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Explore computational optimization methods in this first lecture of a comprehensive course series taught by Professor Di Liu at the Instituto de Matemática Pura e Aplicada. Learn fundamental concepts in one-dimensional optimization methods and gain an introduction to unconstrained optimization techniques including descent methods, linear search, gradient methods, Newton's method, quasi-Newton methods, and conjugate direction methods. Discover convergence globalization strategies and delve into constrained optimization approaches such as projected gradient methods, feasible direction methods, external penalization, internal penalization, augmented Lagrangians, and sequential quadratic programming. Examine non-differentiable optimization methods including subgradient methods, cutting plane methods, and bundle methods. The course follows established references including works by Bertsekas, Bonnans, Dennis & Schnabel, and Izmailov & Solodov, providing a solid theoretical and practical foundation in numerical optimization techniques essential for computational mathematics and applied sciences.
