Sequential linear-quadratic programming

Sequential linear-quadratic programming (SLQP) is an iterative method for nonlinear optimization problems where objective function and constraints are twice continuously differentiable. Similarly to sequential quadratic programming (SQP), SLQP proceeds by solving a sequence of optimization subproblems. The difference between the two approaches is that:


 * in SQP, each subproblem is a quadratic program, with a quadratic model of the objective subject to a linearization of the constraints
 * in SLQP, two subproblems are solved at each step: a linear program (LP) used to determine an active set, followed by an equality-constrained quadratic program (EQP) used to compute the total step

This decomposition makes SLQP suitable to large-scale optimization problems, for which efficient LP and EQP solvers are available, these problems being easier to scale than full-fledged quadratic programs.

It may be considered related to, but distinct from, quasi-Newton methods.

Algorithm basics
Consider a nonlinear programming problem of the form:


 * $$\begin{array}{rl}

\min\limits_{x} & f(x) \\ \mbox{s.t.} & b(x) \ge 0 \\ & c(x) = 0. \end{array}$$

The Lagrangian for this problem is
 * $$\mathcal{L}(x,\lambda,\sigma) = f(x) - \lambda^T b(x) - \sigma^T c(x),$$

where $$\lambda \ge 0$$ and $$\sigma$$ are Lagrange multipliers.

LP phase
In the LP phase of SLQP, the following linear program is solved:
 * $$\begin{array}{rl}

\min\limits_{d} & f(x_k) + \nabla f(x_k)^Td\\ \mathrm{s.t.} & b(x_k) + \nabla b(x_k)^Td \ge 0 \\ & c(x_k) + \nabla c(x_k)^T d = 0. \end{array}$$

Let $${\cal A}_k$$ denote the active set at the optimum $$d^*_{\text{LP}}$$ of this problem, that is to say, the set of constraints that are equal to zero at $$d^*_{\text{LP}}$$. Denote by $$b_{{\cal A}_k}$$ and $$c_{{\cal A}_k}$$ the sub-vectors of $$b$$ and $$c$$ corresponding to elements of $${\cal A}_k$$.

EQP phase
In the EQP phase of SLQP, the search direction $$d_k$$ of the step is obtained by solving the following equality-constrained quadratic program:
 * $$\begin{array}{rl}

\min\limits_{d} & f(x_k) + \nabla f(x_k)^Td + \tfrac{1}{2} d^T \nabla_{xx}^2 \mathcal{L}(x_k,\lambda_k,\sigma_k) d\\ \mathrm{s.t.} & b_{{\cal A}_k}(x_k) + \nabla b_{{\cal A}_k}(x_k)^Td = 0 \\ & c_{{\cal A}_k}(x_k) + \nabla c_{{\cal A}_k}(x_k)^T d = 0. \end{array}$$

Note that the term $$f(x_k)$$ in the objective functions above may be left out for the minimization problems, since it is constant.