The paper deals with convex Semi-In¯nite Programming (SIP) problems. A new
concept of immobility order is introduced and an algorithm of determination of the
immobility orders (DIO algorithm) and so called immobile points is suggested. It is
shown that in the presence of the immobile points SIP problems do not satisfy the
Slater condition. Given convex SIP problem, we determine all its immobile points
and use them to formulate a Nonlinear Programming (NLP) problem in a special
form. It is proved that optimality conditions for the (in¯nite) SIP problem can be
formulated in terms of the analogous conditions for the corresponding (¯nite) NLP
problem. The main result of the paper is the Implicit Optimality Criterion that
permits to obtain new e±cient optimality conditions for the convex SIP problems
(even not satisfying the Slater condition) using the known results of the optimality
theory of NLP
