<p>Surrogate-assisted evolutionary algorithms for expensive constrained optimization have predominantly focused on addressing inequality constraints. However, the role of equality constraints, as well as the simultaneous presence of both equality and inequality constraints, remains underexplored despite its significance in this domain. This research seeks to bridge this gap by addressing all types of constraints in expensive optimization problems. To this end, a deep neural network-based multi-output regression model is developed as a surrogate, specifically designed to tackle the challenges associated with approximating inequality constraints. The model leverages its multi-output capability to establish a consensus-based infill sampling criterion, which enhances objective optimization while ensuring constraint satisfaction. Additionally, a two-stage local search strategy is proposed to refine infeasible yet promising solutions. This strategy combines a surrogate-assisted method with gradient-based mutation to improve solution quality iteratively. Experimental results based on 22 test functions demonstrate the effectiveness of the proposed approach, marking a significant initial effort towards addressing all types of constraints in expensive optimization problems. Furthermore, the empirical results highlight the superiority of the proposed methodology in handling expensive equality constraints, outperforming five recent state-of-the-art surrogate-assisted evolutionary algorithms on finding feasible solutions.</p>