Sequential quadratic programming

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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.

Consider a nonlinear programming problem of the form:

${\displaystyle {\begin{array}{rl}\min \limits _{x}&f(x)\\{\mbox{subject to}}&h(x)\geq 0\\&g(x)=0.\end{array}}}$

The Lagrangian for this problem is^[1]

${\displaystyle {\mathcal {L}}(x,\lambda ,\sigma )=f(x)+\lambda h(x)+\sigma g(x),}$

where ${\displaystyle \lambda }$ and ${\displaystyle \sigma }$ are Lagrange multipliers.

The standard Newton's Method searches for the solution ${\displaystyle \nabla {\mathcal {L}}(x,\lambda ,\sigma )=0}$ by iterating the following equation, where ${\displaystyle \nabla _{xx}^{2}}$ denotes the Hessian matrix:

${\displaystyle {\begin{bmatrix}x_{k+1}\\\lambda _{k+1}\\\sigma _{k+1}\end{bmatrix}}={\begin{bmatrix}x_{k}\\\lambda _{k}\\\sigma _{k}\end{bmatrix}}-\underbrace {{\begin{bmatrix}\nabla _{xx}^{2}{\mathcal {L}}&\nabla h&\nabla g\\\nabla h^{T}&0&0\\\nabla g^{T}&0&0\end{bmatrix}}^{-1}} _{\nabla ^{2}{\mathcal {L}}}\underbrace {\begin{bmatrix}\nabla f+\lambda _{k}\nabla h+\sigma _{k}\nabla g\\h\\g\end{bmatrix}} _{\nabla {\mathcal {L}}}}$.

However, because the matrix ${\displaystyle \nabla ^{2}{\mathcal {L}}}$ is generally singular (and therefore non-invertible), the Newton step ${\displaystyle d_{k}=\left(\nabla _{xx}^{2}{\mathcal {L}}\right)^{-1}\nabla {\mathcal {L}}}$ cannot be calculated directly. Instead the basic sequential quadratic programming algorithm defines an appropriate search direction ${\displaystyle d_{k}}$ at an iterate ${\displaystyle (x_{k},\lambda _{k},\sigma _{k})}$, as a solution to the quadratic programming subproblem

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

where the quadratic form is formed with the Hessian of the Lagrangian. Note that the term ${\displaystyle f(x_{k})}$ in the expression above may be left out for the minimization problem, since it is constant under the ${\displaystyle \min \limits _{d}}$ operator.

Together, the SQP algorithm starts by first choosing the initial iterate ${\displaystyle (x_{0},\lambda _{0},\sigma _{0})}$, then calculating ${\displaystyle \nabla ^{2}{\mathcal {L}}(x_{0},\lambda _{0},\sigma _{0})}$ and ${\displaystyle \nabla {\mathcal {L}}(x_{0},\lambda _{0},\sigma _{0})}$. Then the QP subproblem is built and solved to find the Newton step direction ${\displaystyle d_{0}}$ which is used to update the parent problem iterate using ${\displaystyle \left[x_{k+1},\lambda _{k+1},\sigma _{k+1}\right]^{T}=\left[x_{k},\lambda _{k},\sigma _{k}\right]^{T}+d_{k}}$. This process is repeated for ${\displaystyle k=0,1,2,\ldots }$ until the parent problem satisfies a convergence test.

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:

• The possibility of an infeasible QP subproblem.
• QP subproblem yielding a bad step: one that either fails to reduce the target or increases constraints violation.
• Breakdown of iterations due to significant deviation of the target/constraints from their quadratic/linear models.

To overcome these challenges, various strategies are typically employed:

• Use of merit functions, which assess progress towards a constrained solution, or filter methods.
• Trust region or line search methods to manage deviations between the quadratic model and the actual target.
• Special feasibility restoration phases to handle infeasible subproblems, or the use of L1-penalized subproblems to gradually decrease infeasibility

These strategies can be combined in numerous ways, resulting in a diverse range of SQP methods.

• Sequential linear programming
• Sequential linear-quadratic programming
• Augmented Lagrangian method

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:

• SciPy (de facto standard for scientific Python) has scipy.optimize.minimize(method='SLSQP') solver.
• NLopt (C/C++ implementation, with numerous interfaces including Julia, Python, R, MATLAB/Octave), implemented by Dieter Kraft as part of a package for optimal control, and modified by S. G. Johnson.^[2][3]
• ALGLIB SQP solver (C++, C#, Java, Python API)

and commercial

• LabVIEW
• KNITRO^[4] (C, C++, C#, Java, Python, Julia, Fortran)
• NPSOL (Fortran)
• SNOPT (Fortran)
• NLPQL (Fortran)
• MATLAB
• SuanShu (Java)

• Newton's method
• Secant method
• Model Predictive Control

• Bonnans, J. Frédéric; Gilbert, J. Charles; Lemaréchal, Claude; Sagastizábal, Claudia A. (2006). Numerical optimization: Theoretical and practical aspects. Universitext (Second revised ed. of translation of 1997 French ed.). Berlin: Springer-Verlag. pp. xiv+490. doi:10.1007/978-3-540-35447-5. ISBN 978-3-540-35445-1. MR 2265882.
• Jorge Nocedal and Stephen J. Wright (2006). Numerical Optimization. Springer. ISBN 978-0-387-30303-1.

• Sequential Quadratic Programming at NEOS guide