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# Department of Operations Research

Head of the department: Krasnoshchekov Pavel, Academician of RAS, Professor, Dr.Sc.

Operations research is the science of purposeful behavior which includes the development, study and application of mathematical models in the optimal decision making. The Department is conducting research in traditional areas of the decision-making theory, such as optimization theory and numerical methods, game theory, multicriteria optimization. Results of this research find applications in mathematical modeling of economical systems, sociology, biology, computer-aided design, and ecology.

In 1994 the Department started a new specialization in Actuarial Mathematics. The mathematical models of contemporary market economies, banking and insurance activities, and investment policies are also investigated. The Department cooperates with institutions of the Russian Academy of Sciences, pension funds, and financial institutions.

### Staff Members:

- Evtushenko Yury, Academician of RAS, Professor, Dr.Sc.
- Savin Gennady, Academician of RAS, Professor, Dr.Sc.
- Pospelov Igor, Corresponding Member of RAS, Professor, Dr.Sc.
- Flerov Yury, Corresponding Member of RAS, Professor, Dr.Sc.
- Belolipetsky Alexander, Academician of RANS, Professor, Dr.Sc.
- Vasin Alexander, Academician of RANS, Professor, Dr.Sc., Deputy head of the Department
- Izmailov Aleksey, Professor, Dr.Sc.
- Novikova Natalya, Professor, Dr.Sc.
- Belyankin Georgy, Associate Professor, PhD
- Davidson Michael, Associate Professor, PhD
- Denisov Dmitry, Associate Professor, PhD
- Morozov Vladimir, Associate Professor, PhD, Deputy Dean for professional development
- Pospelova Irina, Associate Professor, PhD
- Fourougian Meran, Associate Professor, PhD
- Reshetov Valery, Associate Professor, PhD
- Belyankina Tatiana, Senior Research Fellow, PhD
- Vrzheshch Valentin, Assistant Professor, PhD
- Shestopalova Vera, Educational Master

### Regular courses:

- Actuarial mathematics by Assoc. Prof. Denisov, 96 lecture hours, 5th – 7th semesters.
- Mathematics of compound interest by Assoc. Prof. Belyankin, 32 lecture hours, 5th semester.
- Actuarial mathematics in property insurance by Assoc. Prof. Pospelova, 32 lecture hours, 5th semester.
- The theory of risk by Assoc. Prof. Denisov, 64 lecture hours, 6th and 7th semesters.
- Discrete optimization by Prof. Flerov, 32 lecture hours, 8th semester.
- Mathematical models in the natural sciences and sociology by Prof. Vasin, 32 lecture hours, 10th semester.
- Mathematics in population by Assoc. Prof. Pospelova, 32 lecture hours, 8th semester.
- Introduction to mathematical economics by Prof. Vasin, 32 lecture hours, 6th semester.
- Additional chapters of operations research by Assoc. Prof. Morozov, 32 lecture hours, 6th semester.
- Optimization theory by Prof. Izmailov, 64 lecture hours, 5th and 6th semesters.
- Contemporary numerical optimization by Prof. Izmailov, 32 lecture hours, 9th semester.

### Special courses:

- Mathematical models of imperfect competition by Prof. Vasin and Dr. Belyankina, 64 lecture hours, 5th and 6th semesters.
- Macro models in economy by Prof. Pospelov, 64 lecture hours, 7th and 8th semesters.

### Special scientific seminars:

- Mathematical methods in economics by Prof. Vasin, Prof. Belolipetsky, Assoc. Prof. Reshetov.
- Optimization and operations research by Prof. Novikova, Assoc. Prof. Pospelova and Assoc. Prof. Davidson.
- The theory of decision-making by Assoc. Prof. Morozov and Assoc. Prof. Belyankin.
- Mathematical programming by Prof. Izmailov, Assoc. Prof. Denisov, Prof. Flerov, and Assoc. Prof. Fourougian.

