AI 中文总结
研究向量优化问题中价值函数的方向导数与误差界，通过推导对偶表示、刻画零方向导数锥等，建立多种全局误差界条件等价性，确定局部误差界常数，为向量优化算法收敛分析提供工具。
AI 中文摘要
本文对与向量优化问题\(\operatorname{Min}_C\{F(x):x\in A\}\)相关的价值函数\(\theta(x)=\sup_{a\in A}\bigl(-\Delta_C(F(x)-F(a))\bigr)\)进行了全面分析，其中\(\Delta_C\)是定向距离函数。首先证明\(\theta\)在整个空间上是凹的且Lipschitz连续，并通过弱\(^*\)紧凸集\(K=\overline{\operatorname{co}}^{w^*}(S(C^+))\)推导其对偶表示。在弱有效解\(\bar x\)处，得到显式公式\(\theta'(\bar x;d)=\min_{y^*\in W(\bar x)}\langle y^*,F(d)\rangle\)，刻画了零方向导数锥。证明在局部误差界条件下，解集的切锥为\(T_{E_w}(\bar x)=T_A(\bar x)\cap T_{\widehat A}(\bar x)=\{d\in T_A(\bar x):\theta'(\bar x;d)=0\}\)。建立了包括线性正则性、全局斜率、渐近条件和扰动稳定性等十三种不同全局误差界条件的等价性。核心结果表明，\(\theta\)在可行集\(A\)上的全局误差界性质由单位球最小方向导数的一致负性条件表征，即\(\sup_{x \in A \setminus E_w} \varphi(x) < 0\)。还精确确定了最优局部误差界常数为\(1/\varphi(\bar x)\)（当\(\varphi(\bar x)>0\)时），并给出反例说明当\(\varphi(\bar x)=0\)时额外方向条件的必要性。这些结果为价值函数的方向导数与解集几何之间提供了完整桥梁，为向量优化算法的收敛分析提供了基本工具。
英文摘要
This paper presents a comprehensive analysis of directional derivatives and error bounds for the merit function $θ(x)=\sup_{a\in A}\bigl(-Δ_C(F(x)-F(a))\bigr)$ associated with the vector optimization problem $\operatorname{Min}_C\{F(x):x\in A\}$, where $Δ_C$ is the oriented distance function. We first prove that $θ$ is concave and Lipschitz continuous on the whole space and derive its dual representation via the weak$^*$ compact convex set $K=\overline{\operatorname{co}}^{w^*}(S(C^+))$. At a weakly efficient solution $\bar x$, we obtain the explicit formula $θ'(\bar x;d)=\min_{y^*\in W(\bar x)}\langle y^*,F(d)\rangle$ with $W(\bar{x})=\{y^*\in K:F^*y^*\in -N_A(\bar{x})\}$, characterize the zero-directional-derivative cone, and prove that, under a local error bound condition, the tangent cone to the solution set is $T_{E_w}(\bar x)=T_A(\bar x)\cap T_{\widehat A}(\bar x)=\{d\in T_A(\bar x):θ'(\bar x;d)=0\}$. We establish the equivalence of thirteen distinct global error bound conditions, including characterizations via linear regularity, the global slope, an asymptotic condition, and perturbation stability. A central result shows that the global error bound property for $θ$ on the feasible set $A$ is characterized by a uniform negativity condition on the unit-sphere minimal directional derivative, namely $\sup_{x \in A \setminus E_w} φ(x) < 0$. We also determine the optimal local error bound constant precisely as $1/φ(\bar x)$ when $φ(\bar x)>0$, and provide a counterexample demonstrating that an additional directional condition is essential when $φ(\bar x)=0$. These results provide a complete bridge between the directional derivative of the merit function and the geometry of the solution set, offering fundamental tools for the convergence analysis of algorithms in vector optimization.
