Necessary optimality conditions for a nonsmooth semi-infinite programming problem

The nonsmooth semi-infinite programming (SIP) is solved in the paper (Mishra et al. in J Glob Optim 53:285-296, 2012) using limiting subdifferentials. The necessary optimality condition obtained by the authors, as well as its proof, is false. Even in the case where the index set is a finite, the result remains false. Two major problems do not allow them to have the expected result; first, the authors were based on Theorem 3.2 (Soleimani-damaneh and Jahanshahloo in J Math Anal Appl 328:281-286, 2007) which is not valid for nonsmooth semi-infinite problems with an infinite index set; second, they would have had to assume a suitable constraint qualification to get the expected necessary optimality conditions. For the convenience of the reader, under a nonsmooth limiting constraint qualification, using techniques from variational analysis, we propose another proof to detect necessary optimality conditions in terms of Karush-Kuhn-Tucker multipliers. The obtained results are formulated using limiting subdifferentials and Frechet subdifferentials.