### Master programs (MSc):

- Mathematical and software support of the economic activity by Prof. Vasin.

## Main Scientific Directions:

### The theory of hierarchical games and collective behavior

(Professors P.S. Krasnoshchekov and A.A. Vasin, Assoc. Professors V.V. Morozov and G.A.Belyankin)

The Department develops and studies mathematical models of collective behavior for further studying problems of optimizing structures of the state control. Results and methods of these studies can be applied in the reform process of hierarchical structures of the government. In addition, the mathematical model of a hierarchical control structure in the state inspection is observed.

It should be noted that nowadays in many countries, including Russia, corruption is regarded as one of the major problems hindering the state's development. For the last 20 years the area of research related to mathematical modeling of processes and practices for the suppression of corruption is rapidly developing. These models solve also the problem of the optimal organization of inspection: a strategy is found that ensures disadvantageous offenses and corrupt deals during the inspection at the lowest cost.

The use of the game-theoretic approach to the suppression of corruption can be considered as an interaction of corrupt agents of the game in which each player (including the state) is trying to maximize their average profit. These studies use an approach known as the "principal-agent-performer".

In models where the state organizes an inspection through a hierarchical structure of tests, the probability of inspections, fines, the amount of the salary levels of checks and inspectors are exogenous parameters. In the case of the tax inspection, it is possible to arrange the process so that the general corruption is possible at a relatively low cost – of only about 4% of the total tax sum. Models that take into account the heterogeneity of the inspectors, different ways to stimulate and the partitioning agents checked on similar groups are being developed.

The Department investigates mathematical models of voting. We obtain estimates of odds for one player to promote their candidate from 4 possible candidates by voting with the veto power. Here the result depends on whether the strict or lax players are in preference, as well as on the ability to control the order of moves (“vetovaniya”) of the selected player. Results of side payments during voting are studied on a rule of “the most pre-emptive first player”.

The stability of coalition structures in heterogeneous populations is considered. Conditions for the occurrence of stable coalition structures (if the original structure is no longer sustainable) are given.

### Operations research in insurance and finance

(Assoc. Professors V.V. Morozov and D.V. Denisov)

The Department studies mathematical models of insurance and finance. This research focuses on the problem of decision-making in the presence of a stochastic uncertainty.

This research direction also observes various formulations of the problem of agents' stimulation, fundamentals and a design of an optimal insurance contract. We construct an optimal strategy and calculate the maximum profit that can be gained in models with an arbitrary incentive scheme and an increasing convex pattern. For the model with an arbitrary incentive scheme the best point scheme is elaborated – in it only one or two specific results are paid while the compensation for any different result equals zero.

Optimal incentive schemes are constructed for an insurance intermediary which acts in deterministic "agent-principal" problems in the case when there are agents of two or more types. The problem of insurance stimulating sets a system of linear-programming problems.

Upper and lower bounds for options on two assets are developed. The upper bound is constructed by a piece-wise linear approximation of the set of immediate execution and the corresponding solutions to integral equations for the value of the option. In order to construct a lower bound, we consider the class of decision rules, each of which is written as a specific boundary-value problem. The solution is obtained in the form of rapidly converging parametric series. Optimization of the series' parameters gives the desired lower bound.

In game problems of interaction between the government and the public, an optimal redistribution function for the state that maximizes the total production is given. The problem is solved under various restrictions, e.g. with a limited minimum level of consumption for each group of citizens, in the continuous and discrete case. The results obtained in the continuous case are used to construct an optimal pension system that maximizes the total production during citizens' lifetime.

### Mathematical models in economics

(Professors A.A. Vasin, N.N. Novikova, and I.I. Pospelov, Assoc. Professor M.R. Davidson)

The Department studies mathematical models of markets with homogeneous goods. The study is focused on a wide range of decision-making problems in a competitive environment.

One of the research themes covers a mathematical optimization problem for a power system that is scheduled in accordance with the rules of the wholesale electricity market in Russia. This research includes an optimal commitment of power stations for operation and planning their load as a result of electricity trades.

