Lecture 1: Lipschitz Global Optimization; 

Lecture 2: Infinity Computing.

In the first lecture, the global optimization problem of a multidimensional function satisfying the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant is considered. It is supposed that the objective function can be “black box”, multiextremal, and non-differentiable. It is also assumed that evaluation of the objective function at a point is a time-consuming operation. 

Several adaptive partition methods and strategies for estimating the Lipschitz constant are analyzed. The main attention is dedicated to two types of algorithms. The first of them is based on using space-filling curves in global optimization. A family of derivative-free numerical algorithms applying space-filling curves to reduce the dimensionality of the global optimization problem is discussed. A number of unconventional ideas, such as adaptive strategies for estimating Lipschitz constant, balancing global and local information to accelerate the search, etc. are presented. Diagonal global optimization algorithms is the second type of methods under consideration. They have a number of attractive theoretical properties and have proved to be efficient in solving applied problems.  

The second lecture presents a recently introduced methodology allowing one to execute numerical computations with finite, infinite, and infinitesimal numbers on a new type of a computer – the Infinity Computer patented in USA and Europe. The new approach is based on the principle ‘The whole is greater than the part’ (Euclid’s Common Notion 5) that is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The new computational methodology is not related to the non-standard analysis and gives the possibility to execute computations of a new type simplifying fields of Mathematics where the usage of infinity and/or infinitesimals is required (e.g., divergent series, limits, derivatives, integrals, measure theory, probability theory, optimization, fractals, etc.). The main attention in the lecture is dedicated to the explanation of how practical numerical computations with infinities and infinitesimals can be executed. Examples regarding optimization, numerical differentiation, and ODEs are given. The Infinity Calculator using the Infinity Computer technology is presented during the talk.