Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization which may be considered a quasi-Newton method. SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable, but not necessarily convex. 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: The Lagrangian for this problem is where \lambda and \sigma are Lagrange multipliers. The standard Newton's Method searches for the solution by iterating the following equation, where denotes the Hessian matrix: . However, because the matrix is generally singular (and therefore non-invertible), the Newton step cannot be calculated directly. Instead the basic sequential quadratic programming algorithm defines an appropriate search direction d_k at an iterate, as a solution to the quadratic programming subproblem where the quadratic form is formed with the Hessian of the Lagrangian. Note that the term f(x_k) in the expression above may be left out for the minimization problem, since it is constant under the operator. Together, the SQP algorithm starts by first choosing the initial iterate, then calculating and. Then the QP subproblem is built and solved to find the Newton step direction d_0 which is used to update the parent problem iterate using. This process is repeated for until the parent problem satisfies a convergence test.

Practical implementations

Practical implementations of the SQP algorithm are significantly more complex than its basic version above. To adapt SQP for real-world applications, the following challenges must be addressed: To overcome these challenges, various strategies are typically employed: These strategies can be combined in numerous ways, resulting in a diverse range of SQP methods.

Alternative approaches

Implementations

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