Comments on: Farkas’ lemma: three decades of generalizations for mathematical optimization
چکیده
The celebrated Farkas lemma originated in Farkas (1902) provides us an attractively simple and extremely useful characterization for a linear inequality to be a consequence of a linear inequality system on the Euclidean space Rn . This lemma has been extensively studied and extended in many directions, including conic systems (linear, sublinear, convex), infinite/semi-infinite convex inequality systems and some classes of nonconvex systems such as the difference of convex (DC) systems and composite convex systems, and the rich results are spread in the literature. A brief survey on earlier extensions was made by Jeyakumar (2008); the authors, Dinh and Jeyakumar, of the present survey and Jeyakumar (2008) have done a good job in drawing a global picture to show us how the celebrated Farkas’ lemma is developed and extended in the last three decades emerging into a major field in the modern convex analysis and the variational analysis theory. This survey contains seven main parts. The first six parts are concerned with the generalized Farkas’ lemma under some mild constraint qualifications for systems from simple cone linear inequality/equality systems in Banach spaces to composite convex inequality systems and some more complicated special nonconvex inequality systems; while the last part is with the sequential forms of the generalized Farkas’ lemma for these systems without any qualification. One main