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SNOPT 525
SQOPT uses a reduced-Hessian active-set method implemented as a reduced-gradient method similar to that in
MINOS [14].
At each minor iteration, the constraints Ax − s = b are partitioned into the form
Bx B + Sx S + Nx N = b,
where the basis matrix B is square and nonsingular and the matrices S, N are the remaining columns of (A −I).
The vectors x B , x S , x N are the associated basic, superbasic, and nonbasic variables components of (x, s).
The term active-set method arises because the nonbasic variables x N are temporarily frozen at their upper or
lower bounds, and their bounds are considered to be active. Since the general constraints are satisfied also, the
set of active constraints takes the form
(
) ⎛ ⎞
x B
( )
B S N ⎜ ⎟ b
⎝ x S ⎠ = ,
I
x N
x N
where x N represents the current values of the nonbasic variables. (In practice, nonbasic variables are sometimes
frozen at values strictly between their bounds.) The reduced-gradient method chooses to move the superbasic
variables in a direction that will improve the objective function. The basic variables “tag along” to keep Ax−s = b
satisfied, and the nonbasic variables remain unaltered until one of them is chosen to become superbasic.
At a nonoptimal feasible point (x, s) we seek a search direction p such that (x, s) + p remains on the set of active
constraints yet improves the QP objective. If the new point is to be feasible, we must have Bp B +Sp S +Np N = 0
and p N = 0. Once p S is specified, p B is uniquely determined from the system Bp B = −Sp S . It follows that the
superbasic variables may be regarded as independent variables that are free to move in any desired direction. The
number of superbasic variables (n S say) therefore indicates the number of degrees of freedom remaining after the
constraints have been satisfied. In broad terms, n S is a measure of how nonlinear the problem is. In particular,
n S need not be more than one for linear problems.
2.5 The reduced Hessian and reduced gradient
The dependence of p on p S may be expressed compactly as p = Zp S , where Z is a matrix that spans the null
space of the active constraints:
⎛ ⎞
−B −1 S
⎜ ⎟
Z = P ⎝ I ⎠ (32.3)
0
where P permutes the columns of (A −I) into the order (B S N). Minimizing q(x, x k ) with respect to p S now
involves a quadratic function of p S :
g T Zp S + 1 2 pT S Z T HZp S , (32.4)
where g and H are expanded forms of g k and H k defined for all variables (x, s). This is a quadratic with
Hessian Z T HZ (the reduced Hessian) and constant vector Z T g (the reduced gradient). If the reduced Hessian is
nonsingular, p S is computed from the system
Z T HZp S = −Z T g. (32.5)
The matrix Z is used only as an operator, i.e., it is not stored explicitly. Products of the form Zv or Z T g are
obtained by solving with B or B T . The package LUSOL[13] is used to maintain sparse LU factors of B as the
BSN partition changes. From the definition of Z, we see that the reduced gradient can be computed from
B T π = g B , Z T g = g S − S T π,
where π is an estimate of the dual variables associated with the m equality constraints Ax − s = b, and g B is the
basic part of g.