The　local　stability　of　a　weakly　dissipative　heat　engine　is　analyzed　and　linked　to　an　energetic　multi-objective　optimization　perspective.　This　constitutes　a　novel　issue　in　the　unified　study　of　cyclic　energy　converters,　opening　the　perspective　to　the　possibility　that　stability　favors　self-optimization　of　thermodynamic　quantities　including　efficiency,　power　and　entropy　generation.　To　this　end,　a　dynamics　simulating　the　restitution　forces,　which　mimics　a　harmonic　potential,　bringing　the　system　back　to　the　steady　state　is　analyzed.　It　is　shown　that　relaxation　trajectories　are　not　arbitrary　but　driven　by　the　improvement　of　several　energetic　functions.　Insights　provided　by　the　statistical　behavior　of　consecutive　random　perturbations　show　that　the　irreversible　behavior　works　as　an　attractor　for　the　energetics　of　the　system,　while　the　endoreversible　limit　acts　as　an　upper　bound　and　the　Pareto　front　as　a　global　attractor.　Fluctuations　around　the　operation　regime　reveal　a　difference　between　the　behavior　coming　from　fast　and　slow　relaxation　trajectories:　while　the　former　are　associated　to　an　energetic　self-optimization　evolution,　the　latter　are　ascribed　to　better　performances.　The　self-optimization　induced　by　stability　and　the　possible　use　of　instabilities　in　the　operation　regime　to　improve　the　energetic　performance　might　usher　into　new　useful　perspectives　in　the　control　of　variables　for　real　engines.　©　2020　American　Physical　Society.