Method of Lagrange Multipliers
One of the first attempts to solve the optimal redundancy problem was based on the classical Lagrange Multiplier Method. This method was invented and developed by great French mathematician Lagrange (see Box). This method allows one to get the extreme value of the function under some specified constraint on another involved function in the form of equality. The Lagrange Multiplier Method is applicable if both functions (optimizing and constraining) are monotone and differentiable.
Joseph-Louis Lagrange (1736–1813)
Lagrange made outstanding contributions to all fields of analysis, to number theory, and to classical and celestial mechanics. He was one of the creators of the calculus of variations, and also introduced the method of Lagrange multipliers where possible constraints were taken into account. He invented the method of solving differential equations known as variation of parameters and applied differential calculus to the theory of probabilities. Lagrange studied the three-body problem for the earth, sun, and moon and the movement of Jupiter's satellites. Above all, he contributed to mechanics, having transformed Newtonian mechanics into a new branch of analysis, Lagrangian mechanics.
Strictly speaking, this method is not appropriate for optimal redundancy problem solving because the system reliability and cost are described by ...