Farkas lemmaFarkas lemma; Fourier–Motzkin elimination method; Fractional programming; Global optimization: Envelope representation; Gröbner bases for polynomial equations; Lagrangian duality: Basics; Least-index anticycling rules; Theorems of the alternative and optimizationFARKAS LEMMA: GENERALIZATIONS

Theorems of the Alternative for Inequality Systems of Real Polynomials

“…Thus, theorems of the alternative have been extensively studied in the literature. For example, many theorems of the alternative for quadratic inequality systems were proposed and analyzed, such as Dine's theorem, Polyak's theorem, Yuan's theorem, and so on [7,13,[20][21][22][23][24][25].…”

Farkas’ lemma for separable sublinear inequalities without qualifications

“…, m. Farkas' lemma is one of the key results in Optimization and it underpins the duality theory of linear programming. Due to its importance in applications, Farkas' lemma has undergone numerous generalizations over a century (see [10] and other references therein). Yet, how to establish the equivalence between (i) and (ii), without qualifications, to a system of completely nonlinear functions, f i , i = 0, 1, .…”

Necessary and sufficient conditions for S-lemma and nonconvex quadratic optimization

“…Our approach was motivated by the recent work on Farkas' lemma for convex inequality systems and its applications to convex programming problems (see Dinh et al 2005Dinh et al , 2006Jeyakumar and Glover 1995;Jeyakumar et al 2003Jeyakumar et al , 2006Jeyakumar 2001Jeyakumar , 2006.…”