This paper addresses the optimal conductor selection (OCS) problem in radial distribution networks, aiming to minimize the total costs associated with conductor investment and energy losses while ensuring voltage regulation and power balance as well as observing thermal limits. The problem is formulated as a mixed-integer nonlinear programming (MINLP) model and solved using a hybrid branch-and-bound (B&B), interior-point optimizer (IPO) approach within the Julia-based JuMP framework. Numerical validations on 27-, 33-, and 69-bus test feeders demonstrate cost-efficient conductor configurations. A multi-objective analysis is employed to construct the Pareto front, offering trade-offs between investment and operating costs. The impact of distributed energy resources (DERs) is also assessed, showing cost reductions when said resources provide reactive power support. The results confirm that the proposed MINLP approach outperforms conventional metaheuristics in terms of accuracy and reliability.