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The paper is devoted to the existence of global optimal solutions for ageneral class of nonsmooth problems of constrained vector optimization withoutboundedness assumptions on constraint sets. The main attention is paid to thetwo major notions of optimality in vector problems: Pareto efficiency andproper efficiency in the sense of Geoffrion. Employing adequate tools ofvariational analysis and generalized differentiation, we first establishrelationships between the notions of properness, $M$-tameness, and thePalais--Smale conditions formulated for the restriction of the vector costmapping on the constraint set. These results are instrumental to deriveverifiable necessary and sufficient conditions for the existence of Paretoefficient solutions in vector optimization. Furthermore, the developed approachallows us to obtain new sufficient conditions for the existence ofGeoffrion-properly efficient solutions to such constrained vector problems.

Existence of efficient and properly efficient solutions to problems of constrained vector optimization