A peculiarity of the electricity auction, as compared with other auctions, is the necessity to schedule the mode of operation of the whole power system so that the corresponding parameters stay within admissible ranges. This feature leads to a complicated optimization problem which takes into account the structure of the grid topology and the nonlinear equations of power flow distribution.

Similar problem has to be solved in an almost real-time mode while the requirements for calculation speed and accuracy are rather high. The related mathematical optimization models that can be applied for planning at different time horizons are consistently described. On a longer time horizon, additional optimization controls include possibility of switching units on and off - that makes this problem even more complicated. Methods for solving the above problems are developed. A mathematical optimization model for different terminal states during unit's start-up and shutdown is developed using linear constraints that involve integer variables. Also a mathematical model for optimization of CHP is developed.

A model is proposed to calculate maximum power transfer between two sets of nodes (zones) in the power system. The purpose of the model is to facilitate power exchange trading without violating the inherent constraints of the system.

Fast methods of assessing the maximum amount of active power that can be delivered to an arbitrary node within an AC power system are becoming increasingly important. A special case when the power at nodes is defined by constant impedance is investigated.

The Department also studies problems of scheduling the natural gas transportation system that includes an electronic market for gas trading.

A network auction model is applied to the study of a possible development of railway cars exchange.

Theoretical research proceeds for two-agent games (sellers and buyers) with vector criteria, as well as for multi-agent games with binding constraints imposed by a network structure.

An influence of a network structure on the hierarchy of control systems (e.g., Gazprom, the Russian Railways, and the Federal Grid Company) is scrutinized.

The properties of classical game theoretical schemes are studied in the setting which is close to actual decision-making conditions (multiple criteria, uncertainty, side payments). This direction of research covers various voting schemes when one candidate is chosen from a set of several candidates (using a majority vote and a veto rule).

### Theory and methods of optimization

(Prof. A.F. Izmailov, Assoc. Prof. M.G. Fourougian)

A wide range of optimization problems is studied, including both classical continuous optimization problems and problems of discrete optimization, especially in scheduling theory. Problems of these classes often arise in economics, programming, and technology.

Optimization problems under relaxed regularity assumptions and constraint qualifications are investigated, with an emphasis on meaningful optimality conditions and effective numerical methods. Convergence of general Newtonian schemes and their specific instances are studied, including the inexact sequential quadratic programming and other related methods. This leads to a new sharp characterization of the local superlinear convergence of the stabilized sequential quadratic programming method and to novel truncated sequential quadratic programming algorithms that exploit the interior point techniques for approximate solution of the related subproblems. We also deal with Newton-type methods for optimization problems with complementary and vanishing constraints, including those that rely on an idea of "lifting" the problem.

Another area of our research deals with the development of algorithms for feasible scheduling problems in multiprocessor systems with interrupts, especially for the case when the duration of jobs depends linearly on the amount of additional resources allocated to them.

### Recent papers

- D.Fernandez, A.F.Izmailov and M.V.Solodov, Sharp primal superlinear convergence results for some Newtonian methods for constrained optimization // SIAM J. Optim., vol. 20, no. 6, pp. 3312-3334, 2010.
- A.F.Izmailov, Solution sensitivity for Karush-Kuhn-Tucker systems with nonunique Lagrange multipliers // Optimization, vol. 59, no. 5, pp. 747-775, 2010.
- A.F.Izmailov and M.V.Solodov, A truncated SQP method based on inexact interior-point solutions of subproblems // SIAM J. Optim., vol. 20, no. 5, pp. 2584-2613, 2010.
- A.F.Izmailov and M.V.Solodov, Inexact Josephy-Newton framework for generalized equations and its applications to local analysis of Newtonian methods for constrained optimization // Comput. Optim. Appl., vol. 46, no. 2, pp. 347-368, 2010.
- A.S.Urazov and A.A.Vasin, Optimal tax enforcement with corruptible auditors // Game Theory and Management. SPb, Russia: Graduate School of Management SPbU, pp. 239-242, 2010.
- A.F.Izmailov, A.L.Pogosyan and M.V.Solodov, Semismooth SQP method for equality-constrained optimization problems with an application to the lifted reformulation of mathematical programs with complementarity constraints // Optim. Meth. Software, vol. 26, no. 4, pp. 847-872, 2011.
- A.F.Izmailov and M.V.Solodov, On attraction of linearly constrained Lagrangian methods and of stabilized and quasi-Newton SQP methods to critical multipliers // Math. Prog., vol. 126, no. 2. pp. 231-257, 2011.

