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

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Consider a nonlinear programming problem of the form:

 

The Lagrangian for this problem is[1]

 

where   and   are Lagrange multipliers.

LP phase

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In the LP phase of SLQP, the following linear program is solved:

 

Let   denote the active set at the optimum   of this problem, that is to say, the set of constraints that are equal to zero at  . Denote by   and   the sub-vectors of   and   corresponding to elements of  .

EQP phase

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In the EQP phase of SLQP, the search direction   of the step is obtained by solving the following equality-constrained quadratic program:

 

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

See also

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Notes

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  1. ^ Jorge Nocedal and Stephen J. Wright (2006). Numerical Optimization. Springer. ISBN 0-387-30303-0.

References

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