Inequality constraint, selection

In the matrix J, only the equality constraints that do not contain variables that appear only in that equation and that are not on a bound constraint, and some (or none of the) active inequality constraints selected from the satisfied inequality constraints can be included. 

The possibility of including in the matrix J only some (or no) active inequality constraints selected from the satisfied inequality constraints may make solving degeneracy problems and the search for linearly independent active constraints... 

We use a method that implements the Unbiased Prediction Risk criterion [13] to provide a data-driven approach for the selection of the regularization parameter. The equality constraints are handled with LQ factorization [14] and an iterative method suggested by Villalobos and Wahba [15] is used to incorporate the inequality constraints [10]. The method is well suited for the relatively large-scale problem associated with analyzing each image voxel as no user intervention is required and all the voxels can be analyzed in parallel. 

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. 

By selecting y = 0 as in our illustration, we notice that the inequality constraint in (6.28) is in fact nonconvex in x2. Hence, application application of OA to (6.28) cannot guarantee global optimality due to having nonconvexity in x2. 

Unconstrained Optimization Unconstrained optimization refers to the case where no inequality constraints are present and all equality constraints can be eliminated by solving for selected dependent variables followed by substitution for them in the objective function. Very few realistic problems in process optimization are unconstrained. However, the availability of efficient unconstrained optimization techniques is important because these techniques must be applied in real time, and iterative calculations may require excessive computer time. Two classes of unconstrained techniques are single-variable optimization and multivariable optimization. 

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. 

At the heart of any RTO system is the optimization subsystem, which may be used to solve a number of problems including data reconciliation, parameter estimation, and the determination of the optimal plant operating conditions or set points. Each of these optimization tasks has similar characteristics and it is these characteristics that guide the selection of appropriate solver technologies. Each of the optimization problems encountered within an RTO system is based on the process model, or at least a portion of it, and most often involves a very small number of degrees of freedom (e.g., less than 100 independent decision variables) in comparison to model size and total number of model variables. Furthermore, the optimization problems encountered within RTO systems are typically, but not always, nonlinear. For the purposes of this discussion, the prototypical optimization problem will be considered to be nonlinear with continuous variables, will be based on a model that may contain hundreds of thousands of variables and equations, will contain inequality constraints and bounds, and will have fewer than 1000 degrees of freedom. 

Similar kinds of constraints involve the reflux ratio in distillation, which must exceed the minimum value for the required separation. If the distillation tower pressure is adjusted, the minimum reflux ratio will change and the actual ratio must be maintained above the minimum value. Even when optimization is not performed, the decision variable values must be selected to avoid violating the inequality constraints. In some cases, the violations can be detected when examining the simulation results. In other cases, the imit subroutines are unable to solve the equations as, for example, when the reflux ratio is adjusted to a value below the minimum value for a specified split of the key components. 

The iteration continues only if the values of for some artificial constraints are nonzero or if some active inequality constraints have Aj < 0 otherwise, the solution is found. Based on the values of k, the elements of a vector h are selected by certain criteria described later and the following linear system is solved ... 

Hz > 1 active inequality constraints are removed. We can decide to remove them either because they became passive or in order to enter the feasible space, exploit the sparsity of the rows to be factorized or prevent degeneracy problems. The selected nz rows are the last of the nw factorized rows. They are replaced by a new active inequality constraint and nz — 1 artificial constraints. 

The solution of the system (10.17) allows the search direction d (depending on the selected vector h) to be calculated. By using this direction, it is possible to obtain both a new feasible point and the new inequality constraint (passive or satisfied)... 

It is worth noting that if we select the values he = 0 for the equality constraints ha = Xa, hj = Xj, and h = 0, the search direction is the one that maximizes the objective function decrease, and simultaneously satisfies all the active equality and inequality constraints. 

The Attic method is based on the idea of introducing one inequality constraint at a time, selecting each one from the most promising ones. Once a vertex of nv constraints is set up, more inequality constraints are simultaneously removed when opportune. From a certain point of view, this strategy is similar to the stepwise method of building the best model in a linear regression problem (Vol. 2 -Buzzi-Ferraris and Manenti, 2010b). A forward method is used to insert constraints and a backward method to remove them. 

In this case, the active inequality constraints are all considered as equality constraints and the iteration is suitable only to provide the correction d. The selection of inequality constraints that must be considered active is carried out outside of this iteration. 

The number of independent variables in a constrained optimization problem can be found by a procedure analogous to the degrees of freedom analysis in Chapter 2. For simplicity, suppose that there are no constraints. If there are Ny process variables (which includes process inputs and outputs) and the process model consists of Ne independent equations, then the number of independent variables is Np = Ny - Ne-This means Np set points can be specified independently to maximize (or minimize) the objective function. The corresponding values of the remaining (Ny - Np) variables can be calculated from the process model. However, the presence of inequality constraints that can become active changes the situation, because the Np set points cannot be selected arbitrarily. They must satisfy all of the equality and inequality constraints. 

A selectivity constraint is imposed, in order to maintain the formation of the secondary products at a low value. All the constraints are introduced as inequality type constraints. 

Select an initial basic feasible solution. If all the constraint equations were inequalities of the dess than or equal to form, the slack variables can be used as the initially feasible basis. 

Let us consider Example 6.3 (p. 159) without the selectivity constraints and impose the constraints Tmin < T < Tmax on the temperature control. Thus, instead of the selectivity constraints, we have the two inequalities... 

---