Directional derivatives for the value-function in semi-infinite programming

For the problemP(λ): Maximizec ^T z subject toz∈Z(λ), whereZ(λ) is defined by an in general infinite set of linear inequalities, it is shown that the value-function has directional derivatives at every point such thatP( ) and its dual are both superconsistent. To compute these directional derivatives a min-max-formula, well-known in convex programming, is derived. In addition, it is shown that derivatives can be obtained more easily by a limit-process using only convergent selections of solutions ofP(λ _n ), λ _n → and their duals.     

A note on an implementation of a method for quadratic semi-infinite programming

For convex quadratic semi-infinite programming problems aFortran-package is described. A first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems. This enables an efficient treatment of large problems with moderate storage requirements. Powell's (1983) numerically stable convex quadratic programming implementation is used to solve the QP-subproblems.