Theory of extremum and extremal problems of classical analysis (Q2777858)

This paper deals with the main principles for getting necessary and sufficient conditions for an extremum, existence of extremal problem solutions and algorithms for their determination, and duality in convex analysis. In the introduction the main definitions of extremum theory and convex analysis are given. Then the Lagrange principle for necessary conditions for an extremum is formulated for smooth convex problems. In particular, mathematical programming problems, problems of classical calculus of variations, Lyapunov problems and optimal control problems are considered. Further, the author studies the perturbations of extremal problems, the extension of extremal problems and existence of solution. Some optimization algorithms are presented: method of convex optimization, simplex method. Kolmogorov's type inequalities for derivatives: \(\|x^{(k)}(\cdot)\|_{L_{q}(T)}\leq K \|x(\cdot)\|_{L_{p}(T)}^{\alpha}\|x^{(n)}(\cdot)\|_{L_{r}(T)}^{\beta}\), where \(0\leq k<n\); \(1\leq p,q,r\leq\infty\); \(\alpha,\beta\geq 0\); \(T=R\) or \(R_{+}\) are presented.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00043].