•Optimization (over continuous domains)
 Convexity
 Convex sets: definition, convex combination and convex hull. Cone, hyperplane, hypersurfaces. 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)
Functional optimization:
 EulerLagrange equations, LegendreHadamard 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.
