Optimality Conditions for Vector Optimisation with Set-valued Maps

In recent years, vector optimisation with set-valued maps in infinite dimensional spaces has been received an increasing amount of attention. See [6, 2, 5, 8, 4, 9] and references therein, for its extensive applications in many fields such as mathematical programming, optimal control, management science. Vector optimisation with setvalued maps, sometimes called set-valued vector optimisation for short, essentially can be considered as an improvement on single-valued vector optimisation. Amongst research topics in optimisation problems, optimality conditions are especially important. For vector optimisation with set-valued maps, many authors have published interesting results on optimality conditions, and most of those results are obtained under different extended cone-convexity assumptions via alternative theorems. For instance, under the supposition of convexlikeness, Li and Chen [6] gave multiplier type and saddle point type optimality conditions for the existence of weak minimisers of set-valued vector optimisation with both inequality and equality constraints. Li [5], under the assumption of cone-subconvexlikeness of set-valued maps, established optimality conditions for setvalued vector optimisation by using the alternative theorem in ordered linear topological spaces.