The authors consider nonlinear semi-infinite optimization problems of type min\(\{\) f(a)\(|\) g(x,a)\(\leq 0\) for all \(x\in X\}\) with a box \(X\subset {\mathbb{R}}^ N\), and \(C^ 2\)-functions f, g. Under appropriate regularity assumptions one may consider the local maxima (w.r.t. variable x) of g(x,a) as functions \(x^ i(a)\) of a, and the problem can be replaced locally around every \(a^ o\) by a finite optimization problem min \(\{\) f(a)\(|\) \(g(x^ i(a),a)\leq 0\), \(i=1,...,k\}\). Here, the number k of functions \(x^ i(a)\) will depend on \(a^ o\), i.e. the number and form of the constraints in the finite problem depends on a. Therefore, each nonlinear programming algorithm may be applied to the semi-infinite problem if it is combined with a suitable active set strategy to detect changes in the set of local maxima of g(x,a). In the paper such an algorithm based on an exact penalty method is presented and discussed by a number of numerical examples.