Explore quantum algorithms for optimization in this comprehensive lecture from the Quantum Colloquium series. Delve into the potential applications of quantum computers for solving optimization problems, including recent advancements in gradient descent and linear and semidefinite program solving. Examine both discrete and continuous optimization, discussing quantum speed-ups and their limitations. Investigate issues such as the requirement for large instance sizes to achieve quadratic quantum speedups and the need for quantum random-access memory (QRAM). Cover topics including graph sparsification, NP-hard optimization, and linear programs while gaining insights from Ronald de Wolf of QuSoft, CWI, and the University of Amsterdam.