# Noghin Vladimir Dmitrievich

D.Sc., Professor of Department of Control Theory Room 223E-mail: noghin@gmail.com |

## Academic degree and appointments

**Specialties**

Mathematics, Applied Mathematics, Operations Research, Mathematical Modelling

**Academic degree**

Candidate of Phys.-Math. Sciences (1978)

Doctor of Phys.-Math. Sciences (1996)

**Academic titles**

Associate Professor (1984)

Professor (1997)

Member of the International Higher Education Academy of Sciences (1997)

**Position**

2003-Present |
Professor, Department of Control Theory, Faculty of Applied Mathematics-Control Processes, St. Petersburg State University |

1993-2003 |
Professor, Department of Mathematics, St. Petersburg State Technical University (former Leningrad Polytechnic Institute) |

1980-1993 |
Associate Professor, Leningrad Polytechnic Institute |

1974-1980 |
Researcher, Leningrad State University |

**Education**

1971-1974 |
Post-graduate, Leningrad State University |

1966-1971 |
Student, Leningrad State University |

**Honors**

State Committee of Public Education prize-winner (1988)

Soros Professor (1999, 2000)

**Publications**

Total number of publications is over 120

## Teaching responsibilities

**Courses for students (1980-present)**

- Multiple Criteria Decision Making
- Mathematical Analysis
- Linear Algebra
- Differential Equations
- Probability Theory
- Elements of Mathematical Logic and Theory of Graphs
- Convex Analysis
- Mathematical Programming
- System Analysis
- Mathematical Control Theory
- Mathematical Theory of Stability
- Pareto set and Pareto principle
- Optimal solutions under uncertainty

## Areas of research

Research interests lie in the field of theoretical, algorithmic and applied aspects of the theory of decision-making with multiple criteria.

Began to analyze multicriteria problems in 1972. During 10 years a number of

- necessary and sufficient conditions for Pareto-optimality (efficiency), proper efficiency, and weak efficiency
- existence theorems
- topological theorems
- duality theorems

for linear as well as nonlinear multicriteria problems were obtained. Moreover, an original concept of **r-optimality** was proposed.

As a result, in 1982 the publishing house 'Nauka' was published a monograph 'Pareto-optimal Decisions in Multicriteria Optimization Problems' [1]. In this book numerous mathematical results from all over the world related to the basic concepts of multicriteria optimization problem were collected, systemized, and stated. Moreover, a plenty of original theorems by the authors there were presented. It was the first of such kind monograph in Russia. Authors of hundreds papers (not only in Russian) have been referring to this book. Unfortunately, this monograph was not translated into English.

Since 1980 began to teach students at St. Petersburg State Technical University (former Polytechnic Institute). The text book 'Fundamentals of Optimization Theory' [2] was published in 1986. From one point of view different topics of optimization theory such as linear programming, geometrical programming, dynamic programming, nonlinear programming, multicriteria optimization, calculus of variations, and optimal control were presented in this text book.

In 1994 the problem concerning to maximal number of partially ordered sets was formulated. This problem is unsolved up to now.

During the second half of the 70th and first half of the 80-ies of the XX century, introduced the concept of r-optimality, obtained a number of necessary and sufficient conditions of optimality, existence theorems, and the design is based dual problems multipurpose programming. Identified certain topological properties of the Pareto set. In the late 80's has been formulated so far unsolved problem in dimension theory of partially ordered sets.

The last three decades of research are mainly associated with the problem of narrowing of the Pareto set, which is the actual practical value. At the beginning of the new millennium, the axiomatic form rendered famous Edgeworth-Pareto principle, according to which the ‘best’ solution should be selected among the Pareto-optimal. It turned out that this principle is valid for a sufficiently broad class of problems, but outside the application of this principle is risky or even unacceptable. In order to narrow the Pareto set and thus facilitate subsequent selection is necessary to have some additional information about the task. For this purpose, along with a vector criterion and the set of feasible solutions (which are generally known in real applications) is considered the preference relation of the decision maker (DM) about which we have only partial information. The problem with the above three objects was named as a multicriteria choice problem.

Information that the DM of the two vectors, each of which at least one component is greater than another, selects one vector and do not select the second, called a **quantum of information **on the preference relation. The existence of such a quantum shows the willingness of the DM to go for a compromise, i.e. surrender in certain criteria, for the sake of winning on other (more significant) criteria, and results in a reduction of the Pareto by single element. If we accept some four ‘reasonable’ axioms of choice, we can expect a much greater reduction. It was found that the using of quantum information can be made as follows: according to a special formula to count the original vector criterion and find the Pareto set with respect to new criterion. The last set will be a narrowing of the initial Pareto set. This narrowing will generally depend on the kind of the quantum (degree of compromise), and on the feasible set of vectors. The greater degree of Pareto set narrowing can be performed by using not one but several quanta of information about preference relation of the DM. Were obtained a number of theorems to make the above reducing. We study the ‘limit’ of possibility for taking into account an arbitrary finite set of quanta. It was proved that under certain additional constraints due to the use of this information we can with any degree of accuracy close to the set of non-dominated (with respect to the initial preference relation) vectors.

