In this paper we consider a general class of variational and programming problems which consist in the minimization of a real functional on a locally convex topological space subject to equality and inequality constraints. LIUSTERNIK [8] and GOLDSTINE [5] seem to have been the first who proved an abstract version of the LAGRANGE multiplier rule for the minimization problem of a real functional on a BANACE-space subject to equality constraints. Later HURWICZ [6] has extended the KUHN-TUCKER-theory to convex optimization problems in locally convex topological spaces. I n the meantime there have appeared several papers by LEVIN [7], DUBOVITSKII and M~LYUTIN [2], GAMKRELIDSE [4], PSHENICHNII [12] and others and above all by NEUSTADT [lo] which are closely related to our topic. In spite of the profundity of KEUSTADT'S new ideas and results, during the preparation of this paper we have taken but little advantage of these results: we still use variations x + E h of the LAGRANGEan type and consider nonscalar side conditons. This paper was mainly stimulated by LIUSTERNIK'S method of construction of a LAGRASGE multiplier and an article of MANCIASARIAN and FROMOVITZ [9], where they use MOTZKIN'S transposition theorem to derive a LAGRANQE multiplier rule for programming problems in spaces of b i t e dimension.
Concerning the contents of this paper in our opinion novel are (i) the distinction of three types of so called regularity conditions, (ii) a statement of cases where any one of these regularity conditions holds, (iii) a LAGRANGE multiplier rule (for the above optimization problems) based on the method of LIUSTERNIK, (iv) some generalized versions of MOTZKTN'S transposition theorem in a locally convex topological space, (v) multiplier rules immediately derived by the mentioned generalized versions, (vi) a sufficiency theorem in case of a "pseudoconvex" optimization problem, (vii) certain applications to variational or control problems. Above all the results (v) are easily applicable. Results (vii) will appear in part 11.