Superbasic variables

To solve this first reduced problem, follow the steps of the descent algorithm outlined at the start of this section with some straightforward modifications that account for the bounds on x and y. When a nonbasic variable is at a bound, we must decide whether it should be allowed to leave the bound or be forced to remain at that bound for the next iteration. Those nonbasic variables that will not be kept at their bounds are called superbasic variables [this term was coined by Murtaugh and Saunders (1982)]. In step 1 the reduced gradient off(x,y) is... 

The variable y becomes superbasic. Because s is at its lower bound of zero, consider whether s should be allowed to leave its bound, that is, be a superbasic variable. Because its reduced gradient term is , increasing s (which is the only feasible change for s) increases the objective value. Because we are minimizing F, fix s at zero this corresponds to staying on the line x = y. The search direction d = and new values for y are generated from... 

First, the first element in the reduced gradient with respect to the superbasic variable y is zero. Second, because the reduced gradient (the derivative with respect to s) is 1, increasing s (the only feasible change to s) causes an increase in the objective value. These are the two necessary conditions for optimality for this reduced problem and the algorithm terminates at (1.5, 1.5) with an objective value of 2.0. 

GAMS output Infeasible solution. There are no superbasic variables.) ... 

The feasibility approach FA rounds the relaxed NLP solution to an integer solution with the least local degradation by successively forcing the superbasic variables to become nonbasic based on the reduced cost information. 

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