FARKASEVA LEMA: ELEMENTARNI DOKAZ I EKONOMSKE PRIMJENE
FARKAS' LEMMA: ELEMENTARY PROOF AND ECONOMIC APPLICATIONS

Summary/Abstract: In this paper, we provide elementary mathematical proof of Farkas’ lemma. In mathematics, Farkas’ lemma is a very important fact used in the theory of optimization, for example in derivation Karush-Khun-Tucker’s optimum conditions in the case of inequality constraints in nonlinear programming, and in proving dual theorems for linear programming. Although the statement of the Farkas’ lemma is easy, its proof is not trivial (most of the existing proofs are based on non-trivial results in the field of optimization and (linear) algebra), which is said to have been proven by many in many ways before 1972 to date, by overcoming who will offer simpler proof. In this paper, we prove the Farkas’ lemma in an elementary way by using mathematical induction. The proof of this lemma by mathematical induction is known in foreign, but not in domestic literature. Therefore, the purpose of this paper is to revise this proof, correct the existing deficiencies and errors, and explain in detail each and every step of the proof, not using complicated terms and facts from the area of optimization and algebra. Apart from the evidence of Farkas’ lemma, we also give its two applications in the economics. On the one hand, we want to explain Farkas’ lemma in a comprehensible way to the readers who are not mathematicians but they use it in their work, and on the other hand, to contribute to understanding of Farkas’ lemma statement through particular examples

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