GAMS — The Solver Manuals - Available Software

526 SNOPT
By analogy with the elements of Z T g, we define a vector of reduced gradients (or reduced costs) for all variables
in (x, s):
( )
A T
d = g − π, so that d S = Z T g.
−I
At a feasible point, the reduced gradients for the slacks s are the dual variables π.
The optimality conditions for subproblem QP k (32.2) may be written in terms of d. The current point is optimal
if d j ≥ 0 for all nonbasic variables at their lower bounds, d j ≤ 0 for all nonbasic variables at their upper bounds,
and d j = 0 for all superbasic variables (d S = 0). In practice, SNOPT requests an approximate QP solution
(̂x k , ŝ k , ̂π k ) with slightly relaxed conditions on d j .
If d S = 0, no improvement can be made with the current BSN partition, and a nonbasic variable with nonoptimal
reduced gradient is selected to be added to S. The iteration is then repeated with n S increased by one.
At all stages, if the step (x, s)+p would cause a basic or superbasic variable to violate one of its bounds, a shorter
step (x, s) + αp is taken, one of the variables is made nonbasic, and n S is decreased by one.
The process of computing and testing reduced gradients d N is known as pricing (a term introduced in the context
of the simplex method for linear programming). Pricing the jth variable means computing d j = g j − a T j π, where
a j is the jth column of (A −I). If there are significantly more variables than general constraints (i.e., n ≫ m),
pricing can be computationally expensive. In this case, a strategy known as partial pricing can be used to compute
and test only a subset of d N .
Solving the reduced Hessian system (32.5) is sometimes expensive. With the option QPSolver Cholesky, an
upper-triangular matrix R is maintained satisfying R T R = Z T HZ. Normally, R is computed from Z T HZ at
the start of phase 2 and is then updated as the BSN sets change. For efficiency the dimension of R should not
be excessive (say, n S ≤ 1000). This is guaranteed if the number of nonlinear variables is “moderate”. Other
QPSolver options are available for problems with many degrees of freedom.
2.6 The merit function
After a QP subproblem has been solved, new estimates of the NLP solution are computed using a linesearch on
the augmented Lagrangian merit function
M(x, s, π) = f(x) − π T ( F (x) − s N
)
+
1
2(
F (x) − sN
) T
D
(
F (x) − sN
)
, (32.6)
where D is a diagonal matrix of penalty parameters. If (x k , s k , π k ) denotes the current solution estimate and
(̂x k , ŝ k , ̂π k ) denotes the optimal QP solution, the linesearch determines a step α k (0 < α k ≤ 1) such that the new
point
⎛ ⎛
⎜
⎝
x k+1
s k+1
⎞
⎟
⎠ =
π k+1
⎜
⎝
x k
s k
⎞ ⎛ ⎞
̂x k − x k
⎟ ⎜ ⎟
⎠ + α k ⎝ ŝ k − s k ⎠ (32.7)
π k ̂π k − π k
gives a sufficient decrease in the merit function. When necessary, the penalties in D are increased by the minimumnorm
perturbation that ensures descent for M [11]. As in NPSOL, s N is adjusted to minimize the merit function
as a function of s prior to the solution of the QP subproblem. For more details, see [9, 3].
2.7 Treatment of constraint infeasibilities
SNOPT makes explicit allowance for infeasible constraints.
solving a problem of the form
Infeasible linear constraints are detected first by
FLP
minimize e T (v + w)
x,v,w
(
)
x
subject to l ≤
≤ u, v ≥ 0, w ≥ 0,
A L x − v + w