Second order tangent vectors to sets and minimality conditions for points of subsets of ordered normed spaces

Using the extended second order tangent cone as a local approximation of a subset at a neighborhood of the point under consideration, we obtain both necessary and sufficient (in finite-dimensional spaces) conditions for locally minimal points of subsets of ordered normed spaces. The gap between necessary and sufficient conditions obtained in the paper is irremovable in the setting of second order local approximations.