Optimizing fuzzy portfolio selection problems by parametric quadratic programming

This paper develops a robust method to describe fuzzy returns by employing parametric possibility distributions. The parametric possibility distributions are obtained by equivalent value (EV) reduction methods. For common type-2 triangular and trapezoidal fuzzy variables, their reduced fuzzy variabl...

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Published in:	Fuzzy optimization and decision making Vol. 11; no. 4; pp. 411 - 449
Main Authors:	Wu, Xiao-Li, Liu, Yan-Kui
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01.12.2012 Springer Science + Business Media B.V Springer Nature B.V
Subjects:	Artificial Intelligence Calculus of Variations and Optimal Control; Optimization Credibility Decision making Equivalence Expected values Fuzzy Fuzzy logic Fuzzy set theory Fuzzy sets Integrals Mathematical analysis Mathematical Logic and Foundations Mathematical models Mathematics Mathematics and Statistics Methods Operations Research/Decision Theory Optimization Portfolio management Probability Theory and Stochastic Processes Quadratic programming Random variables Representations Studies Parametric possibility distribution Portfolio selection Moment Parametric programming Reduced fuzzy variable
ISSN:	1568-4539, 1573-2908
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Abstract	This paper develops a robust method to describe fuzzy returns by employing parametric possibility distributions. The parametric possibility distributions are obtained by equivalent value (EV) reduction methods. For common type-2 triangular and trapezoidal fuzzy variables, their reduced fuzzy variables are studied in the current development. The parametric possibility distributions of reduced fuzzy variables are first derived, then the second moment formulas for the reduced fuzzy variables are established. Taking the second moment as a new risk measure, the reward-risk and risk-reward models are developed to optimize fuzzy portfolio selection problems. The mathematical properties of the proposed optimization models are analyzed, including the analytical representations for the second moments of linear combinations of reduced fuzzy variables as well as the convexity of second moments with respect to decision vectors. On the basis of the analytical representations for the second moments, the reward-risk and risk-reward models can be turned into their equivalent parametric quadratic convex programming problems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling ideas and the efficiency of solution method.
AbstractList	This paper develops a robust method to describe fuzzy returns by employing parametric possibility distributions. The parametric possibility distributions are obtained by equivalent value (EV) reduction methods. For common type-2 triangular and trapezoidal fuzzy variables, their reduced fuzzy variables are studied in the current development. The parametric possibility distributions of reduced fuzzy variables are first derived, then the second moment formulas for the reduced fuzzy variables are established. Taking the second moment as a new risk measure, the reward-risk and risk-reward models are developed to optimize fuzzy portfolio selection problems. The mathematical properties of the proposed optimization models are analyzed, including the analytical representations for the second moments of linear combinations of reduced fuzzy variables as well as the convexity of second moments with respect to decision vectors. On the basis of the analytical representations for the second moments, the reward-risk and risk-reward models can be turned into their equivalent parametric quadratic convex programming problems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling ideas and the efficiency of solution method. This paper develops a robust method to describe fuzzy returns by employing parametric possibility distributions. The parametric possibility distributions are obtained by equivalent value (EV) reduction methods. For common type-2 triangular and trapezoidal fuzzy variables, their reduced fuzzy variables are studied in the current development. The parametric possibility distributions of reduced fuzzy variables are first derived, then the second moment formulas for the reduced fuzzy variables are established. Taking the second moment as a new risk measure, the reward-risk and risk-reward models are developed to optimize fuzzy portfolio selection problems. The mathematical properties of the proposed optimization models are analyzed, including the analytical representations for the second moments of linear combinations of reduced fuzzy variables as well as the convexity of second moments with respect to decision vectors. On the basis of the analytical representations for the second moments, the reward-risk and risk-reward models can be turned into their equivalent parametric quadratic convex programming problems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling ideas and the efficiency of solution method.[PUBLICATION ABSTRACT]
Author	Wu, Xiao-Li Liu, Yan-Kui
Author_xml	– sequence: 1 givenname: Xiao-Li surname: Wu fullname: Wu, Xiao-Li organization: Key Laboratory in Machine Learning and Computational Intelligence, College of Mathematics & Computer Science, Hebei University – sequence: 2 givenname: Yan-Kui surname: Liu fullname: Liu, Yan-Kui email: yliu@hbu.edu.cn organization: Key Laboratory in Machine Learning and Computational Intelligence, College of Mathematics & Computer Science, Hebei University
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Cites_doi	10.1109/TFUZZ.2002.800692 10.1016/S0377-2217(00)00298-8 10.1007/978-1-4612-1174-7 10.1016/j.cam.2010.08.031 10.1007/s12190-010-0409-4 10.1007/s00500-009-0461-x 10.1016/0020-0255(75)90036-5 10.1016/0165-0114(78)90029-5 10.1016/S0019-9958(65)90241-X 10.1016/S0020-0255(01)00069-X 10.1016/S0019-9958(76)80011-3 10.1109/91.995115 10.1016/j.ins.2004.02.027 10.1016/j.eswa.2011.01.071 10.1007/s10700-011-9101-x 10.1137/0108053 10.1016/j.amc.2009.06.040 10.1016/S0019-9958(79)90730-7 10.1016/j.ins.2006.05.003 10.1287/mnsc.37.5.519 10.1007/s10700-009-9064-3 10.21314/JOR.2000.038 10.1007/978-3-540-39987-2 10.1007/978-3-642-11214-0
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Keywords	Parametric possibility distribution Portfolio selection Moment Parametric programming Reduced fuzzy variable
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Snippet	This paper develops a robust method to describe fuzzy returns by employing parametric possibility distributions. The parametric possibility distributions are...
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SubjectTerms	Artificial Intelligence Calculus of Variations and Optimal Control; Optimization Credibility Decision making Equivalence Expected values Fuzzy Fuzzy logic Fuzzy set theory Fuzzy sets Integrals Mathematical analysis Mathematical Logic and Foundations Mathematical models Mathematics Mathematics and Statistics Methods Operations Research/Decision Theory Optimization Portfolio management Probability Theory and Stochastic Processes Quadratic programming Random variables Representations Studies
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Title	Optimizing fuzzy portfolio selection problems by parametric quadratic programming
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