On the Principle of Fermat-Lagrange for Mixed Smooth-Convex Extremal Problems

A necessary condition - the Principle of Fermat-Lagrange - is offered for mixed smooth-convex optimization problems. This generalizes and unifies most of the known necessary conditions for concrete finite and infinite dimensional optimization problems of interest. The new idea in comparison with the unified version of Tikhomirov and others ([I-T], [A-T-F] and [T]) is that a geometrical construction of the principle is given. In the present set-up constraints are not mentioned explicitly, the feasibility set is allowed to vary in a non-standard way and the objective function is also allowed to vary. An equivalent analytical formulation is given as well; we propose a new standard form for optimization problems which allows greater flexibility.