Optimization equality constraints

In this case, there are n process variables with equality constraints and inequahty constraints. Such problems pose a serious challenge to performing optimization calculations in a reasonable amount of time. Typical constraints in chemical process optimization include operating conditions (temperatures, pressures, and flows have limits), storage capacities, and produc t purity specifications. 

If the mass flowrate of the cold stream through the exchanger had not been fixed, there would have been one fewer equality constraint, and this would have provided an additional degree of freedom and the optimization would have been a two-dimensional optimization. Each degree of freedom provides an opportunity for optimization. 

In this case, there are n design variables, with p equality constraints and q inequality constraints. The existence of such constraints can simplify the optimization problem by reducing the size of the problem to be searched or avoiding problematic regions of the objective function. In general though, the existence of the constraints complicates the problem relative to the problem with no constraints. 

An important class of the constrained optimization problems is one in which the objective function, equality constraints and inequality constraints are all linear. A linear function is one in which the dependent variables appear only to the first power. For example, a linear function of two variables x and x2 would be of the general form ... 

The objective of the optimization would be for the equality constraint given by Equation 3.37 to be satisfied as nearly as possible. There are a number of ways in which this objective can be made specific for optimization. Three possibilities are12... 

Thirdly, the inlet and outlet concentrations were specified such that one was fixed directly and the other determined by mass balance using flowrate and mass load. However, a number of variations are possible in the way that the process constraints on quantity (or flowrate) present themselves. For instance, it could happen that there is no direct specification of the water quantity (or flow) in a particular stream, as long as the contaminant load and the outlet concentration are observed. Furthermore, the vessel probably has minimum and maximum levels for effective operation. In that case the water quantity falls away as an equality constraints, to become an inequality constraints, thereby changing the nature of the optimization problem. 

Now consider the imposition of inequality [g(x) < 0] and equality constraints 7i(x) = 0] in Fig. 3-55. Continuing the kinematic interpretation, the inequality constraints g(x) < 0 act as fences in the valley, and equality constraints h(x) = 0 act as "rails. Consider now a ball, constrained on a rail and within fences, to roll to its lowest point. This stationary point occurs when the normal forces exerted by the fences [- Vg(x )] and rails [- V/i(x )] on the ball are balanced by the force of gravity [— Vfix )]. This condition can be stated by the following Karush-Kuhn-Tucker (KKT) necessary conditions for constrained optimality ... 

For each of the following six problems, formulate the objective function, the equality constraints (if any), and the inequality constraints (if any). Specify and list the independent variables, the number of degrees of freedom, and the coefficients in the optimization problem. 

Constraints in optimization arise because a process must describe the physical bounds on the variables, empirical relations, and physical laws that apply to a specific problem, as mentioned in Section 1.4. How to develop models that take into account these constraints is the main focus of this chapter. Mathematical models are employed in all areas of science, engineering, and business to solve problems, design equipment, interpret data, and communicate information. Eykhoff (1974) defined a mathematical model as a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in a usable form. For the purpose of optimization, we shall be concerned with developing quantitative expressions that will enable us to use mathematics and computer calculations to extract useful information. To optimize a process models may need to be developed for the objective function/, equality constraints g, and inequality constraints h. 

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. 

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,... 

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... 

Chapter 1 presents some examples of the constraints that occur in optimization problems. Constraints are classified as being inequality constraints or equality constraints, and as linear or nonlinear. Chapter 7 described the simplex method for solving problems with linear objective functions subject to linear constraints. This chapter treats more difficult problems involving minimization (or maximization) of a nonlinear objective function subject to linear or nonlinear constraints ... 

One method of handling just one or two linear or nonlinear equality constraints is to solve explicitly for one variable and eliminate that variable from the problem formulation. This is done by direct substitution in the objective function and constraint equations in the problem. In many problems elimination of a single equality constraint is often superior to an approach in which the constraint is retained and some constrained optimization procedure is executed. For example, suppose you want to minimize the following objective function that is subject to a single equality constraint... 

In problems in which there are n variables and m equality constraints, we could attempt to eliminate m variables by direct substitution. If all equality constraints can be removed, and there are no inequality constraints, the objective function can then be differentiated with respect to each of the remaining (n — m) variables and the derivatives set equal to zero. Alternatively, a computer code for unconstrained optimization can be employed to obtain x. If the objective function is convex (as in the preceding example) and the constraints form a convex region, then any stationary point is a global minimum. Unfortunately, very few problems in practice assume this simple form or even permit the elimination of all equality constraints. 

