Necessary optimality conditions for a nonsmooth semi-infinite programming problem

The nonsmooth semi-infinite programming $$\left( {\textit{SIP}}\right) $$ 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 Fréchet subdifferentials.