• 2013

- Arutyunov A.V., Izmailov A.F., Shvartsman I. Optimality conditions for 2-regular problems with nonsmooth objective functions // Nonlinear Analysis: Theory, Methods & Applications. 2013. 90. N 1. P. 37-45.
- Izmailov A.F., Kurennoy A.S., Solodov M.V. A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems // Math. Prog. 2013. 142. N 1. P. 591-604.
- Izmailov A.F., Kurennoy A.S., Solodov M.V. The Josephy-Newton method for semismooth generalized equations and semismooth SQP for optimization // Set-Valued and Variational Analysis. Theory and Applications. 2013. 21. N 1. P. 17-45.
- Mashechkin A.I. Estimating the winning chances of a solution suggested by the first player under conditions of voting by veto // Moscow University Comput. Mathematics and Cybern. 2013. 37. N 2. P. 572-596.
- Mashechkin A.I., Novikova N.M. An additional player in the voting by veto problem // Computational Mathematics and Modeling. 2013. 24. N 1. P. 153-166.
- Mashechkin A.I., Novikova N.M. Controlling the order of moves in voting by veto. I. Conditions for making the given decision // J. Computer Systems and Sci. Intern. 2013. 52. N 1. P. 66-79.
- Mashechkin A.I., Novikova N.M. Controlling the order of moves in voting by veto. II. Algorithms for constructing an optimal order of moves // J. Computer Systems and Sci. Intern. 2013. 52. N 2. P. 186-214.
- Morozov V.V., Khishnyak K.V. An upper bound on the value of an infinite american call option on difference and sum of two assets // Computational Mathematics and Modeling. 2013. 24. N 1. P. 54-64.
- Vasin A. Models for capacity and electricity market design // Central Europ. J. Operations Research. 2013. 21. N 3. P. 651-661.
- Vasin A.A. Russian electricity market: variants of development // The Oxford Handbook of the Russian Economy. Oxford, USA: Oxford university press, 2013. P. 478-526.

• 2012

- Izmailov A.F., Solodov M.V. Optimizacao.V.2. Metodos computacionais. Segunda edicao. Rio de Janeiro, Brasil: IMPA, 2012. 435 p.
- Izmailov A.F., Pogosyan A.L. Active-set Newton methods for mathematical programs with vanishing constraints // Comput. Optim. Appl. 2012. 53. N 2. P. 425-452.
- Izmailov A.F., Pogosyan A.L., Solodov M.V. Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints // Comput. Optim. Appl. 2012. 51. N 1. P. 199-221.
- Izmailov A.F., Solodov M.V. Stabilized SQP revisited // Math. Prog. 2012. 122. N 1. P. 93-120.
- Morozov V.V., Khizhnyak K.V. An upper bound on the value of an infinite American call option on two assets // Computational Mathematics and Modeling. 2012. 23. N 4. P. 478-486.
- Morozov V.V., Muravey D.L. A lower bound on the value of an infinite American call option on two assets // Computational Mathematics and Modeling. 2012. 23. N 4. P. 79-87.
- Morozov V.V., Shalbuzov K.D. One modification of Gomory’s algorithm // Moscow University Comput. Mathematics and Cybern. 2012. 36. N 1. P. 1-7.