Fouling is a major source of energy inefficiencies that decreases the performance of process units. Fouling mitigation alternatives are required to ensure a sustainable, profitable, and safe operation. In heat exchanger networks, two mitigation alternatives are: flow distribution control, and periodical cleanings. This paper addresses the optimal control and optimal cleaning scheduling, individually and simultaneously, for heat exchangers in refining operations. The problem is formulated as a MINLP, which uses an accurate model of the process, logic disjunctions, and a continuous time representation. To solve it efficiently, it is reformulated as a mathematical program with complementarity constraints (MPCC). The modelling and solution strategies are demonstrated in several case studies, from small to moderately large networks for realistic applications. The formulation is versatile, and large network problems are solved in a reasonable computational time. The integration of optimal scheduling and control decreases the operational cost substantially relative to independent mitigation alternatives.