In this paper, a mathematical programming with equilibrium constraints (MPEC) model for the optimal operation of unbalanced three-phase electrical distributed systems is presented. First, the problem is formulated as a non-linear programming (NLP) problem, considering photo-voltaic (PV) generation, dispatchable distributed generation (DG) units, and day-ahead minimization of total active power losses. Then, through a set of efficient linearization techniques, the unbalanced three-phase AC power flow is transformed into a linear programming (LP) model. Finally, through Karush-Kuhn-Tucker (KKT) conditions, the proposed MPEC model is formulated as the combination of the LP primal constraints, the LP dual constraints and complementary constraints. MPEC models are able to optimize without an explicit objective function. Thus, they could be used as inner constraints in bi-level problems, for formulating robust optimization models and in electricity market design for distribution systems. Results show that the proposed MPEC model obtains the same optimal solution of the primal LP problem, without an explicit objective function.