression-based machine learning model and further optimizes with strategic improvement of the training dataset. Optimization needs cost and benefit criteria are used to base its decision of success, and cost is minimized with maximum benefit within given constraints. This study minimizes the coolant flow (cost) while satisfying the constraints on average metal temperature and metal temperature variations (benefits) that limit the useful life of turbine components. The proposed NORS methodology provides a scientific basis for selecting design parameters in a nonlinear design space. This model is also a potential academic tool to be used in thesis works without demanding extensive computing resources.