Constrained multiobjective optimization problems (CMOPs) widely exist in real-world applications, but posing stiff challenges due to the constraints and multiple conflicting objectives. Although many CMOEAs and CHTs have been proposed trying to balance convergence, diversity, and feasibility, most ignore the relaxation of objectives. Thus, it is difficult to preserve promising dominated solutions that can assist in detecting feasible regions under complex constraints while unfortunately, CMOPs in modern industry and daily life can contain very complex constraints. This work develops a new CHT that relaxes objectives to address the limitations of existing CHTs and a new co-evolutionary algorithm that integrates the objective relaxation technique to solve complex CMOPs. The objective relaxation technique introduces a new perspective of multiobjective constraint-handling by considering not only the constraints but also the objectives, leading to significantly improved performance.

Since most multiobjective optimization problems in real-world applications contain constraints, constraint-handling techniques (CHTs) are necessary for a multiobjective optimizer. However, existing CHTs give no relaxation to objectives, resulting in the elimination of infeasible dominated solutions that are promising (potentially useful but inferior) for detecting feasible regions and the constrained Pareto front (CPF). To overcome this drawback, in this work, we propose an objective relaxation technique that can preserve promising by relaxing objective function values, i.e., convergence, through an adaptively adjusted relaxation factor. Further, we develop a new constrained multiobjective optimization evolutionary algorithm (CMOEA) based on relaxations on both constraints and objectives. The proposed algorithm evolves one population by the constraint relaxation technique to preserve promising infeasible solutions and the other population by both objective and constraint relaxation tec...