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Towards historical roots of necessary conditions of optimality: Regula of Peano

Dolecki S., Greco G. H.

Control and Cybernetics

2007

Vol. 36, no 3

491-518

At the end of 19th century Peano discerned vector spaces, differentiability, convex sets, limits of families of sets, tangent cones, and many other concepts, in a modern perfect form. He applied these notions to solve numerous problems. The theorem on necessary conditions of optimality (Regula) is one of these. The formal language of logic that he developed, enabled him to perceive mathematics with great precision and depth. Actually he built mathematics axiomatically based exclusively on logical and set-theoretic primitive terms and properties, which was a revolutionary turning point in the development of mathematics.

On semialgebraic tangent cones

Fortuna E.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 4

383-390

The paper deals with the following question: which semialgebraic subsets of [R^n] can be realized as tangent cones to real algebraic subsets of [R^n]? At first we prove that the answer is positive for every closed semialgebraic cone in [R^n] of dimension [is less than or equal to 2]. Then, for closed semialgebraic cones A in [R^n] admitting a sufficiently nice algebraic presentation, we explicitely give equations of an algebraic subset V [...] [R^n] having A as its tangent come.