The $E$-optimality criterion for a regression model maximizes the smallest eigenvalue of the information matrix and becomes non-differentiable when this eigenvalue has multiplicity greater than one. Working in the $2$-Wasserstein space, we show that the Wasserstein gradient at an empirical measure coincides, up to a constant factor, with the Euclidean particle gradient for smooth criteria such as $D$- and $L$-optimality, and that the approximation gap for equal-weight $N$-particle designs vanishes at an explicit rate. The main challenge is the nonsmooth $E$-criterion, for which the Wasserstein gradient does not exist. We replace it with a constrained Wasserstein steepest-ascent field obtained by maximizing feasible directional derivatives over the tangent cone of the design space, and prove that the resulting flow satisfies an exact energy identity and that every limit point is first-order stationary. The particle ascent computation reduces to a convex semidefinite programme whose dimension equals the multiplicity of the smallest eigenvalue. In numerical comparisons on second-order response surface models and a seven-dimensional logistic regression model, the constrained Wasserstein steepest-ascent method attains near-optimal $E$-criterion values and is markedly more reliable than particle swarm optimization in higher-dimensional settings. The framework applies more broadly to other nonsmooth minimax criteria in optimal design, and a numerical experiment on the minimax-single-parameter criterion confirms that the method attains the theoretical optimum.

翻译：摘要：对于回归模型的$E$-最优性准则，其目标是最大化信息矩阵的最小特征值，当该特征值的重数大于1时，该准则变得不可微。在$2$-Wasserstein空间中，我们证明：对于$D$-最优性和$L$-最优性等光滑准则，经验测度处的Wasserstein梯度（至多差一个常数因子）与欧几里得粒子梯度一致，且等权$N$粒子设计的逼近间隙以显式速率消失。主要挑战在于非光滑的$E$-准则，其Wasserstein梯度不存在。我们将其替换为受约束的Wasserstein最陡上升场，该场通过在设计空间的切锥上最大化可行方向导数得到，并证明由此产生的流满足精确的能量恒等式，且每个极限点都是一阶驻点。粒子上升计算可简化为一个凸半定规划，其维数等于最小特征值的重数。在二阶响应曲面模型和七维逻辑回归模型的数值比较中，受约束的Wasserstein最陡上升方法获得了接近最优的$E$-准则值，且在更高维设定下显著比粒子群优化更为可靠。该框架更广泛地适用于最优设计中其他非光滑极小极大准则，关于极小极大单参数准则的数值实验证实了该方法能达到理论最优值。