# sequential quadratic programming

**sequential quadratic programming** A widely used and successful approach to solving constrained optimization problems, that is minimize *F*(*x*), *x* = (*x*_{1},*x*_{2},…,*x _{n}*)

^{T},

where

*F*(

*x*) is a given objective function of

*n*real variables, subject to the

*t*nonlinear constraints on the variables,

*c*(

_{i}*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*) – Σλ

*(*

_{i}c_{i}*x*), λ = (λ

_{1},λ

_{2},…,λ

*)*

_{t}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**.

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