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FARKAS-TYPE THEOREMS FOR GENERAL INTERVAL LINEAR SYSTEMS

Li Wei 1, Xia Mengxue1 , Li Haohao2   

  1. 1.Institute of Operational Research and Cybernetics, Hangzhou Dianzi University, Hangzhou 310018;2. School of Data Sciences, Zhejiang University of Finance and Economics, Hangzhou 310018
  • Online:2016-11-25 Published:2017-01-18

Li Wei,Xia Mengxue,Li Haohao. FARKAS-TYPE THEOREMS FOR GENERAL INTERVAL LINEAR SYSTEMS[J]. Journal of Systems Science and Mathematical Sciences, 2016, 36(11): 1959-1971.

In interval linear systems, combining weak and strong solvability or feasibility of interval linear equations or inequalities, we have eight different decision problems, including weak solvability of equations, strong feasibility of inequalities and so on. All Farkas-type theorems for eight decisions problems of interval linear systems are established separately, since equivalent transformation between the interval linear systems are generally impossible due to dependency. In this paper, we consider about the general interval linear systems consisting of mixed equations and inequalities with mixed free and sign-restricted variables, and generalize the Farkas-type necessary and sufficient conditions for weak and strong solvability to a unified framework. Each particular result of Farkas-type results established separately for weak and strong solvability and feasibility of interval linear systems is a special case of our general approach. Besides, a concrete example is presented to illustrate our main results. It is known that the classical Farkas lemma and its various generalized versions are often used to characterize the optimality conditions of optimization problems. Thus, the interval version Farkas lemma may apply to characterize the optimality conditions of interval optimization problems.

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[1] LI Haohao, XIA Mengxue, JIN Jianghong. Interval Solutions of Interval Linear Systems [J]. Journal of Systems Science and Mathematical Sciences, 2021, 41(12): 3395-3404.
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