Image Space Analysis to the generalized optimization problem on a local sphere

This paper introduces and studies the generalized optimization problem (for short, GOP) defined by the conic order relation on a local sphere. The existence of solution to this problem is studied by using image space analysis (for short, ISA), and a class of regular weak separation functions on the local sphere is established. Moreover, a Lagrangian-type sufficient optimality condition and a saddle-point-type necessary optimality condition for GOP is obtained by a second class of weak separation functions, which are based on the Gerstewitz function and the directional distance function. The problem is transformed into a solvable real-valued optimization problem using scalarization methods.