The key to an approach to vector optimization based on infimum and supremum is to consider set-based objective functions and to extend the partial ordering of the original objective space to a suitable subspace of the power set. In this new space the infimum and supremum exist under the usual assumptions.
Took my public-school training in three jails and a plenty of pool-rooms, went to college in a gang of tea-pads, earned my Ph.D. in more creep joints and speakeasies and dancehalls than the law allows.
Having almost filled a vial, capable of containing near a pound of human blood, with a mixture of that, and some rectify'd spirit of wine, by guess a fourth, or an eighth part; at the end of above three years, looking upon the same glass, stopt with nothing but a cork, we found it coagulated, or of a consistent form: when the vessel being unstopp'd, there appeared no sign of putrefaction in the blood; and having smelt to it, we could not perceive that it was fetid: so balsamic a vertue has dephlegmed spirit of wine to preserve it.
Nice seeing you both, a woman at the check-in said. Hey, I love you, another crooned.