Directional Derivatives and Error Bounds of Merit Functions in Vector Optimization

Abstract

This paper presents a comprehensive analysis of directional derivatives and error bounds for the merit function $\theta (x)={\sup }_{a\in A}(-{\Delta }_C(F(x)-F(a)))$ associated with the vector optimization problem ${\mathrm{Min}}_C\{F(x):x\in A\}$, 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${}^{\ast }$ compact convex set $K={\overline{\mathrm{co}}}^{w^{\ast }}(S(C^{+}))$. At a weakly efficient solution $\bar{x}$, we obtain the explicit formula ${\theta }^{\prime }(\bar{x};d)={\min }_{y^{\ast }\in W(\bar{x})}⟨y^{\ast },F(d)⟩$ with $W(\bar{x})=\{y^{\ast }\in K:F^{\ast }{y}^{\ast }\in -N_A(\bar{x})\}$, characterize the zero-directional-derivative cone, and prove that, under a local error bound condition, the tangent cone to the solution set is $T_{E_w}(\bar{x})=T_A(\bar{x})\cap T_{\hat{A}}(\bar{x})=\{d\in T_A(\bar{x}):{\theta }^{\prime }(\bar{x};d)=0\}$. 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 ${\sup }_{x\in A\setminus E_w}\phi (x)<0$. We also determine the optimal local error bound constant precisely as $1/\phi (\bar{x})$ when $\phi (\bar{x})>0$, and provide a counterexample demonstrating that an additional directional condition is essential when $\phi (\bar{x})=0$. 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.