The class of optimization problems involving both continuous and discrete variables is known as mixed-integer programming (MIP), which emerges in many fields of applications. Due to their inherent combinatorial nature, solving such a class of problems in real-time poses a major challenge, especially in embedded applications where computational and memory resources are limited. This thesis mainly focuses on novel solution methods tailored to small-scale Mixed- Integer Quadratic Programming (MIQP) problems, such as those that typically arise in embedded hybrid Model Predictive Control (MPC) and estimation problems. With an emphasis on algorithm simplicity, efficient solution techniques to solve MIQP problems are developed in the thesis based on first-order methods, specialized to find both exact and approximate solutions. In addition, a numerically robust algorithm is proposed in order to tackle MIQP problem with positive semidefinite Hessian matrices, often encountered in hybrid MPC formulations. The proposed techniques, being library-free and relatively simple to code, are specifically tailored to real-time embedded applications. Such techniques are also employed in a novel algorithm for the MIPbased PieceWise Affine (PWA) regression, as well as in new approaches for energy disaggregation using binary quadratic programming that are particularly suitable for smart energy meters.