  Many complex systems obey to optimality conditions that are usually not
simple. Conflicting traits often interact making a Multi Objective Optimization
(MOO) approach necessary. Recent MOO research on complex systems report about
the Pareto front (optimal designs implementing the best trade-off) in a
qualitative manner. Meanwhile, research on traditional Simple Objective
Optimization (SOO) often finds phase transitions and critical points. We
summarize a robust framework that accounts for phase transitions located
through SOO techniques and indicates what MOO features resolutely lead to phase
transitions. These appear determined by the shape of the Pareto front, which at
the same time is deeply related to the thermodynamic Gibbs surface. Indeed,
thermodynamics can be written as an MOO from where its phase transitions can be
parsimoniously derived; suggesting that the similarities between transitions in
MOO-SOO and Statistical Mechanics go beyond mere coincidence.
