Higher-order cr-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 cr-cone arcwise connectedness is described as an entirely new type of generalized higherorder arcwise connectedness for set -valued optimization problems. Under the higher -order contingent epiderivative and higher -order cr-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 cr-cone arcwise connectedness supposition. In order to demonstrate that higher -order cr-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.