This chapter develops the optimization foundations of machine unlearning (MU). It begins with the design of forget objectives, covering formulations such as gradient difference (GradDiff), negative preference optimization (NPO), its simple variant SimNPO, and representation misdirection unlearning (RMU). These span output-level suppression and representation-level disruption, each with distinct trade-offs in stability, boundedness, and structural guarantees. Building on these objectives, the chapter introduces unlearning-aware optimizers, linking influence-function methods to second-order optimization. Classical influence-based unlearning is extended into an iterative second-order framework that accelerates convergence and stabilizes the forgetting—retaining trade-off. Finally, the chapter examines bi-level optimization (BLO) as a principled means of balancing forgetting with utility. By disentangling lower-level forget objectives from upper-level retain objectives, BLO achieves state-of-the-art performance, enabling effective unlearning without undermining general utility. Collectively, these advances establish a cohesive foundation for optimization-driven unlearning, transforming MU from an ad hoc practice into a systematic methodology for safe, reliable, and governable AI.