We analyze the optimal value function v associated with a general parametric optimization problem via the theory of viscosity solutions. The novelty is that we obtain regularity properties of v by showing that it is a viscosity solution to a set of first-order equations. As a consequence, in Banach spaces, we provide sufficient conditions for local and global Lipschitz properties of v. We also derive, in finite dimensions, conditions for optimality through a comparison principle. Finally, we study the relationship between viscosity and Clarke generalized solutions to get further differentiability properties of v in Euclidean spaces.

中文翻译：

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最优值函数正则性的粘度解法

我们通过粘度解理论分析与一般参数优化问题相关的最优值函数v。新颖之处在于，我们通过证明v是一组一阶方程的粘度解来获得v的正则性。因此，在 Banach 空间中，我们为v 的局部和全局 Lipschitz 性质提供了充分条件。我们还通过比较原理在有限维度中推导出最优性条件。最后，我们研究了粘度和克拉克广义解之间的关系，以获得v在欧几里德空间中的进一步可微性。