GAMS — The Solver Manuals - Available Software

MINOS 337
to find an approximate solution to the one-dimensional problem
minimize F (x + αp)
α
subject to 0 < α < β
where β is determined by the bounds on the variables. Another important piece in GAMS/MINOS is a step-length
procedure used in the linesearch to determine the step-length α (see [6]). The number of nonlinear function
evaluations required may be influenced by setting the Linesearch tolerance, as discussed in Section 9.
As a linear programming solver, an equation B T π = gB is solved to obtain the dual variables or shadow prices π
where gB is the gradient of the objective function associated with basic variables. It follows that gB − B T π = 0.
The analogous quantity for superbasic variables is the reduced-gradient vector Z T g = gs−s T π; this should also be
zero at an optimal solution. (In practice its components will be of order r||π|| where r is the optimality tolerance,
typically 10 −6 , and ||π|| is a measure of the size of the elements of π.)
3.3 Problems with Nonlinear Constraints
If any of the constraints are nonlinear, GAMS/MINOS employs a project Lagrangian algorithm, based on a method
due to [13], see [9]. This involves a sequence of major iterations, each of which requires the solution of a linearly
constrained subproblem. Each subproblem contains linearized versions of the nonlinear constraints, as well as the
original linear constraints and bounds.
At the start of the k th major iteration, let x k be an estimate of the nonlinear variables, and let λ k be an estimate
of the Lagrange multipliers (or dual variables) associated with the nonlinear constraints. The constraints are
linearized by changing f(x) in equation (2) to its linear approximation:
or more briefly
f ′ (x, x k ) = f(x k ) + J(x k )(x − x k )
f ′ = f k + J k (x − x k )
where J(x k ) is the Jacobian matrix evaluated at x k . (The i-th row of the Jacobian is the gradient vector of the
i-th nonlinear constraint function. As for the objective gradient, GAMS calculates the Jacobian using symbolic
differentiation).
The subproblem to be solved during the k-th major iteration is then
minimize F (x) + c T x + d T y − λ T
x,y
k (f − f ′ ) + 0.5ρ(f − f ′ ) T (f − f ′ ) (5)
subject to f ′ + A 1 y ∼ b 1 (6)
A 2 x ( + A 3 y ∼ b 2 (7)
x
l ≤ ≤ u (8)
y)
The objective function (5) is called an augmented Lagrangian. The scalar ρ is a penalty parameter, and the term
involving ρ is a modified quadratic penalty function.
GAMS/MINOS uses the reduced-gradient algorithm to minimize (5) subject to (6) – (8). As before, slack variables
are introduced and b 1 and b 2 are incorporated into the bounds on the slacks. The linearized constraints take the
form ( ) ( ) ( ) ( ) ( )
Jk A 1 x I 0 s1 Jk x
+
= k − f k
A 2 A 3 y 0 I s 2 0
This system will be referred to as Ax+Is = 0 as in the linear case. The Jacobian J k is treated as a sparse matrix,
the same as the matrices A 1 , A 2 , and A 3 .
In the output from GAMS/MINOS, the term Feasible subproblem indicates that the linearized constraints have
been satisfied. In general, the nonlinear constraints are satisfied only in the limit, so that feasibility and optimality
occur at essentially the same time. The nonlinear constraint violation is printed every major iteration. Even if
it is zero early on (say at the initial point), it may increase and perhaps fluctuate before tending to zero. On
well behaved problems, the constraint violation will decrease quadratically (i.e., very quickly) during the final few
major iteration.