### Final results

The final results lists the final values for the problem variables and the computed values of the constraint functions and the objective function. The Functions table shows the values of the constraint functions computed using the final values of the variables.

The GRG algorithm uses the binding constraints to solve for some variables, classified as basic, in terms of the remaining variables, thus reducing the number of independent variables. The variables that are not basic are classified as nonbasic, if they are at one of their bounds and superbasic, otherwise. The classification is given in the Status column in the Final Results table. The reduced objective function is the objective function treated as a function of the nonbasic and superbasic variables. The gradient of the reduced objective function is called the reduced gradient.

The Reduced Gradient column gives the value of the reduced gradient for each nonbasic and superbasic variable. Reduced gradient values for basic variables are 0, by definition. Since nonbasic variables are at one of their bounds, the reduced gradient, with respect to a nonbasic variable, should have the correct sign. For example, when the problem is a minimization and the nonbasic variable is at its lower bound, the reduced gradient should be ≥ 0. This is an indication that a small increase in the value of the variable results in an increase in the value of the objective (that is, a move away from the optimum).

The reduced gradient of a nonbasic variable that is at its upper bound should be ≤ 0. Reduced gradient values are affected by the scale of the variable and the scale of the objective function.  Therefore, Embed scales these reduced gradient values by the value of the variable and by the reciprocal of the value of the objective function and then computes the K-T factor as the maximum over all the superbasics. The K-T factor is printed in the solution process section of the report in the Norm of Red.Grad column. A small K-T factor is a good indication that a local optimum has been located. If this value is ≤ epstop, the stopping criteria has been satisfied and Embed terminates with inform = 0. Embed also terminates, with inform = 1, when the objective function converges to a relative error of epstop for nstop consecutive iterations.

Final Results

Functions

 Distance Initial Final from Lagrange No. Name Value Value Status Nearest Multiplier Bound ­­­­___ ______ ________ ______ _________ ______________ __________ 1 G 0.17805 0.62148 Objective 2 G 0.24999 0.4 UpperBnd 6.43e-009  :U -0.51091

Variables

 Distance Initial Final from Reduced No. Name Value Value Status Nearest Gradient Bound ­­­­___ ______ ________ ______ _________ ______________ __________ 1 X 1 1.7278 Basic 8.272  :U 2 X 1 1.8531 SupBasic 8.147  :U 0.000114