| In recent years, since Semi-Infinite Programming (SIP) problems have wide application background, many efforts have been made in the researches of SIP. Especially since 1990s, the research of SIP has made a great development both in algorithm theory and performing algorithm. Many authors proposed various effective approaches to solve this type of problems. In particular, the method of discretization is among these algorithms, which can approximate constraint function by means of progressively finer discretization of an interval of continuous variable. Accordingly, solving SIP problem can be substituted by solving the Discretized Semi-Infinite (DSI) problem, which has a amount of constraints. As to DSI problem, some authors suitably choose constraint indices at each iteration while preserving the global and local convergence, which may lead to substantial computational saving. On the other hand, the Method of Feasible Directions (MFD) is an important method for solving inequality constrained optimization. Since norm-relaxed method of feasible direction is presented, it has been further studied and applied widely. Moreover, it is shown that the norm-relaxed method of feasible direction has good convergence speed.In this paper, combining the technique of updating discretization index set at each iteration and the idea of the norm-relaxed MFD, we present a algorithm for solving inequality constrainted DSI problems from SIP. At each iteration, the iteration point is feasible and the main search direction is computed by solving only one direction finding subproblem, i.e., a quadratic programming, and some appropriate constraints are chosen to reduce the computational cost. A high-order correction direction can be obtained by solving another quadratic subproblem with only equality constraints. Under suitable conditions, the proposed algorithm is proved to possess the global convergence and superlinear convergence. Finally, some elementary numerical experiments are reported.
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