Characterizations of ε-duality gap statements for composed optimization problems

In this paper we present different regularity conditions that equivalently characterizeε-duality gap statements for optimization problems consisting of minimizing the sum of a function with the precomposition of a cone-increasing function to a vector function. These regularity conditions are formulated by using epigraphs and ε-subdifferentials. Taking ε=0 one can rediscover recent results on stable strong and total duality and zero duality gap from the literature. Moreover, we deliver as byproducts ε-optimality conditions and (ε,η)-saddle point statements for the aforementioned kind of problems, and ε-Farkas statements involving the sum of a function with the precomposition of a cone-increasing function to a vector function.