This paper proposes algorithms for minimizing a continuously differentiable function f(x): \({\mathbb{R}}^ n\to {\mathbb{R}}\) subject to the constraint that x does not lie in specified bounded subsets of \({\mathbb{R}}^ n\). Such problems arise in a variety of applications, such as tolerance design of electronic circuits and obstacle avoidance in the selection of trajectories for robot arms. Such constraints have the form \(\psi\) (x)\(\triangleq \min \{g^ j(x)| j\in J\}\leq 0\). The function \(\psi\) is not continuously differentiable. Algorithms based on the use of generalized gradients have considerable disadvantages because of the local concavity of \(\psi\) at points where the set \(\{j| g^ j(x)=\psi (x)\}\) has more than one element. Algorithms which avoid these disadvantages are presented, and their convergence is established.