This is an open-label, single-arm intra-patient dose escalating pilot study of ALT-803, a recombinant human super agonist interleukin-15 (IL-15) complex with an expansion cohort at the maximum tolerated dose (MTD). The primary aim is to investigate the safety and tolerability of ALT-803 in HIV-infected people receiving potent and optimized antiretroviral therapy. All infusions will occur on a Monday. Subjects will be monitored for 24 hours after the first infusion and then for 6 hours after each subsequent infusion, provided no unacceptable toxicity.