Saturday, 11:10 - 12:50 - Room 105

*Stream: Linear and nonlinear optimization*

Chair:

- On the real Jacobian Conjecture and Newton polytopes

We discuss the relationship between the global diffeomorphism property of polynomial maps and the non-vanishing determinant of the corresponding Jacobian matrix by analysing the coercivity property of some specific sum of squares polynomials via their Newton polytopes.

- On stability of nonlinear constraint systems and applications

We consider a nonlinear constraint system depending on some parameter. It is well known that Robinson's CQ is sufficient for a certain stable behavior of the solution map of the constraint system when the parameter varies. In terms of modern variational analysis Robinson's CQ is equivalent to metric regularity of the multifunction associated with the constraint system at some reference parameter, whereas the implied stability property can be interpreted as some kind of uniform metric subregularity. In this talk we analyze this stability property from the viewpoint of metric subregulariy.

- Application of p-regularity theory to nonlinear optimization problems.

We present the main concept and results of the p-regularity theory (also known as p-factor analysis of nonlinear mappings) applied to nonlinear optimization problems. This approach is based on the construction of p-factor operator. The main result of this theory gives a detailed description of the structure of the zero set of an irregular nonlinear mappings. Applications include a new numerical methods for solving nonlinear optimization problems and p-order necessary and sufficient optimality conditions. We substantiate the rate of convergence of p-factor method.

- Partial Order Relations for Set Optimization

We define new order relations on family of sets by using Pontryagin (Minkowski) difference in order to construct a model for set optimization. Firstly, we investigate relationships between these order relations and well known upper and lower set less order relations. We obtain some properties of these order relations based on the properties of the cone. Furthermore, we show that, depending on the cone, these order relations are partial order on the family of nonempty, closed, convex and bounded sets. Also, we examine minimal and maximal sets of a family with respect to these partial orders.

- Optimization and Control Reenters under Stochastic Uncertainty, Jumps, Regime Switches and Paradigms Shifts

We introduce hybrid stochastic differential equations with jumps and to its optimal control. These hybrid systems allow for the representation of "random" and impulsive regime switches or paradigm shifts in economies and societies, and they are of growing importance in the areas of finance, science, development and engineering. We present special approaches to this stochastic optimal control: one is based on the finding of optimality conditions and closed-form solutions. We further discuss aspects of information asymmetries, given by delay or insider information.