The above was the content of the axiomatic approach to the Pareto set reducing, which has recently been successfully extended to the case of a fuzzy preference relation.

## Conferences

- IX International Conference on MCDM, Washington D.C., USA (1990)
- International Congress on Computer Systems and Applied Mathematics, St. Petersburg, Russia (1993)
- XV International Conference on MCDM, Ankara, Turkey (2000)
- 52nd Meeting of the European Working Group MCDA, Vilnius, Lithuania (2000)
- 11 IFAC International Workshop .Control Applications of Optimization Theory., St. Petersburg, Russia (2000)
- Workshop 2001 .Decision Theory and Optimization in Theory and Practice, Kloster Banz, Germany (2001)
- International Conference on Soft Computing and Measurements, St. Petersburg, Russia (2001)
- MCDM Winter Conference 2002 (16th MCDM world conference), Semmering, Austria (2002)
- 58nd Meeting of the European Working Group MCDA, Moscow, Russia (2003)
- Russian-Finnish Graduate School Seminar "Dynamic Games and Multicriteria Optimization", Petrozavodsk (Russia) (2006)
- V Moscow International Conference on Operations Research (ORM2007), Moscow (2007)
- XXI International Conference on MCDM, Finland (2011)
- International Conference 'Information - Interaction - Intellect', Varna, Bulgaria (2012)
- International Conference 'Information - Interaction - Intellect', Varna, Bulgaria (2013)
- Operation Research (ORM2013), Moscow, Russia (2013)

## Publications

**Monographs, text books, handbooks**

- Podinovskii V.V., Noghin V.D. (1982, 2007- 2-nd edition) Pareto-optimal Decisions in Multicriteria Optimization Problems, Moscow, Nauka, 256 pp. (in Russian).
- Noghin V.D. et al. (1986) Fundamentals of Optimization Theory. Moscow, Higher School, 384 pp. (in Russian).
- Noghin V.D. (1988) Elements of Optimization Theory and Mathematical Economics. St. Petersburg, Polytechnic Institute, 96 pp. (in Russian).
- Noghin V.D. (1994) Introduction to Mathematical Analysis. St. Petersburg, State Technical University, 68 pp. (in Russian).
- Noghin V.D., Chystaykov S.V. (1998) Application of Linear Algebra for Decision Making. St. Petersburg, State Technical University, 40 pp. (in Russian).
- Noghin V.D. (2002,2005) Decision Making in Multicriteria Environment: a Quantitative Approach. Moscow, Fizmatlit, 176 pp.(in Russian).
- Noghin V.D. et al. (2004) System Analysis and Decision Making (handbook). Moscow, Visshaya Shkola, 616 pp. (in Russian).
- Noghin V.D. et. al. (2006) Systems theory and systems analysis in management (handbook). Moscow: Finance and Statistics, 848 pp. (in Russian).
- Noghin V.D. (2007) Decision making under several criteria. - Saint-Petersburg, UTAS, 104 pp. (in Russian).
- Noghin V.D. (2008) Introduction to optimal control. - Saint-Petersburg, UTAS, 92 pp. (in Russian).
- Noghin V.D. (2002, 2005 - 2-nd edition) Decision making in multicriteria environment: a quantitative approach. Moscow: FUZMATLIT, 176 pp. (in Russian).
- Noghin V.D. (2018) Reduction of the Pareto set: an axiomatic approach. Springer International Publishing, 232 pp.

