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DOI: 10.46698/o9578-0948-6676-e
Farkas Lemma for Multilinear Operators
Kusraev, A. G.
Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 1.
Abstract: Farkas's lemma is a classic result underlying the duality of linear programming, and it played a central role in the development of mathematical optimization. Numerous generalizations of this lemma are known, including various linear and nonlinear operator versions. However, Farkas's lemma is generally false for multilinear operators and even for bilinear forms in a finite-dimensional space. In this paper, we identify a class of orthoregular multilinear operators for which Farkas's lemma holds true. Consider vector lattices \(E\) and \(G\) with \(E\) uniformly complete and \(G\) universally complete. The main result is worded as follows. Theorem 3.2. For \(n\)-linear orthoregular operators \(S_1,\dots,S_N,S:E^n\to G\) the following are equivalent: (1) The inequalities \(\pi S_1(x_1,\dots,x_n)\leq0,\dots,\pi S_N(x_1,\dots,x_n)\leq0\) imply \(\pi S(x_1,\dots,x_n)\leq0\) for all members \(x_1,\dots,x_n\in E\) and for every band projection \(\pi\) in \(G\). (2) There exists positive orthomorphisms \(\alpha_1,\dots,\alpha_N\in Orth(G^u)\) such that \(S=\alpha_1S_1+\dots+\alpha_NS_N\). The proof relies on Kutateladze's stratification principle. A similar result is established when the domain of the operators under considerations is a vector space equipped with a disjointness relation satisfying certain additional conditions. Some open questions are also formulated.
Keywords: simultaneous linear inequalities, Farkas Lemma, vector lattice, Kutateladze's stratification principle, orthoregular multilinear operator, disjointness relation.
Fund name: The work was carried out at the North Caucasus Center for Mathematical Research of the Russian Academy of Sciences with the support of the Ministry of Education and Science of Russia, agreement No. 075-02-2026-738.
For citation: Kusraev, A. G. Farkas Lemma for Multilinear Operators, Vladikavkaz Math. J., 2026, vol. 28, no. 1, pp. 82-97 (in Russian). DOI 10.46698/o9578-0948-6676-e
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