The computational cost of modern simulation-based optimization tends to be prohibitive in practice. Complex design problems often involve expensive constraints evaluated through Finite Element Analysis or other computationally intensive procedures. To speed up the optimization process and deal with expensive constraints, a new dimension-selection based constrained multi-objective optimization (MOO) algorithm is developed combining LASSO regression, artificial neural networks, and Grey Wolf Optimizer, named L-ANN-GWO. Instead of considering all variables at each iteration during the optimization, the proposed algorithm only adaptively retains the variables that are highly influential on the objectives. The un-selected variables are adjusted to satisfy the constraints through a local search. With numerical benchmark problems and a simulation-based engineering design problem, L-ANN-GWO outperforms state-of-the-art constrained MOO algorithms. The method is then applied to solve a highly complex optimization problem, the design of a high-temperature superconducting magnet. The optimal solution shows significant improvement as compared to the baseline design.