A widely used and successful approach to solving constrained optimization problems, that is minimize F(x), x = (x1,x2,…,xn)T, where F(x) is a given objective function of n real variables, subject to the t nonlinear constraints on the variables, ci(x) = 0, i = 1,2,…,t Inequality constraints are also possible. A solution of this problem is also a stationary point (a point at which all the partial derivatives vanish) of the related function of x and λ, L(x,λ) = F(x) − Σλici(x), λ = (λ1,λ2,…,λt) A quadratic approximation to this function is now constructed that along with linearized constraints forms a quadratic programming problem – i.e., the minimization of a function quadratic in the variables, subject to linear constraints. The solution of the original optimization problem, say x✻, is now obtained from an initial estimate and solving a sequence of updated quadratic programs; the solutions of these provide improved approximations, which under certain conditions converge to x✻.
minimize F(x), x = (x1,x2,…,xn)T,
ci(x) = 0, i = 1,2,…,t
L(x,λ) = F(x) − Σλici(x),
λ = (λ1,λ2,…,λt)