Algebraic equality constraints

The number of equality constraints I is less than the number of controls, i. e., m. 

The reason for the above provision is that if / = m, then the constraints would uniquely determine the control, and there would not be any optimal control problem remaining. 

According to the Lagrange Multiplier Rule, the above problem is equivalent to the minimization of the augmented functional 

Considering that both final state and time are free, i.e., unspecified, the variation of M upon simplification is given by [compare with Equation (6.9) on p. 156] 

Since SM should be zero at the minimum, the following equations are necessary at the minimum of M, and equivalently of J and /  

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F = objective function, g = algebraic inequality constraint vector, c = algebraic equality constraint vector,... 

To the base problem we add the following algebraic equality constraints ... 

Consider the batch reactor problem in Example 7.1 (p. 192) subject to the following algebraic equality constraint throughout the time interval [0,tf] ... 

This problem is similar to that in Section 7.2.3 (p. 209) except that the algebraic equality constraints are replaced with the inequalities... 

The set of 33 DAEs (systems dynamics), acting as differential-algebraic equality constraints. 

In some cases besides the governing algebraic or differential equations, the mathematical model that describes the physical system under investigation is accompanied with a set of constraints. These are either equality or inequality constraints that must be satisfied when the parameters converge to their best values. The constraints may be simply on the parameter values, e.g., a reaction rate constant must be positive, or on the response variables. The latter are often encountered in thermodynamic problems where the parameters should be such that the calculated thermophysical properties satisfy all constraints imposed by thermodynamic laws. We shall first consider equality constraints and subsequently inequality constraints. 

If the equality constraint involves independent variables and parameters in an algebraic model, i.e.. it is of the form, model equations reduces the number of unknown parameters by one. 

Models can be written in a variety of mathematical forms. Figure 2.3 shows a few of the possibilities, some of which were already illustrated in Section 2.1. This section focuses on the simplest case, namely models composed of algebraic equations, which constitute the bulk of the equality constraints in process optimization. Emphasis here is on estimating the coefficients in simple models and not on the complexity of the model. 

The model involves four variables and three independent nonlinear algebraic equations, hence one degree of freedom exists. The equality constraints can be manipulated using direct substitution to eliminate all variables except one, say the diameter, which would then represent the independent variables. The other three variables would be dependent. Of course, we could select the velocity as the single independent variable of any of the four variables. See Example 13.1 for use of this model in an optimization problem. 

Alternative (i) can be applied successfully to certain classes of problems (e.g., design of batch processes Vaselenak etal. (1987) synthesis of gas pipelines (e.g., Duran and Grossmann (1986a)). However, if the number of nonlinear equality constraints is large, then the use of algebraic elimination is not a practical alternative. 

Remark 5 If instead of the above presented relaxation procedure we had selected the alternative of algebraic elimination of X, from the equality constraint, then the resulting MINLP is... 

The GBD algorithm can address nonlinear equality constraints explicitly without the need for algebraic or numerical elimination as in the case for OA. 

In contrast to the sequential solution method, the simultaneous strategy solves the dynamic process model and the optimization problem at one step. This avoids solving the model equations at each iteration in the optimization algorithm as in the sequential approach. In this approach, the dynamic process model constraints in the optimal control problem are transformed to a set of algebraic equations which is treated as equality constraints in NLP problem [20], To apply the simultaneous strategy, both state and control variable profiles are discretized by approximating functions and treated as the decision variables in optimization algorithms. 

For the PDF model of the SMB process, full discretization was used, that is, both temporal and spatial variables were discretized leading to a huge system of algebraic equations. The standard SMB optimization problem has 33 997 decision variables and 33 992 equality constraints while the superstructure SMB optimization problem has 34 102 decision variables and 34 017 equality constraints. Note that there are many more degrees of freedom in the superstructure formulation (altogether 85) than in the standard SMB formulation (5 degrees of freedom). 

The representation of different types of reactor units in the approach proposed by Kokossis and Floudas (1990) is based on the ideal CSTR model, which is an algebraic model, and on the approximation of plug flow reactor, PFR units by a series of equal volume CSTRs. The main advantage of such a representation is that the resulting mathematical model consists of only algebraic constraints. At the same time, however, we need to introduce binary variables to denote the existence or not of the CSTR units either as single units or as a cascade approximating PFR units. As a result, the mathematical model will consist of both continuous and binary variables. 

In the reactive case, r is not equal to zero. Then, Eq. (3) represents a nonhmoge-neous system of first-order quasilinear partial differential equations and the theory is becoming more involved. However, the chemical reactions are often rather fast, so that chemical equilibrium in addition to phase equilibrium can be assumed. The chemical equilibrium conditions represent Nr algebraic constraints which reduce the dynamic degrees of freedom of the system in Eq. (3) to N - Nr. In the limit of reaction equilibrium the kinetic rate expressions for the reaction rates become indeterminate and must be eliminated from the balance equations (Eq. (3)). Since the model Eqs. (3) are linear in the reaction rates, this is always possible. Following the ideas in Ref. [41], this is achieved by choosing the first Nr equations of Eq. (3) as reference. The reference equations are solved for the unknown reaction rates and afterwards substituted into the remaining N - Nr equations. 

Algebraic optimization with equality and inequality constraints... 

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