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Some new Farkas-type results for inequality systems with DC functions. (English) Zbl 1182.90071
The authors study the DC (difference of convex functions) optimization problem (P): Inf(g(x)-h(x)) s.t. xX, g i (x)-h i (x)0 (i=1,,m), where x n is a non-empty convex set, g,h: n ¯ are two proper convex functions and g i ,h i : n ¯ (i=1,,m) are proper convex functions such that i=1 m r i (dom(g i ))r i (dom(g))r i (X)φ. It is assumed that h is lower semicontinuous and h i (i=1,...,m) are subdifferentiable on the feasible set of (P). A Fenchel-Lagrange dual problem for (P) is constructed and using the technique of J.-E. Martinez-Legaz and M. Volle [J. Math. Anal. Appl. 237, No. 2, 657–671 (1999; Zbl 0946.90064)] a dual problem to (P) is associated. The authors then use these results to derive Farkas-type results for inequality systems involving DC functions. It is claimed that some equivalent formulations of known results are obtained.
MSC:
90C26Nonconvex programming, global optimization
90C46Optimality conditions, duality
References:
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