Solving constrained multi-objective optimization problems with evolutionary algorithms has attracted considerable attention. Various constrained multi-objective optimization evolutionary algorithms (CMOEAs) have been developed with the use of different algorithmic strategies, evolutionary operators, and constraint-handling techniques. The performance of CMOEAs may be heavily dependent on the operators used, however, it is usually difficult to select suitable operators for the problem at hand. Hence, improving operator selection is promising and necessary for CMOEAs. This work proposes an online operator selection framework assisted by Deep Reinforcement Learning. The dynamics of the population, including convergence, diversity, and feasibility, are regarded as the state; the candidate operators are considered as actions; and the improvement of the population state is treated as the reward. By using a Q-Network to learn a policy to estimate the Q-values of all actions, the proposed approach can adaptively select an operator that maximizes the improvement of the population according to the current state and thereby improve the algorithmic performance. The framework is embedded into four popular CMOEAs and assessed on 42 benchmark problems. The experimental results reveal that the proposed Deep Reinforcement Learning-assisted operator selection significantly improves the performance of these CMOEAs and the resulting algorithm obtains better versatility compared to nine state-of-the-art CMOEAs.

The goal of constrained multiobjective evolutionary optimization is to obtain a set of well-converged and welldistributed feasible solutions. To complete this goal, there should be a tradeoff among feasibility, diversity, and convergence. However, it is nontrivial to balance these three elements simultaneously by using a single tradeoff model since the importance of each element varies in different evolutionary phases. As an alternative, we adapt different tradeoff models in different phases and propose a novel algorithm called ATM-R. In the infeasible phase, ATM-R takes the tradeoff between diversity and feasibility into account, aiming to move the population toward feasible regions from diverse search directions. In the semi-feasible phase, ATM-R promotes the transition from "the tradeoff between feasibility and diversity" to "the tradeoff between diversity and convergence", which can facilitate the discovering of enough feasible regions and speed up the search for the feasible Pareto optima in succession. In the feasible phase, the tradeoff between diversity and convergence is considered to attain a set of well-converged and well-distributed feasible solutions. It is worth noting that the merits of reference points are leveraged in ATM-R to accomplish these tradeoff models. Also, in ATM-R, a multiphase mating selection strategy is developed to generate promising solutions beneficial to different evolutionary phases. Systemic experiments on a wide range of benchmark test functions demonstrate that ATM-R is effective and competitive, compared against five state-of-the-art constrained multiobjective optimization evolutionary algorithms.