Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization. SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable.

SQP methods solve a sequence of optimization subproblems, each of which optimizes a quadratic model of the objective subject to a linearization of the constraints. If the problem is unconstrained, then the method reduces to Newton's method for finding a point where the gradient of the objective vanishes. If the problem has only equality constraints, then the method is equivalent to applying Newton's method to the first-order optimality conditions, or Karush–Kuhn–Tucker conditions, of the problem.

Algorithm basics

Consider a nonlinear programming problem of the form:

${\displaystyle {\begin{array}{rl}\min \limits _{x}&f(x)\\{\mbox{s.t.}}&b(x)\geq 0\\&c(x)=0.\end{array}}}$

The Lagrangian for this problem is[1]

${\displaystyle {\mathcal {L}}(x,\lambda ,\sigma )=f(x)-\lambda ^{T}b(x)-\sigma ^{T}c(x),}$

where ${\displaystyle \lambda }$ and ${\displaystyle \sigma }$ are Lagrange multipliers. At an iterate ${\displaystyle x_{k}}$, a basic sequential quadratic programming algorithm defines an appropriate search direction ${\displaystyle d_{k}}$ as a solution to the quadratic programming subproblem

${\displaystyle {\begin{array}{rl}\min \limits _{d}&f(x_{k})+\nabla f(x_{k})^{T}d+{\tfrac {1}{2}}d^{T}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\lambda _{k},\sigma _{k})d\\\mathrm {s.t.} &b(x_{k})+\nabla b(x_{k})^{T}d\geq 0\\&c(x_{k})+\nabla c(x_{k})^{T}d=0.\end{array}}}$

Note that the term ${\displaystyle f(x_{k})}$ in the expression above may be left out for the minimization problem, since it is constant.

Implementations

SQP methods have been implemented such well known numerical environments as MATLAB and GNU Octave. There also exist numerous software libraries, including open source

• SciPy (de facto standard for scientific Python) has scipy.optimize.minimize(method=’SLSQP’) solver
• NLopt (C/C++ implementation, numerous interfaces including Python, R, MATLAB/Octave)

and proprietary/commercial ones