Personalized　federated　learning　(PFL)　is　an　exciting　approach　that　allows　machine　learning　(ML)　models　to　be　trained　on　diverse　and　decentralized　sources　of　data,　while　maintaining　client　privacy　and　autonomy.　However,　PFL　faces　several　challenges　that　can　deteriorate　the　performance　and　effectiveness　of　the　learning　process.　These　challenges　include　data　heterogeneity,　communication　overhead,　model　privacy,　model　drift,　client　heterogeneity,　label　noise　and　imbalance,　federated　optimization　challenges,　and　client　participation　and　engagement.　To　address　these　challenges,　researchers　are　exploring　innovative　techniques　and　algorithms　that　can　enable　efficient　and　effective　PFL.　These　techniques　include　several　optimization　algorithms.　This　research　survey　provides　an　overview　of　the　challenges　and　motivations　related　to　the　model　optimization　strategies　for　PFL,　as　well　as　the　state-of-the-art　(SOTA)　methods　and　algorithms　which　seek　to　provide　solutions　of　these　challenges.　Overall,　this　survey　can　be　a　valuable　resource　for　researchers　who　are　interested　in　the　emerging　field　of　PFL　as　well　as　its　potential　for　personalized　machine　learning　in　a　federated　environment.