Higher-order σ-cone arcwisely connectedness in optimization problems associated with difference of set-valued maps

In this paper, an optimization problem (DP) is studied where the objective maps and the constraints are the difference of set-valued maps (abbreviated as SVMs). The higher-order $\sigma$-cone arcwise connectedness is described as an entirely new type of generalized higher-order arcwise connectedness for set-valued optimization problems. Under the higher-order contingent epiderivative and higher-order $\sigma$-cone arcwise connectedness suppositions, the higher-order sufficient Karush–Kuhn–Tucker (KKT) optimality requirements are demonstrated for the problem (DP). The higher-order Wolfe ($WD$) form of duality is investigated and the corresponding higher-order weak, strong, and converse theorems of duality are established between the primary (DP) and the corresponding dual problem by employing the higher-order $\sigma$-cone arcwise connectedness supposition. In order to demonstrate that higher-order $\sigma$-cone arcwise connectedness is more generalized than higher-order cone arcwise connectedness, an example is also constructed. As a special case, the results coincide with the existing ones available in the literature.