•Optimization (over continuous domains)
- Convex sets: definition, convex combination and convex hull. Cone, hyperplane, hyper-surfaces. Jensen inequality. Operations that preserve convexity.
- Convex Functions: Definition of convex function, strict and strong convexity. Examples. Theorems of convex functions. Convexity along all lines. First and second order characterization of convex functions.
- Convex Optimization: standard form, optimality, local optimality.
- Duality theory: Lagrangian, Lagrange dual function, weak and strong duality, geometric interpretation, complementary slackness, KKT conditions, constrain qualification (for strong duality)
- Convergence and uniqness of major (iterative) optimization algorithms
- Complexity of optimization algorithms (on example of linear programming)
- Euler-Lagrange equations, Legendre-Hadamard conditions, their role in classical mechanics, and connections with convexity, lack of convexity, oscillation phenomena and absence of minimizers.
- Examples of infinite dimensional optimization problems in physics.