Optimization problems are relevant in solving many practical problems. Optimization studies the methods for choosing optimal parameters in various problems arising in applications. For example, the task of building an optimal route in order to get from one point of the city to another in the shortest time, which is solved by the navigator in a smartphone. The tasks of scientists in the field of optimization are to develop software and the methods themselves, which are later applied in practice, for example, in production.
Optimization tasks are divided into different classes that reflect their features. Special solution methods are developed for each class. Convex optimization problems are the most studied. For convex optimization problems, many very effective methods and software packages have been developed that implement these methods on a computer. Non-convex optimization problems are more complicated.
One of the effective approaches to non-convex optimization problems is DC optimization (from the English Difference of Convex functions) — optimization of the difference of convex functions.
The main idea of this approach lies in the fact that, since it is possible to solve convex problems efficiently and quickly, it is worth trying to apply convex optimization methods to non-convex problems in some way. There are different ways to do this. As part of DC optimization, this is done by reducing a non-convex problem to some specially constructed set of convex problems. Earlier, Maxim Dolgopolik, an IPMash RAS leading researcher, proved the possibility of applying DC optimization methods to solving some classes of such problems.
«In the new work, we are talking about the extension of the classical DCA method (from the English Difference of Convex functions Algorithm) to the case of problems with matrix constraints (in particular, problems of non-smooth optimization on matrix manifolds). The article is devoted to a detailed analysis of this method for such tasks. It also provides examples of using this method in solving several applied problems, which demonstrated the high efficiency of the proposed method.
The new article shows how, with the help of the theoretical apparatus developed in the first part of the study, it is possible to build an applied numerical method (algorithm) and use it to solve applied problems on a computer,» Maxim Dolgopolik said.
A new numerical method (algorithm) for solving some classes of complex non-convex optimization problems, namely, non-smooth semidefinite programming problems and non-smooth optimization problems on matrix manifolds, is proposed in the scientific work. In the course of the work, the software implementation of the developed algorithm was realized and numerical experiments were carried out on a computer, which showed the high efficiency of the developed method.