1. Speaker: Prof. Andrei Dmitruk, CEMI RAS and MSU, Russia
Title: Lagrange Multipliers Rule for a General Extremum Problem with an Infinite Number of Constraints
We consider a general optimization problem with equality and inequality constraints in a Banach space. The first is given by a level set of a nonlinear operator into another Banach space, and the latter by inclusions of images of smooth operators into closed convex sets (possibly cones) with nonempty interiors lying in some other Banach spaces. This statement covers a wide range of optimization problems both in pure mathematics and in applications. Some of its particular cases were considered earlier by many authors. We prove a first-order necessary optimality condition in the form of Lagrange multipliers rule, where the multipliers at the inequality constraints are elements of the normal cones at the corresponding points of these sets. This form is transparent for learning and convenient for application. The proof is self-contained, it uses basic facts of functional analysis and follows the line of Dubovitskii— Milyutin approach. As an application of the result, we consider an optimal control problem with state constraints, in which we obtain necessary conditions for a weak minimum.
This is joint work with Nikolai Osmolovskii.
Published in "Recent Advances of the Russian Operations Research Society" (F.Aleskerov and A.Vasin eds.), Cambridge Scholars Publishing, 2020, p. 212—232. ISBN-13: 978-1-5275-4792-6.
2. Speaker: Prof. Nikolai Osmolovskii, Systems Research Institute,
Polish Academy of Sciences, Warsaw, Poland
Title: Quadratic optimality conditions for broken extremals and discontinuous
The talk is devoted to second-order conditions for broken extremals in variational calculus problems and for discontinuous controls in optimal control problems. A characteristic feature of the conditions under discussion is the absence of a gap between necessary and sufficient conditions. The conditions are formulated as sign-definiteness of a quadratic form on the so-called critical cone. In the first part of the talk, quadratic conditions for broken extremals are formulated in the simplest problem of the calculus of variations. In the second, we consider the optimal control problem with regular mixed constraints on the state variable and control, and the quadratic conditions for a strong local minimum are formulated for it in the case of piecewise continuous control.
3. Speaker: Prof. Panos M. Pardalos, University of Florida, USA
Title: Sustainable interdependent networks
Sustainable interdependent networks have a wide spectrum of applications in computer science, electrical engineering, and smart infrastructures. We are going to discuss the next generation sustainability framework as well as smart cities with special emphasis on energy, communication, data analytics and financial networks. In addition, we will discuss solutions regarding performance and security challenges of developing interdependent networks in terms of networked control systems, scalable computation platforms, and dynamic social networks.
Amini, M.H., Boroojeni, K.G., Iyengar, S.S., Pardalos, P., Blaabjerg, F., Madni, A.M. (Eds.), "Sustainable Interdependent Networks: From Theory to Application," Springer (2018)
Amini, M.H., Boroojeni, K.G., Iyengar, S.S., Pardalos, P., Blaabjerg, F., Madni, A.M. (Eds.), "Sustainable Interdependent Networks: From Smart
Power Grids to Intelligent Transportation Networks," Springer (2019)
Rassia, Stamatina Th., Pardalos, Panos M. (Eds.) , "Smart City Networks: Through the Internet of Things," Springer (2017)
Kalyagin, Valery A., Pardalos, Panos M., Rassias, Themistocles M. (Eds.),
"Network Models in Economics and Finance," Springer (2014).
Laura Carpi, Tiago Schieber, Panos M. Pardalos, Gemma Marfany, Cristina Masoller, Albert Diaz-Guilera, and Martín Ravetti, "Assessing diversity in multiplex networks," Nature Scientific Reports (2019).
Tiago A. Schieber, Laura Carpi, Albert Dı´az-Guilera, Panos M. Pardalos, Cristina Masoller & Martı´n G. Ravetti, "Quantification of network structural dissimilarities," Nature Communications 8, Article number: 13928 (2017)
4. Speaker: Prof. Boris T. Polyak, Institute for Control Science, Moscow, Russia
Title: Static feedback in linear control systems as optimization problem
The linear quadratic regulator is the fundamental problem of optimal control. Its state feedback version was set and solved in the early 1960s. However, the static output feedback problem has no explicit-form solution. It is suggested to look at both of them from another point of view as a matrix optimization problem, where the variable is a feedback matrix gain. The properties of such a function are investigated, it turns out to be non-convex, with the possible non-connected domain. Moreover, it is not L-smooth on the entire domain but has this property on sublevel sets. Nevertheless, a specially adopted gradient method for its minimization converges to the optimal solution in the state feedback case and to a stationary point in the output feedback case. The results can be extended for the general framework of the reduced gradient method for optimization with equality-type constraints. Directions for future research are addressed.
This is joint work with Ilyas Fatkhullin.
5. Speaker: Prof. Alexey Tret'yakov, Siedlce University of Natural Sciences and Humanities, Poland
Title: P-regularity Theory: Applications to Optimization
We present recent advances in the analysis of nonlinear structures and their applications to nonlinear optimization problems with constraints given by nonregular mappings or other singularities obtained within the framework of the p-regularity theory developed over the last twenty years. In particular, we address the problem of the description of the tangent cone to the solution set of the operator equation, optimality conditions, and solution methods for optimization problems.
This is joint work with Yuri Evtushenko and Vlasta Malkova. The investigation was supported by the Russian Foundation for Basic Research (project no. 17-07-00510).