**Selected papers**

- Noghin V.D. (1976) A New Approach to Reduce a Compromise Set //Technicheskaya Kibernetika, No. 5, pp. 10-14 (in Russian).
- Noghin V.D. (1977) Duality in Multipurpose Programming //Computational Mathematics and Mathematical Physics, No. 1, pp. 254-258 (in Russian).
- Noghin V.D. (1980) Existence Criteria of Decisions in Finite Dimensional Multipurpose Optimization Problem //Vestnik St. Petersburg State University (Ser.: Mathematics, Mechanics, Astronomy), No. 7, pp. 27-32 (in Russian).
- Noghin V.D. (1991) Estimation of the Set of Nondominated Solutions // Numerical Functional Analysis and Applications, V. 12, No. 5&6, pp. 507-515.
- Noghin V.D. (1994) Problem 214// Discrete Mathematics, 135, p. 394.
- Noghin V.D. (1994) Upper Estimate for a Fuzzy Set of Nondominated Solutions. //Fuzzy Sets and Systems, V. 67, pp. 303-315.
- Noghin V.D. (1997) Relative Importance of Criteria: a Quantitative Approach //J. Multi-Criteria Decision Analysis, V. 6, pp. 355-363.
- Noghin V.D. (2000) Completeness Theorems in the Theory of Relative Importance of Criteria //Vestnik St. Petersburg State University (Ser.: Mathematics, Mechanics, Astronomy), 40 (25), pp. 13-18 (in Russian).
- Noghin V.D., Tolstych I.V. (2000) Using Quantitative Information on the Relative Importance of Criteria for Decision Making //Computational Mathematics and Mathematical Physics, V. 40, No. 11, pp. 1529-1536.
- Noghin V.D. (2001) What is the Relative Importance of Criteria and How to Use It in MCDM //Lecture Notes in Economics and Mathematical Systems, V. 507: Multiple Criteria Decision Making in the New Millennium (Proceedings of the XV International Conference on MCDM, Ankara, Turkey, 2000), pp. 59-68.
- Noghin V.D. (2001) A Logical Justification of the Edgeworth-Pareto Principle//Computational Mathematics and Mathematical Physics, V. 42, No. 7, pp. 915-920.
- Noghin V.D. (2003) The Edgeworth-Pareto Principle and the Relative Importance of Criteria in the Case of a Fuzzy Preference Relation// Computational Mathematics and Mathematical Physics, V. 43, No. 11, pp. 1604-1612.
- Noghin V.D. (2004) A Simplified Variant of the Analytic Hierarchy Processes Based on a Nonlinear Scalarizing Function. //Computational Mathematics and Mathematical Physics, V. 44, No. 7, pp. 1194-1202.
- Noghin V.D. (2005) Restriction of a Pareto Set Based on Information about a DecisionMaker’s Preferences of the PointMultiple Type//Computational Mathematics and Mathematical Physics, V 46, No 4, P. 554-563.
- Noghin V.D. (2006) The Edgeworth-Pareto Principle in Decision Making. Tutorial Presentation for the Russian-Finnish Graduate School Seminar "Dynamic Games and Multicriteria Optimization", Petrozavodsk (Russia),
- Noghin V.D. (2006) Axiomatization of the Generalized Edgeworth-Pareto Principle in Terms of Choice Functions//Mathematical Social Sciences, v. 52, No 2, pp. 210-216.
- Noghin V.D. (2006) The Edgeworth-Pareto principle in terms of a fuzzy choice function//Computational Mathematics and Mathematical Physics. Vol. 46, No. 4, pp. 554-562.
- Klimova O.N., Noghin V.D. (2006) Using inderpendent information on the relative importance of criteria in decision making // Computational Mathematics and Mathematical Physics, Vol. 46, No. 12, pp. 2080–2091.
- Noghin V.D. (2008) Pareto set reducing problem: approaches// Artificial Intellect and Decisioin Making. No 1, p. 98-112 (in Russian)
- Noghin V.D. (2011) Pareto set reducing based on point-set information// Scientific and Technical Information Processing, Vol. 38, No. 5, pp. 1–5.
- Noghin V.D. (2011) Pareto set reducing based on set-point information// Scientific and Technical Information Processing, Vol. 38, No. 6, pp. 435–439.
- Noghin V.D., Prasolov A.V. (2011) The quantitative analysis of trade policy: a strategy in global competitive conflict// Int. J. Business Continuity and Risk Management, Vol. 2, No. 2, pp. 167-182
- Noghin V.D., Baskov O.V. (2011) Pareto Set Reduction Based on an Arbitrary Finite Collection of Numerical Information on the Preference Relation//Doklady Mathematics, Vol. 83, No. 3, pp. 418–420.
- Noghin V.D. (2012) Pareto Set Reducing based on Fuzzy Information//Int. J. Information Technologies&Knowledge, Vol. 6, No. 2, pp. 157-168 (in Russian)
- Noghin V.D. (2014) Reducing of the Pareto Set Algorithm Based on an Arbitrary Finite Set of Information “Quanta // Scientific and Technical Information Processing, V. 41, No 5, pp. 1-5.
- Noghin V.D. (2015) Generalized Edgeworth–Pareto Principle// Computational Mathematics and Mathematical Physics, 2015, Vol. 55, No. 12, pp. 1975–1980 .
- Noghin V.D. (2015) Linear Scalarizaion of Criteria in Multi-Criterion Optimization // Scientific and Technical Information Processing, 2015, Vol. 42, No. 6, pp. 463–469.
- Noghin V.D. (2016) Composed methods to reduce the Pareto set. - in Proc. VIII Moscow Intern. Conf. on Operations Research, Moscow, 2016, pp.79-81.
- Noghin V.D. (2017) Pareto set reduction based on an axiomatic approach with application of some metrics// Computational Mathematics and Mathematical Physics, Vol. 57, No 4, pp.645-652.

## Grants

**Russian Fund for Basic Research (RFBR)**

(1998-2000)

(2001-2003)

(2005-2007)

(2008-2010) Reducing the Pareto set based on collection numerical information on the DM's preferens relation

(2011-2013) Computational methods of reducing the Peroto set

(2014-2016) Combined methods of reducing the Pareto set

(2017-2019) Combined methods of choice in multicriteria environment

## Scientific publications

- Noghin V.D. Reduction of the Pareto set: an axiomatic approach. 2018. 1-232.