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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): $\text{Inf}\left(g\left(x\right)-h\left(x\right)\right)$ s.t. $x\in X$, ${g}_{i}\left(x\right)-{h}_{i}\left(x\right)\le 0$ $\left(i=1,\cdots ,m\right)$, where $x\subseteq {ℝ}^{n}$ is a non-empty convex set, $g,h:{ℝ}^{n}\to \overline{ℝ}$ are two proper convex functions and ${g}_{i},{h}_{i}:{ℝ}^{n}\to \overline{ℝ}$ $\left(i=1,\cdots ,m\right)$ are proper convex functions such that ${\bigcap }_{i=1}^{m}{r}_{i}\left(\text{dom}\left({g}_{i}\right)\right)\cap {r}_{i}\left(\text{dom}\left(g\right)\right)\cap {r}_{i}\left(X\right)\ne \phi$. It is assumed that $h$ is lower semicontinuous and ${h}_{i}$ $\left(i=1,...,m\right)$ 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:
 90C26 Nonconvex programming, global optimization 90C46 Optimality conditions, duality
##### Keywords:
conjugate duality
##### References:
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