Optimization (Theory)

•Optimization (over continuous domains)

  • Convexity
    • 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)

Functional optimization:

  • 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.