A predictive controller for controlling a system, the predictive controller comprising: at least one processor; and memory having instructions stored thereon that, when executed by the at least one processor, cause the predictive controller to: accept feedback signal including measurements of a state of the system; solve a mixed-integer optimal control optimization problem using branch-and-bound, B&B, optimization that searches for a global optimal solution within a search space to produce a control signal, wherein the B&B optimization iteratively partitions the search space into a nested tree of regions, and prunes at least one region from the nested tree of regions before finding a local optimal solution for each region when a dual objective value of a projection of a sub-optimal dual solution estimate for each region into a dual feasible space is greater than an upper bound or lesser than a lower bound of the global optimal solution maintained by the B&B optimization; and control the system based on the control signal to change the state of the system; wherein the nested tree of regions is formed by different convex relaxations of integer variables of the mixed-integer optimal control optimization problem, and wherein the local optimal solution within each region of the nested tree of regions is searched by an interior-point method over multiple iterations, such that the sub-optimal dual solution estimate for each region is produced by an intermediate iteration of the interior-point method; wherein the local optimal solution is based on an early termination procedure, wherein the early termination procedure allows the interior-point method to terminate its iterative procedure before the local optimal solution for the at least one region is found in case when: a projection step is performed after one or multiple iterations of the interior point method to compute a dual feasible solution estimate and the corresponding dual objective value is greater than the upper bound of the global optimal solution maintained by the B&B optimization; characterized in that, the projection step computes an update to the sub-optimal dual solution estimate by solving an equality-constrained optimization problem that enforces the dual feasibility constraints and minimizes a norm of the update for Lagrange multiplier variables. accept feedback signal including measurements of a state of the system; solve a mixed-integer optimal control optimization problem using branch-and-bound, B&B, optimization that searches for a global optimal solution within a search space to produce a control signal, wherein the B&B optimization iteratively partitions the search space into a nested tree of regions, and prunes at least one region from the nested tree of regions before finding a local optimal solution for each region when a dual objective value of a projection of a sub-optimal dual solution estimate for each region into a dual feasible space is greater than an upper bound or lesser than a lower bound of the global optimal solution maintained by the B&B optimization; and control the system based on the control signal to change the state of the system; wherein the nested tree of regions is formed by different convex relaxations of integer variables of the mixed-integer optimal control optimization problem, and wherein the local optimal solution within each region of the nested tree of regions is searched by an interior-point method over multiple iterations, such that the sub-optimal dual solution estimate for each region is produced by an intermediate iteration of the interior-point method; wherein the local optimal solution is based on an early termination procedure, wherein the early termination procedure allows the interior-point method to terminate its iterative procedure before the local optimal solution for the at least one region is found in case when: a projection step is performed after one or multiple iterations of the interior point method to compute a dual feasible solution estimate and the corresponding dual objective value is greater than the upper bound of the global optimal solution maintained by the B&B optimization; characterized in that, the projection step computes an update to the sub-optimal dual solution estimate by solving an equality-constrained optimization problem that enforces the dual feasibility constraints and minimizes a norm of the update for Lagrange multiplier variables. a projection step is performed after one or multiple iterations of the interior point method to compute a dual feasible solution estimate and the corresponding dual objective value is greater than the upper bound of the global optimal solution maintained by the B&B optimization; characterized in that, the projection step computes an update to the sub-optimal dual solution estimate by solving an equality-constrained optimization problem that enforces the dual feasibility constraints and minimizes a norm of the update for Lagrange multiplier variables.