Nonlinear optimization
Cetraro, July 1-7, 2007

Global optimization

Prof. Immanuel M. Bomze, University of Vienna, Austria


The course will concentrate on a particular class of smooth constrained optimization which allows for enough flexibility to illustrate problems and methods of Global Optimization: standard quadratic optimization.
Moreover, the approach chosen enables students to easily establish connections to topics covered by the other lectures of this course as well: contact points with the presentation by V.Demyanov (Non-smooth optimization) can be found in parts 1 and 4 below; with that by R.Fletcher (Sequential Quadratic Programming) in parts 1 and 2; and with that by T.Terlaky (Interior Point Methods) in parts 3 and 4.

1. The simplest global optimization problems: quadratic optimization
1a. Local versus global optimality; complexity issues
1b. Extremal increments and optimality conditions
1c. Standard Quadratic Problems (StQPs)
1d. Some applications of StQPs
2. Some basic techniques, illustrated by StQPs
2a. Escape directions
2b. D.c. decompositions and bounding
2c. Branch-and-bound: principles and special cases
3. Reformulation, relaxation, approximation (again of StQPs)
3a. Unconstrained reformulations
3b. Convex conic reformulations
3c. Copositive programming with applications
3d. Complexity issues, revisited
4. Approaches to copositivity
4a. Role of copositivity in optimality conditions
4b. Copositivity detection
4c. SDP relaxations and bound hierarchies
4d. SDP relaxation bounds for StQPs, revisited: d.c. again