With advances in additive manufacturing of metal components, commercial production of complex turbine components is becoming feasible. Thus, designers are not constrained to the limitations of conventional manufacturing methods. A new conjugate optimization technique is proposed, which is not computationally demanding and can be used when several heat transfer modes are working simultaneously. For this study, film cooling holes in the leading edge of a gas turbine airfoil are optimized without trial and error simulations. Since the machine learning technique is not dependent on thermal analysis, the optimization technique can be applied to any nonlinear problem. Film hole sizes are optimized to minimize coolant flow rate while reducing the temperature variations in the stationary vane. The technique used a transfer function based iterative optimization process with unsupervised machine learning that has been termed nonlinear optimization with replacement strategy (NORS). It uses a grading metric to replace the worst performing hole combinations with one that has been optimized with a given objective and several constraints. Optimized results show significant reductions in vertical temperature variations along the leading edge while minimizing coolant flow rate. Reduced temperature variation results in reduced thermal stresses. The finite element (FE) model and the associated correlations are not part of the unsupervised machine learning technique; therefore, the proposed optimization model can be generalized for any engineering design with multiple inputs for learning and multiple outputs for grading.