Pareto set topology refers to the geometry formed in decision space by Pareto optimal solutions from continuous multi-objective optimization problems. Recent studies have shown that problems with difficult Pareto set topology can present a tough challenge for evolutionary algorithms to find a good approximation of the optimal set of solutions, well-distributed in decision and objective space. One important challenge optimizing these problems is to keep or restore diversity in decision space. In this work, we present a method that learns a model of the topology of solutions from evolutionary algorithm's population by performing parametric cubic interpolations for all variables in decision space. The model uses Catmull-Rom parametric curves as they allow us to deal with any dimension in decision space. According to the Karush-Kuhn-Tucker condition, this method is appropriated for bi-objective problems since their optimal set is a one-dimensional curve. We couple this method with four different evolutionary algorithm approaches by promoting restarts from solutions generated by the model. We argue and discuss the algorithm's behavior and its implications for model building.

For air-conditioning systems in office buildings, it is crucial to reduce power consumption while maintaining office workers' thermal comfort. This paper proposes a simulation-based evolutionary multi-objective air-conditioning schedule optimization system for office buildings. In the proposed system, a target office building is modeled and simulated by EnergyPlus building simulator which is one of the practical simulators widely used in the building construction field. To obtain the temperature schedules which dynamically change the temperature setting over time, we use an improved multi-objective particle swarm optimization algorithm, OMOPSO, to simultaneously optimize the thermal comfort of office workers in the building and the power consumption of the air-conditioning system. Experimental results show that the proposed system can obtain temperature schedules better than the conventional schedule with constant temperature settings from viewpoints of both the thermal comfort and the power consumption. Also, we show experimental results that the multi-objective search in the proposed system acquires better temperature schedules than single objective particle swarm optimization and differential evolution algorithms using ε-constraint method as one option of single objective optimization approaches. Furthermore, we show that OMOPSO obtains temperature schedules widely approximating the optimal tradeoff between the thermal comfort and the power consumption compared with other evolutionary multi-objective optimizers, NSGA-II, NSGA-III, MOEA/D-DE.