zfolwick said:I can't for the life of me understand this topic. given a point x =(1,1,1,1,1,1,1) and a primal LP, does the point solve the primal? An internet search revealed no answer to my question, only criteria which involves knowing y. I am aware that [itex]\sum[/itex]aijxj < b then yi = 0, so I at least know which elements of y are zero, but the rest of the steps elude me.
"Testing optimality via complementary slackness" is a mathematical principle used in optimization problems to determine the optimal solution. It states that if a pair of variables satisfy certain conditions known as complementary slackness, then they are both optimal.
Complementary slackness works by comparing the values of the constraints and the objective function in an optimization problem. If the value of a constraint is equal to its upper or lower bound, then its corresponding variable must be zero. This is known as complementary slackness.
The conditions for complementary slackness are that the optimal value of a constraint must be equal to its upper or lower bound, and the optimal value of the corresponding variable must be zero. In other words, the constraint must be either active or inactive at the optimal solution, and the corresponding variable must be either positive or zero.
Complementary slackness is used in optimization problems to determine the optimal solution. By checking the conditions of complementary slackness, we can determine which variables are active and which are inactive at the optimal solution. This helps in simplifying the problem and finding the optimal solution more efficiently.
The benefits of using complementary slackness in optimization problems include simplifying the problem and reducing the number of variables to be considered. It also helps in identifying the optimal solution more efficiently and can be used to check the optimality of a solution found through other methods. Additionally, it provides insight into the sensitivity of the optimal solution with respect to changes in the constraints and objective function.