Optimal and Robust Control

Knowledge (I remember and understand)

1. Give a rigorous definition of a local minimum (and maximum) and explain how it differs from a global minimum (maximum).
2. Define convex function, convex set and convex optimization problem and explain the impact the convexity has on the optimization.
3. State the Weierstrass (extreme value) theorem on existence of minimum (maximum).
4. Explain the concepts of big O() and little o() within the framework of truncated Taylor series.
5. Give first-order necessary conditions of optimality for a scalar function of a scalar argument and define critical (or stationary) point. Extend these to the vector argument case. Formulate them both using a Fréchet and Gateaux derivatives. Specialize the result to quadratic functions.
6. Give second-order sufficient conditions of optimality for a scalar function of a scalar argument. How can we distinguish between a minimum, maximum, and an inflection point? Extend these to the vector case. Define Hessian and show how it can be used to classify the critical points into minimum, maximum, saddle point, and singularity point. Specialize the results to quadratic functions.
7. Give first-order necessary condition of optimality for an equality-constrained optimization problem using Lagrange multipliers. Specialize the results to quadratic cost functions and linear constraints.
8. Characterize the regular point (for a given set of equality constraints). Give an example of equality constraints lacking regularity.
9. Give second-order sufficient conditions of optimality for an equality-constrained optimization problem using the concept of a projected Hessian.
10. State and explain the Karush-Kuhn-Tucker (KKT) conditions for inequality-constrained optimization problems.

Skills (I can use the knowledge to solve a problem)

1. Formulate a provided problem as an instance of mathematical optimization: identify the cost function, the constraints, decide if the problems fits into one of the (numerous) families of optimization problems such as linear program, quadratic program (with linear constraints, with quadratic constraints), (general) nonlinear program, ...
2. Solve a provided linear and/or quadratic programming problem using a solver of your choice.