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. ... 

Many real problems do not satisfy these convexity assumptions. In chemical engineering applications, equality constraints often consist of input-output relations of process units that are often nonlinear. Convexity of the feasible region can only be guaranteed if these constraints are all linear. Also, it is often difficult to tell if an inequality constraint or objective function is convex or not. Hence it is often uncertain if a point satisfying the KTC is a local or global optimum, or even a saddle point. For problems with a few variables we can sometimes find all KTC solutions analytically and pick the one with the best objective function value. Otherwise, most numerical algorithms terminate when the KTC are satisfied to within some tolerance. The user usually specifies two separate tolerances a feasibility tolerance Sjr and an optimality tolerance s0. A point x is feasible to within if... 

Luus, R. arid T. Jaakola. Optimization of Nonlinear Function Subject to Equality Constraints. Chem Process Des Develop 12 380-383 (1973). 

Table 9.1 shows how outer approximation, as implemented in the DICOPT software, performs when applied to the process selection model in Example 9.3. Note that this model does not satisfy the convexity assumptions because its equality constraints are nonlinear. Still DICOPT does find the optimal solution at iteration 3. Note, however, that the optimal MILP objective value at iteration 3 is 1.446, which is not an upper bound on the optimal MINLP value of 1.923 because the convexity conditions are violated. Hence the normal termination condition that the difference between upper and lower bounds be less than some tolerance cannot be used, and DICOPT may fail to find an optimal solution. Computational experience on nonconvex problems has shown that retaining the best feasible solution found thus far, and stopping when the objective value of the NLP subproblem fails to improve, often leads to an optimal solution. DICOPT stopped in this example because the NLP solution at iteration 4 is worse (lower) than that at iteration 3. 

This example focuses on the design and optimization of a steady-state staged column. Figure El 2.1 shows a typical column and some of the notation we will use, and Table El2.1 A lists the other variables and parameters. Feed is denoted by superscript F. Withdrawals take the subscripts of the withdrawal stage. Superscripts V for vapor and L for liquid are used as needed to distinguish between phases. If we number the stages from tihe bottom of the column (the reboiler) upward with k= 1, then V0 = L1 = 0, and at the top of the column, or the condenser, Vn = Ln+l = 0. We first formulate the equality constraints, then the inequality constraints, and lastly the objective function. 

To this point we isolated four variables D, v, Ap, and/, and have introduced three equality constraints—Equations (d (e), and (/)—leaving 1 degree of freedom (one independent variable). To facilitate the solution of the optimization problem, we eliminate three of the four unknown variables (Ap, v, and/) from the objective function using the three equality constraints, leaving D as the single independent variable. Direct substitution yields the cost equation... 

Suppose the inequality constraints on ethylene and propylene production were changed to equality constraints (ethylene = 50,000 propylene = 20,000). The optimal solution for these conditions is shown as case 2 in Table E14.1B. This specification forces the use of DNG as well as ethane. 

Process simulators contain the model of the process and thus contain the bulk of the constraints in an optimization problem. The equality constraints ( hard constraints ) include all the mathematical relations that constitute the material and energy balances, the rate equations, the phase relations, the controls, connecting variables, and methods of computing the physical properties used in any of the relations in the model. The inequality constraints ( soft constraints ) include material flow limits maximum heat exchanger areas pressure, temperature, and concentration upper and lower bounds environmental stipulations vessel hold-ups safety constraints and so on. A module is a model of an individual element in a flowsheet (e.g., a reactor) that can be coded, analyzed, debugged, and interpreted by itself. Examine Figure 15.3a and b. 

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. 

The redundant constraints and variables can be regarded as 24 + 9 = 33 additional equality constraints in the optimization problem. 

Infeasible path algorithms. The equality constraints and active inequality constraints are satisfied only at the stage on which the optimal solution is reached. 

Finally, a great advantage to SQP is that it does not require convergence of the equality constraints, h(x) = 0, at intermediate points. Consequently, the process model (or at least the part directly incorporated into the optimization problem) can be solved simultaneously with the optimization problem. In the next section we discuss the application of the SQP algorithm to flowsheet optimization. Here, if the number of variables in the optimization problem is small, application is straightforward. On the other hand, when the number of variables, n, becomes large (n > 100, say), special-purpose extensions to SQP are required. These are discussed in the remainder of this section. 

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