Some Optimality Conditions For Set Optimization Problems

Set-valued optimization theory has been widely used in a variety of fields. Atpresent, there are two ways to study set-valued optimization problems: vectorialoptimality and set relation optimality.Vectorial set-valued optimization problem aims to find Pareto or weak Paretooptimal solutions. While in our daily life, there are some cases unfit for selecting Paretopoints as the optimal solutions. According to this, Kuroiwa introduce a conceptcalled”set relation”. Set optimization method takes the whole image sets of objectivefunctions as an element. By establishing set relations, we compare the elements anddefine the optimal solutions.In this paper, we mainly develop constrained set-valued optimization problemswith set relations. At first, we introduce the basic theory of this paper—set relations andthe important tools—the derivatives of set-valued maps. We give the relations ofl minimal solution and l weak minimal solution of set-valued maps with setrelations. We study the property of several kinds of derivatives of set-valued maps. Wegive some examples to verify the relations and properties. Next, by the concepts andproperties of different kinds of derivatives of set-valued maps, we get the sufficient andnecessary conditions for the existence of the l minimal solution of constrainedset-valued optimization problems with set relations. Finally, according to the relationbetween l minimal solution and l weak minimal solution, the sufficient andnecessary conditions for the existence of l strict minimal solution of this constrainedset-valued optimization problem are concluded.