An approach to characterizing $$\epsilon $$ ϵ -solution sets of convex programs

In this paper, we propose an approach to characterizing $${\epsilon} $$ ϵ -solution sets of convex programs with a given $${\epsilon} >0$$ ϵ > 0 . The results are divided into two parts. The first one is devoted to establishing the expressions of $${\epsilon} $$ ϵ -solution sets of a class of convex infinite programs. The representation is given based on the study of relationships among the following three sets: the set of Lagrange multipliers corresponding to a given $${\epsilon} $$ ϵ -solution, the set of $${\epsilon} $$ ϵ -solutions of the dual problem corresponding, and the set of $${\epsilon} $$ ϵ -Kuhn–Tucker vectors associated with the problem in consideration. The second one is devoted to some special cases: the $${\epsilon} $$ ϵ -solution sets of convex programs that have set constraints and the almost $${\epsilon} $$ ϵ -solution sets of convex programs that have finite convex constraints. Several examples are given.