- Algorithm basics LP phase EQP phase
- See also
- Notes
- References
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-fledge quadratic programs.
Algorithm basics
Consider a nonlinear programming problem of the form:
The Lagrangian for this problem is[1]
where 
and 
are Lagrange multipliers.
LP phase
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
In the EQP phase of SLQP, the search direction 
of the step is obtained by solving the following 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
- Newton's method
- Secant method
- Sequential linear programming
- Sequential quadratic programming
Notes
References
- {{ cite book | year=2006|url=http://www.ece.northwestern.edu/~nocedal/book/num-opt.html| title= Numerical Optimization| publisher=Springer.|isbn=0-387-30303-0| author=Jorge Nocedal and Stephen J. Wright}}
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