Inequality constraints inactive

Then at each point, check which of the inequahty constraints are active, or exactly equal to zero. These can be placed into the active set and treated as equahties. The remaining can be put aside to be used only for testing. A step can then be proposed using the GRG algorithm. If it does not cause one to violate any of the inactive inequality constraints, the step is taken. Otherwise one can add the closest inactive inequality constraint to the active set. Finding the closet inactive equahty will almost certainly require a hne search in the direc tion proposed by the GRG algorithm. 

Complementarity Inequality constraints are either strictly satisfied (active) or inactive, in which case they are irrelevant to the solu-... 

If this constraint is inactive, that is, the optimum value of xu is less than 40,000 kg/day, then, in effect, there are still 3 degrees of freedom. If, however, the optimization procedure yields a value of xn = 40,000 (the optimum lies on the constraint, such as shown in Figure 1.2), then inequality constraint/becomes an equality constraint, resulting in only 2 degrees of freedom that can be used for optimization. You should recognize that it is possible to add more inequality constraints, such as constraints on materials supplies, in the model, for example,... 

We can state these ideas precisely as follows. Consider any optimization problem with n variables, let x be any feasible point, and let act(x) be the number of active constraints at x. Recall that a constraint is active at x if it holds as an equality there. Hence equality constraints are active at any feasible point, but an inequality constraint may be active or inactive. Remember to include simple upper or lower bounds on the variables when counting active constraints. We define the number of degrees of freedom (dof) at x as... 

These results may be restated to include all constraints by defining the multiplier uj to be zero if gj(x ) previous example uj, the multiplier of the inactive constraint g3, is zero. Then we can say that uj > 0 if g/x ) = c-, and uj = 0 if gj(x ) < Cp thus the product uj gj(x) - cj is zero for all j. This property, that inactive inequality constraints have zero multipliers, is called complementary slackness. Conditions (8.21) and (8.22) then become... 

Let the starting point be (1, 0), at which the objective value is 6.5 and the inequality is satisfied strictly, that is, its slack is positive (s = 1). At this point the bounds are also all satisfied, although y is at its lower bound. Because all of the constraints (except for bounds) are inactive at the starting point, there are no equalities that must be solved for values of dependent variables. Hence we proceed to minimize the objective subject only to the bounds on the nonbasic variables x and y. There are no basic variables. The reduced problem is simply the original problem ignoring the inequality constraint. In solving this reduced problem, we do keep track of the inequality. If it becomes active or violated, then the reduced problem changes. 

The vector of Kuhn-Tucker multipliers for the active inequality constraints is negative Kuhn-Tucker multipliers for the inactive constraints are zero. 

Modification of Constraint Sets at X°. At the calculated feasible point X°, some of the inactive constraints (i.e. will become active. In order to preserve the determinancy features of the system of equations, some of the previously active constraints (i.e. cp t) will become inactive. Starting at the current optimum X where cgpt is the set of active inequality constraints,... 

Case 2 Here the strict inequality K u) < ko is in effect and the inequality constraint is said to be inactive. Let K u) = kc where kc is some real number less than ko- Then Equation (4.20) for K u) = kc is... 

In deriving the above equation, we have considered the fact that all equality constraints are satisfied at the optimal steady state, and the multipliers are zero for inactive inequality constraints (see Section 4.5, p. 109). 

Note that V L = 0 gives Eqs. (18.21), which are the definitions of the slack variables and need not be expressed in the KKT conditions. Note also that L = 2A.jZ, = 0, and, using Eqs. (18.21), Eqs. (18.26) result. These are the so-called complementary slackness equations. For constraint i, either the residual of the constraint is zero, g, = 0, or the Kuhn-Tucker multiplier is zero, X., = 0, or both are zero that is, when the constraint is inactive (gj > 0), the Kuhn-Tucker multiplier is zero, and when the Kuhn-Tucker multiplier is greater than zero, the constraint must be active (g, = 0). Stated differently, there is slackness in either the constraint or the Kuhn-Tucker multiplier. Finally, it is noted that V c x is the Jacobian matrix of the equality constraints, J x, and V g i is the Jacobian matrix of the inequality constraints, K[x). 

What about the multipliers for die inactive inequality constraints Let us rewrite (5.88) as... 

---