Numerical Methods Library for OCTAVE

3.7.3 Lagrange multipliers[XMIN, LAMBDAMIN, FMIN] = OPT_LAGRANGE(F, GRADF, G, JACG, X0) computed theminimum of the function FUN subject to ’G(X) = 0’ with the lagrange multipliermethod. Function GRADF defines the gradient vector of F. Function G representsequality-constrained and function JACG defines its jacobian matrix. F andG accept a real vector input and return a real vector. F, GRADF, G and JACGcan also be inline object.ParametersF evaluated function.GRADF F’s gradient function.G equality-constrained function : ’G(X) = 0’.X0 initial point.ReturnsXMIN computed solution of min(FUN).LAMBDAMIN vector of Lagrange multipliers on XMIN.FX value of FUN(X) with X computed solution.Consider the minimization of a nonlinear function subject to nonlinear equality constraints,where f : R n → R and g : R n → R m , with m ≤ n.The Lagrangian function, L : R n+m → R, is given byminxf (x) subject to g(x) = 0, (14)L(x, λ) = f (x) + λ T g(x), (15)whose gradient and Hessian are given by[ ] [ ]Lx (x, λ) ▽ f (x) + JT▽L(x, λ) = =g (x)L λ (x, λ) g(x)where J g is the Jacobian matrix of g and λ is an m-vector of Lagrange multipliers.Together, the necessary condition and the requirement of feasibility say that we are lookingfor a critical point of the Lagrangian function, which is expressed by the system of nonlinearequations[ ]▽ f (x) + JTg (x)= 0 (17)g(x)3.8 Initial value problems for Ordinary differential EquationsWe determine a unique solution to the ODE y ′ (t) = f (y, t) with y(t 0 ) = t 0 , provided that f iscontinuously differentiable. Because the independent variable t usually represents time, wethink of t 0 as the initial time and y 0 as the initial value. Hence, this is termed an initial valueproblem. The ODE governs the dynamic evolution of the system in time from its initial state y 0at time t 0 onward, and we seek a function y(t) that describes the state of the system as a functionof time.20(16)