Computational Discretization Algorithms for Functional Inequality Constrained Optimization

In this paper, a functional inequality constrained optimization problem is studied using a discretization method and an adaptive scheme. The problem is discretized by partitioning the interval of the independent parameter. Two methods are investigated as to how to treat the discretized optimization problem. The discretization problem is firstly converted into an optimization problem with a single nonsmooth equality constraint. Since the obtained equality constraint is nonsmooth and does not satisfy the usual constraint qualification condition, relaxation and smoothing techniques are used to approximate the equality constraint via a smooth inequality constraint. This leads to a sequence of approximate smooth optimization problems with one constraint. An adaptive scheme is incorporated into the method to facilitate the computation of the sum in the inequality constraint. The second method is to apply an adaptive scheme directly to the discretization problem. Thus a sequence of optimization problems with a small number of inequality constraints are obtained. Convergence analysis for both methods is established. Numerical examples show that each of the two proposed methods has its own advantages and disadvantages over the other.