This work investigates the application of a Model Order Reduction technique based on the Cauer ladder network method to the air cored solenoid geometry of the number 35 benchmark problem from the Testing electromagnetic analysis methods series, approaching both source and discrete radius optimization tasks. A projection-based reduced-order model was developed to accelerate the optimization process, achieving an average computational speedup factor of 35 compared to full-order finite element simulations. For discrete radius optimization, the problem was reformulated as a source optimization, enabling effective model order reduction application. The multiport Cauer ladder network approach reduced to a superposition of magnetic fields due to the simplicity of the problem. The reduced order model preserved consistency with the full order model when a small enough element size was chosen in the finite element model during its creation process. The error of the proposed methodology is an order of magnitude lower than the lowest objective function values in the optima, therefore it is suitable for multi-objective optimization.