# Optimization (Methods)

Intro

- Optimization in Sciences and Engineering: Running Examples (fluid mechanics, bio-, energy systems)
- Examples of formulations, flavors: discrete-continuous, convex-non-convex, multi-objective, stochastic, chance-constrained, multi-level (e.g. games), function –> functionals (functions of functions)= calculus of variations
- What does solving optimization problem mean? Complexity. Exact vs Approximate. Analytic, automatic, numerical.

Convex Optimization (continuous)

- Examples – Linear, Quadratic, Geometric, Semidefinite
- Duality, change of variables, transformations and other tricks
- Methods of solutions (main ideas), how to use it in non-convex world (link to next section)
- Link to discrete optimization studies (in the “statistics” portion)

Non-convex Optimization

- find local solution, count local solutions, feasibility
- convexification/approximations (regularizations) – examples.
- change of variables … other tricks, examples

More complex optimization problems

- Robust and Stochastic Optimizations - formulations, methods of solutions
- Optimization of functionals = Calculus of variations. Example: optimization over paths.

Bonus

- Lyapunov function and stability: conditions of stability, examples of Lyapunov functions , more on dynamical systems, Hamiltonian systems, ODEs, linear systems, linearization
- Optimal Control. Bellman-Hamilton-Jacobi. Dynamic Programming. Linear-Quadratic-Gaussian control problems.