Solution output

PRINT file in accordance with the Solution keyword. Some header information appears first to identify the problem and the final state of the optimization procedure. A ROWS section and a COLUMNS section then follow, giving one line of information for each row and column. The format used is similar to certain commercial systems, though there is no industry standard.

An example of the printed solution is given in Examples. In general, numerical values are output with format f16.5. The maximum record length is 111 characters, including the first (carriage-control) character.

To reduce clutter, a dot ".'' is printed for any numerical value that is exactly zero. The values 1 are also printed specially as 1.0 and -1.0. Infinite bounds (1020 or larger) are printed as None.

Note : If two problems are the same except that one minimizes an objective f(x) and the other maximizes –f(x), their solutions will be the same but the signs of the dual variables p_i and the reduced gradients d_j will be reversed.

The ROWS section

General linear constraints take the form l Ax u. The ith constraint is therefore of the form

a a^Tx b,

and the value of a^Tx is called the row activity. Internally, the linear constraints take the form Ax - s = 0, where the slack variables s should satisfy the bounds lsu. For the ith "row", it is the slack variable s_i that is directly available, and it is sometimes convenient to refer to its state. Slacks may be basic or nonbasic (but not superbasic).

Nonlinear constraints a F_i(x) + a^Tx b are treated similarly, except that the row activity and degree of infeasibility are computed directly from F_i(x) + a^Tx rather than from s_i.

Label

Description

Number

The value n+i. This is the internal number used to refer to the ith slack in the iteration log.

Row

The name of the ith row.

State

The state of the ith row relative to the bounds a and b. The various states possible are as follows.

LL	The row is at its lower limit, a.
UL	The row is at its upper limit, b.
EQ	The limits are the same (a = b).
BS	The constraint is not binding. s_i is basic.

A key is sometimes printed before the State to give some additional information about the state of the slack variable.

A	Alternative optimum possible. The slack is nonbasic, but its reduced gradient is essentially zero. This means that if the slack were allowed to start moving from its current value, there would be no change in the objective function. The values of the basic and superbasic variables might change, giving a genuine alternative solution. The values of the dual variables might also change.
D	Degenerate. The slack is basic, but it is equal to (or very close to) one of its bounds.
I	Infeasible. The slack is basic and is currently violating one of its bounds by more than the Feasibility tolerance.
N	Not precisely optimal . The slack is nonbasic. Its reduced gradient is larger than the Optimality tolerance .

Note : If Scale option > 0, the tests for assigning A, D, I, N are made on the scaled problem, since the keys are then more likely to be meaningful.

Activity

The row value a^Tx (or F_i(x) + a^Tx for nonlinear rows).

Slack activity

The amount by which the row differs from its nearest bound. (For free rows, it is taken to be minus the Activity.)

Lower limit

a, the lower bound on the row.

Upper limit

b, the upper bound on the row.

Dual activity

The value of the dual variable p_i, often called the shadow price (or simplex multiplier) for the ith constraint. The full vector p always satisfies B^Tp = g_B, where B is the current basis matrix and g_B contains the associated gradients for the current objective function.

The constraint number, i.

The COLUMNS section

Here we talk about the "column variables" x_j, j = 1:n. We assume that a typical variable has bounds a x_j b.

Label

Description

Number

The column number, j. This is the internal number used to refer to x_j in the iteration log.

Column

The name of x_j.

State

The state of the ith row relative to the bounds a and b. The various states possible are as follows.

LL	x_j is nonbasic at its lower limit, a.
UL	x_j is nonbasic at its upper limit, b.
EQ	x_j is nonbasic and fixed at the value a = b.
FR	x_jis nonbasic at some value strictly between its bounds: a < x_j < b.
BS	x_j is basic. Usually a < x_j < b.
SBS	x_j is superbasic. Usually a < x_j< b.

A key is sometimes printed before the State to give some additional information about the state of x_j.

A	Alternative optimum possible. The slack is nonbasic, but its reduced gradient is essentially zero. This means that if x_jwere allowed to start moving from its current value, there would be no change in the objective function. The values of the basic and superbasic variables might change, giving a genuine alternative solution. The values of the dual variables might also change.
D	Degenerate. x_j is basic, but it is equal to (or very close to) one of its bounds.
I	Infeasible. x_j is basic and is currently violating one of its bounds by more than the Feasibility tolerance.
N	Not precisely optimal. x_j is nonbasic. Its reduced gradient is larger than the Optimality tolerance .

Note : If Scale option > 0, the tests for assigning A, D, I, N are made on the scaled problem, since the keys are then more likely to be meaningful.

Activity

The value of the variable x_j.

Obj Gradient

g_j, the jth component of the gradient of the (linear or nonlinear) objective function. (If any x_j is infeasible, g_j is the gradient of the sum of infeasibilities.)

Lower limit

a, the lower bound on x_j.

Upper limit

b, the upper bound on x_j.

Reduced gradnt

The reduced gradient d_j = g_j–p^Ta_j, where a_j is the jth column of the constraint matrix (or the jth column of the Jacobian at the start of the final major iteration).

M+J

The value m+j .