Algebraic inequality constraints

Consider the optimal control problem in which the objective is to minimize the functional 

The augmented objective functional for this problem is M, which is given by Equation (6.15) on p. 163. From the John Multiplier Theorem (Section 4.5.1, p. 113), the following equations must be satisfied at the minimum of M, and equivalently of /  

The number of inequality constraints, /, can be greater than m, the number of controls. That the number of active constraints at any time does not exceed the m is assured by the constraint qualification. It requires that if p inequality constraints are active (i.e., / = 0 for f = 1,2. p), then p should be the rank of the matrix of partial derivatives of f with respect to u. Note that p is the number of linearly independent rows or columns of the matrix (see Section 4.3.2.1, p. 97). 

Let us consider Example 6.7 with its equality constraint replaced with the inequality 

This is one of the four constraint qualifications, any one of which must be satisfied to obtain the necessary conditions for the minimum in inequality constrained problems (Arrow et al., 1961). See Takayama (1985) for a thorough exposition. 

---

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

Algebraic inequality constraints These usually specify the practical operating limits of certain variables within the process. They are written in mathematical form as... 

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. 

The ingredients of formulating optimization problems include a mathematical model of the system, an objective function that quantifies a criterion to be extremized, variables that can serve as decisions, and, optionally, inequality constraints on the system. When represented in algebraic form, the general formulation of discrete/continu-ous optimization problems can be written as the following mixed integer optimization problem ... 

All of the various optimization techniques described in previous chapters can be applied to one or more types of reactor models. The reactor model forms a set of constraints so that most optimization problems involving reactors must accommodate steady-state algebraic equations or dynamic differential equations as well as inequality constraints. 

Algebraic optimization with equality and inequality constraints... 

In summary, condition 1 gives a set of n algebraic equations, and conditions 2 and 3 give a set of m constraint equations. The inequality constraints are converted to equalities using h slack variables. A total of M + m constraint equations are solved for n variables and m Lagrange multipliers that must satisfy the constraint qualification. Condition 4 determines the value of the h slack variables. This theorem gives an indirect problem in which a set of algebraic equations is solved for the optimum of a constrained optimization problem. 

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

As mentioned earlier, the developed algorithm employs dynopt to solve the intermediate problems associated with the local interaction of the agents. Specifically, dynopt is a set of MATLAB functions that use the orthogonal collocation on finite elements method for the determination of optimal control trajectories. As inputs, this toolbox requires the dynamic process model, the objective function to be minimized, and the set of equality and inequality constraints. The dynamic model here is described by the set of ordinary differential equations and differential algebraic equations that represent the fermentation process model. For the purpose of optimization, the MATLAB Optimization Toolbox, particularly the constrained nonlinear rninimization routine fmincon [29], is employed. 

Based on a multiple shooting method for parameter identification in differential-algebraic equations due to Heim [4], a new implementation of a direct multiple shooting method for optimal control problems has been developed, which enables the solution of problems that can be separated into different phases. In each of these phases, which might be of unknown length, the control behavior due to inequality constraints, the differential equations, even the dimensions of the state and/or the control space can differ. For the optimal control problems under investigation, the different phases are concerned with the different steps of the recipes. 

Path constraints g represent conditions that must be fulfilled throughout the entire integration horizon. These inequality constraints augment the algebraic equations ft. 

It is important to note that two functionally equivalent process models Mi and M2 of equal quality with respect to some quality indices may or may not be algebraically equivalent. This depends on the invariance properties of the performance and quality indices with respect to algebraic transformation and also on the equality-inequality constraints in the formulation of the modelling goal. 

Theorem A.5.5 (which is algebraic only) may be applied to the thermodynamics of our book, namely in the admissibility principle used on the models of differential type as we show in the examples below. The X are here the time or space derivatives of deformation and temperature fields other than those contained in the independent variables of the constitutive equations and therefore al a, /3, Aj, Aj, Bj are functions of these independent variables. Constraint conditions (A.99) usually come from balances (of mass, momentum, energy) and (A. 100) from the entropy inequality. 

Subject to Differential-Algebraic Process Model, Inequality Path Constraints Control Scheme Equations, Process Design Equations Feasibility of Operation (over time), Process Variability Constraints... 

S( — 1 if all the feed enters tray k, and is zero otherwise, and = 1 if the reflux enters tray k, and is zero otherwise. Additionally, a set of control binary variables 6f are introduced that are associated with each MV-CV pairing and are unity when the pairing exists and zero otherwise. The modelling of the control structure selection is carried our similarly to Narraway and Perkins [15]. These features lead to a mixed-integer dynamic distillation model. The principal differential-algebraic equations (DAEs) for the trays are given below. A full list of nomenclature, values of the parameters, details of the DAEs for the reboiler, condenser, reflux drum and control scheme, cost correlations for the objective function and inequality path constraints, can be found in Bansal et al, [7]. 

---