On error bounds for quasinormal programs (Q535085)

Let \(I\) and \(I_{0}\) be finite index sets, \(h_{i}:\mathbb{R}^{m}\rightarrow \mathbb{R}\) \((i\in I\cup I_{0})\) be continuously differentiable functions, and \(C:=\{y\in \mathbb{R}^{m}:h_{i}(y)\leq 0\) \((i\in I),\) \(h_{i}(y)=0\) \((i\in I_{0})\}\). The main result states that, assuming that the gradients \(\nabla h_{i}(y)\) \((i\in I\cup I_{0})\) are locally Lipschitz near \(y^{0}\in C\), if this point is quasinormal in the sense of \textit{M. R. Hestenes} [Optimization theory. The finite dimensional case. New York etc.: John Wiley\&Sons (1975; Zbl 0327.90015)] then the system defining \(C\) has the local error bound property at \(y^{0}\). An easy example for \(m=2\) shows that the converse does not hold true.