The history of nonlinear constrained optimization problems in XVIII-XIX centuries

Constrained optimization problems which take place in different science areas were always of interest to mathematicians and engineers. The derivation of differential and variations calculus methods has allowed solving such problems since XVIII century. Such well-known mathematicians as J.-L. Lagrange, A.O. Cournot, J. B. Fourier, C.F. Gauss, and M.V. Ostrogradsky made the substantial contribution to nonlinear optimization development. Lagrange paid special attention to nonlinear constrained optimization problems. In his paper «Théorie des fonctions analytiques» Lagrange considered a case of problems on continuously differentiable functions with equality constraints and offered special method for such problems, called Lagrange multipliers method. Multipliers method is described there as a tool for finding a mechanical system equilibrium. Fourier was first to solve the nonlinear optimization problem with inequality constraints (φj(X) ≥ 0) in his paper «Memoire sur la statique». Cournot in «Extension du principe des vitesses virtuelles au cas ou les conditions de liaison du systeme sont experimees par des inegalites» formulated the necessary condition for equilibrium without proof for special cases in 1827. In 1829, Gauss in his work «Über ein neues allgemeines Grundgesetz der Mechanik» enunciated the inequality principle for equilibrium. In 1834, Ostrogradsky also enunciated the inequality principle. Ostrogradsky formulated the necessary condition for equilibrium for the general case. In his paper, «The general concepts respectively to moments of force» complemented a solution of some problems that were set in the first chapter of «Mecanique analytique».