Directional Derivatives and Error Bounds of Merit Functions in Vector Optimization

This paper presents a comprehensive analysis of directional derivatives and error bounds for the merit function $\theta(x)=\sup_{a\in A}\bigl(-\Delta_C(F(x)-F(a))\bigr)$ associated with the vector optimization problem $\operatorname{Min}_C\{F(x):x\in A\}$, where $\Delta_C$ is the oriented distance function. We first prove that $\theta$ 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 $\theta'(\bar x;d)=\min_{y^*\in W(\bar x)}\langle y^*,F(d)\rangle$ 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 $T_{E_w}(\bar x)=T_A(\bar x)\cap T_{\widehat A}(\bar x)=\{d\in T_A(\bar x):\theta'(\bar x;d)=0\}$. 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 $\theta$ on the feasible set $A$ is characterized by a uniform negativity condition on the unit-sphere minimal directional derivative, namely $\sup_{x \in A \setminus E_w} \varphi(x) < 0$. We also determine the optimal local error bound constant precisely as $1/\varphi(\bar x)$ when $\varphi(\bar x)>0$, and provide a counterexample demonstrating that an additional directional condition is essential when $\varphi(\bar x)=0$. 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.