Saddle Point Optimality Conditions in Fuzzy Optimization Problems

The fuzzy-valued Lagrangian function of fuzzy optimization problem via the concept of fuzzy scalar (inner) product is proposed. A solution concept of fuzzy optimization problem, which is essentially similar to the notion of Pareto solution in multiobjective optimization problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. Under these settings, the saddle point optimality conditions along with necessary and sufficient conditions for the absence of a duality gap are elicited.

[1]  J. Cooper,et al.  Theory of Approximation , 1960, Mathematical Gazette.

[2]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[3]  José L. Verdegay,et al.  Fuzzy Optimization: Recent Advances , 1994 .

[4]  Richard Bellman,et al.  Decision-making in fuzzy environment , 2012 .

[5]  C. Hwang,et al.  Fuzzy Multiple Objective Decision Making: Methods And Applications , 1996 .

[6]  S. Vajda,et al.  Probabilistic Programming , 1972 .

[7]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[8]  Alberto Isidori,et al.  Nonlinear control systems: an introduction (2nd ed.) , 1989 .

[9]  J Figueira,et al.  Stochastic Programming , 1998, J. Oper. Res. Soc..

[10]  Chuen-Chien Lee.  FUZZY LOGIC CONTROL SYSTEMS: FUZZY LOGIC CONTROLLER - PART I , 1990 .

[11]  I. M. Stancu-Minasian,et al.  Stochastic Programming: with Multiple Objective Functions , 1985 .

[12]  Dimitar P. Filev,et al.  A generalized defuzzification method via bad distributions , 1991, Int. J. Intell. Syst..

[13]  Ronald R. Yager,et al.  Implementing fuzzy logic controllers using a neural network framework , 1992 .

[14]  Masahiro Inuiguchi,et al.  Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem , 2000, Fuzzy Sets Syst..

[15]  Peter Kall,et al.  Stochastic Linear Programming , 1975 .

[16]  Dimitar Filev,et al.  SLIDE: A simple adaptive defuzzification method , 1993, IEEE Trans. Fuzzy Syst..

[17]  R. Yager,et al.  On the issue of defuzzification and selection based on a fuzzy set , 1993 .

[18]  T. Runkler,et al.  A set of axioms for defuzzification strategies towards a theory of rational defuzzification operators , 1993, [Proceedings 1993] Second IEEE International Conference on Fuzzy Systems.

[19]  S. Mabuchi.  A proposal for a defuzzification strategy by the concept of sensitivity analysis , 1993 .

[20]  P. Huard,et al.  Point-to-set maps and mathematical programming , 1979 .

[21]  Lotfi A. Zadeh,et al.  Outline of a New Approach to the Analysis of Complex Systems and Decision Processes , 1973, IEEE Trans. Syst. Man Cybern..

[22]  Qiang Song,et al.  Defuzzification Filters and Applications to Power System Stabilization Problems , 2000 .

[23]  B. Chissom,et al.  Forecasting enrollments with fuzzy time series—part II , 1993 .