Inequality constraints active

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. 

Assume certain inequality constraints will be active at the final solution. The necessaiy conditions for optimality are... 

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

Constraint Qualification For a local optimum to satisfy the KKT conditions, an additional regularity condition is required on the constraints. This can be defined in several ways. A typical condition is that the active constraints at x be linearly independent i.e., the matrix [Vh(x ) I VgA(x )] is full column rank, where gA is the vector of inequality constraints with elements that satisfy g x ) = 0. With this constraint qualification, the KKT multipliers (X, v) are guaranteed to be unique at the optimal solution. 

In Section 1.5 we briefly discussed the relationships of equality and inequality constraints in the context of independent and dependent variables. Normally in design and control calculations, it is important to eliminate redundant information and equations before any calculations are performed. Modem multivariable optimization software, however, does not require that the user clearly identify independent, dependent, or superfluous variables, or active or redundant constraints. If the number of independent equations is larger than the number of decision variables, the software informs you that no solution exists because the problem is overspecified. Current codes have incorporated diagnostic tools that permit the user to include all possible variables and constraints in the original problem formulation so that you do not necessarily have to eliminate constraints and variables prior to using the software. Keep in mind, however, that the smaller the dimensionality of the problem introduced into the software, the less time it takes to solve the problem. 

As mentioned in Chapter 1, the occurrence of linear inequality constraints in industrial processes is quite common. Inequality constraints do not affect the count of the degrees of freedom unless they become active constraints. Examples of such constraints follow ... 

These can also become active constraints if the optimum lies on the constraint boundary. Note that we can also place inequality constraints on production of E, F, and G in order to satisfy market demand or sales constraints... 

Now the analysis is much more complex, and it is clear that more potential equality constraints exist than variables if all of the inequality constraints become active. It is possible that optimization could lead to a situation where no degrees of freedom would be left—one set of the inequality constraints would be satisfied as equalities. This outcome means no variables remain to be optimized, and the optimal solution reached would be at the boundaries, a subset of the inequality constraints. 

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... 

The KTC comprise both the necessary and sufficient conditions for optimality for smooth convex problems. In the problem (8.25)-(8.26), if the objective fix) and inequality constraint functions gj are convex, and the equality constraint functions hj are linear, then the feasible region of the problem is convex, and any local minimum is a global minimum. Further, if x is a feasible solution, if all the problem functions have continuous first derivatives at x, and if the gradients of the active constraints at x are independent, then x is optimal if and only if the KTC are satisfied at x. ... 

Despite the exactness feature of Pv no general-purpose, widely available NLP solver is based solely on the Lx exact penalty function Pv This is because Px also has a negative characteristic it is nonsmooth. The term hj(x) has a discontinuous derivative at any point x where hj (x) = 0, that is, at any point satisfying the y th equality constraint in addition, max 0, gj (x) has a discontinuous derivative at any x where gj (x) = 0, that is, whenever the yth inequality constraint is active, as illustrated in Figure 8.6. These discontinuities occur at any feasible or partially feasible point, so none of the efficient unconstrained minimizers for smooth problems considered in Chapter 6 can be applied, because they eventually encounter points where Px is nonsmooth. 

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. 

Although, as explained in Chapter 9, many optimization problems can be naturally formulated as mixed-integer programming problems, in this chapter we will consider only steady-state nonlinear programming problems in which the variables are continuous. In some cases it may be feasible to use binary variables (on-off) to include or exclude specific stream flows, alternative flowsheet topography, or different parameters. In the economic evaluation of processes, in design, or in control, usually only a few (5-50) variables are decision, or independent, variables amid a multitude of dependent variables (hundreds or thousands). The number of dependent variables in principle (but not necessarily in practice) is equivalent to the number of independent equality constraints plus the active inequality constraints in a process. The number of independent (decision) variables comprises the remaining set of variables whose values are unknown. Introduction into the model of a specification of the value of a variable, such as T = 400°C, is equivalent to the solution of an independent equation and reduces the total number of variables whose values are unknown by one. 

