E-variables enable safe and anytime-valid inference, with log-optimal e-variables given by the likelihood ratio of the least favorable distributions (LFDs) when they exist in composite settings. While this unconstrained theory is well understood, one may need/wish to impose additional structural constraints, including differential privacy, quantization, boundedness, or moment restrictions. We show that under these constraints, log-optimal constrained e-variables can often be constructed by a simple \emph{optimize-then-constrain} principle: first compute the unconstrained log-optimal e-variable, then impose the constraint via an appropriate transformation. Thus, the constrained growth-rate optimization problem does not require solving for a different LFD pair; the constrained optimal solution is just a post-processing of the unconstrained optimal solution.

翻译：E变量能够实现安全且始终有效的推断，在复合假设设定下，当最不利分布（LFD）存在时，对数最优e变量由似然比给出。尽管这一无约束理论已得到充分理解，但实践中可能需要或希望施加额外的结构性约束，包括差分隐私、量化、有界性或矩条件。我们证明，在这些约束下，对数最优约束e变量通常可通过简单的“先优化再约束”原则构建：首先计算无约束对数最优e变量，再通过适当的变换施加约束。因此，约束增长率优化问题无需求解不同的LFD对；约束最优解仅是无约束最优解的后处理结果。