High-Dimensional Evaluation of Central Composite Designs Under Classical and Regularized Optimality Criteria

Central Composite Design (CCD) is often used in Response Surface Methodology (RSM) to fit second-order models. However, little is known about their behavior as the number of factors increases. The study develops a cross-dimensional evaluation methodology to investigate the behavior of three types of CCDs as the number of factors increases from $k=3$ to $k=10$. The CCD families considered in this study include Face-Centered CCD (FCCD), Rotatable CCD (RCCD), and Spherical CCD (SCCD). The designs are evaluated using the alphabetic optimality criteria of D-, A-, and G-optimality. A total of 1080 design configurations are generated by varying the replication levels of factorial points, axial points, and center points. The results show that classical CCDs encounter structural limitations in high-dimensional settings. When $k\ge 7$, the quadratic model matrix becomes rank deficient due to the fixed factorial core of 32 runs, which takes on a budget-constrained approach and uses regularization as a diagnostic tool to evaluate the stability of CCDs in the rank-deficient regime. In lower dimensions ($k\le 6$), RCCD consistently provides the highest efficiency, followed by SCCD and FCCD. In higher-dimensional settings, fixed regularization becomes less effective as the scale of the information matrix increases. To address this limitation, scaled regularization is introduced by adjusting the regularization parameter according to the average eigenvalue of the information matrix. The results indicate that rotatable designs provide greater efficiency and stability in high-dimensional settings compared with spherical and face-centered designs.