Feasible path algorithms. The equality constraints and active inequality constraints are satisfied at the end of every intermediate stage of the calculations. 

This form is convenient in that the active inequality constraints can now be replaced in the QP by all of the inequalities, with the result that Sa is determined directly from the QP solution. Finally, since second derivatives may often be hard to calculate and a unique solution is desired for the QP problem, the Hessian matrix, is approximated by a positive definite matrix, B, which is constructed by a quasi-Newton formula and requires only first-derivative information. Thus, the Newton-type derivation for (2) leads to a nonlinear programming algorithm based on the successive solution of the following QP subproblem ... 

However the accurate treatment of state variable inequality constraints presents a few problems. Parameter optimization problems obtained by discretizing the control profile generally allow inequality constraints to be active only at a finite set of points, simply because a finite set of decisions cannot influence an infinite number of values (i.e., keeping the state fixed at every point in a finite time period). 

However, if we are interested in an accurate representation of the exact optimal control profile, the problem becomes more complicated. First, from (16) we recognize the possibility that the optimal solution can cause time-dependent inequality constraints to become active for a finite period of time. 

Therefore, for large optimal control problems, the efficient exploitation of the structure (to obtain 0(NE) algorithms) still remains an unsolved problem. As seen above, the structure of the problem can be complicated greatly by general inequality constraints. Moreover, the number of these constraints will also grow linearly with the number of elements. One can, in fact, formulate an infinite number of constraints for these problems to keep the profiles bounded. Of course, only a small number will be active at the optimal solution thus, adaptive constraint addition algorithms can be constructed for selecting active constraints. 

Remark 4 It should be emphasized that we need to include only the linearizations of the inequality constraints that are active at the solution of the primal problem P(yk). This is important because it reduces the number of linearizations that need to included in the master problem. 

Activation and deactivation of inequality and equality constraints can be obtained in a similar way. For instance, let us consider the model of a process unit i that consists of one inequality g(x) < 0 and one equality h(x) = 0. If the process unit i does not exist (i.e., yi = 0), then both the equality and inequality should be relaxed. If, however, the process unit i exists (i.e., yi = 1), then the inequality and equality constraints should be activated. This can be expressed by introducing positive slack variables for the equality and inequality constraint and writing the model as... 

Equality and inequality constraints on any proportion or any group of proportions (e.g., 80% < proportion of support within the catalyst <90%, the proportion of Mg within the active components + the proportion of Mn within the active components =50%). 

Equality and inequality constraints on the number of simultaneously present components or on the number of those among them that belong to a given class of the component types hierarchy (e.g., the number of all component lies between 3 and 6, the number of active components is at least twice that of dopants). 

Selection of Process Controlled and Manipulated Variables. At the calculated optimum X (x, m, d ) of the above problem (Pi), some of the inequality constraints will be active. The regulatory control objectives and the active design constraints (i.e. g ) at the current optimum will constitute the class of primary controlled variables denoted by c, i.e. 

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,... 

The situation is quite different when inequality constraints are included in the MPC on-line optimization problem. In the sequel, we will refer to inequality constrained MPC simply as constrained MPC. For constrained MPC, no closed-form (explicit) solution can be written. Because different inequahty constraints may be active at each time, a constrained MPC controller is not linear, making the entire closed loop nonlinear. To analyze and design constrained MPC systems requires an approach that is not based on linear control theory. We will present the basic ideas in Section III. We will then present some examples that show the interesting behavior that MPC may demonstrate, and we will subsequently explain how MPC theory can conceptually simplify and practically improve MPC. 

Linear programs always solve to a global optimum. The optimum must lie on the boundary at an intersection between constraints, which is known as a vertex of the feasible region. The inequality constraints that intersect at the optimum are said to be active and have fi(x) = 0, where x is the vector of decision variables. 

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