Equality constraints nonlinear

Section 6.8 compares the GBD approach and the OAbased approaches with regard to the formulation, handling of nonlinear equality constraints, nonlinearities in y and joint x — y, the primal problem, the master problem, and the quality of the lower bounds. 

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. 

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... 

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... 

Table 8.5 summarizes the relative merits of SLP, SQP, and GRG algorithms, focusing on their application to problems with many nonlinear equality constraints. One feature appears as both an advantage and a disadvantage—whether or not the algorithm can violate the nonlinear constraints of the problem by relatively large amounts during the solution process. 

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. 

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. 

Equality constraints. The equality constraints (30 in all) are the linear and nonlinear material and energy balances and the phase relations. 

In summary, the problem consists of 34 bounded variables (both upper bound and lower bounds) associated with the process, 12 linear equality constraints, 18 nonlinear equality constraints, and 3 linear inequality constraints. 

Solution of the problem. It was not possible to use analytical derivatives in the nonlinear programming code because the energy balance equality constraints... 

An optimization problem is a mathematical model which in addition to the aforementioned elements contains one or multiple performance criteria. The performance criterion is denoted as objective function, and it can be the minimization of cost, the maximization of profit or yield of a process for instance. If we have multiple performance criteria then the problem is classified as multi-objective optimization problem. A well defined optimization problem features a number of variables greater than the number of equality constraints, which implies that there exist degrees of freedom upon which we optimize. If the number of variables equals the number of equality constraints, then the optimization problem reduces to a solution of nonlinear systems of equations with additional inequality constraints. 

The Lagrange multipliers in a constrained nonlinear optimization problem have a similar interpretation to the dual variables or shadow prices in linear programming. To provide such an interpretation, we will consider problem (3.3) with only equality constraints that is,... 

The OA/ER algorithm, extends the OA to handle nonlinear equality constraints by relaxing them into inequalities according to the sign of their associated multipliers. 

Note that the objective function is convex since it has linear and positive quadratic terms. The only nonlinearities come from the equality constraint. By introducing three new variables w1,w2)w3, and three equalities ... 

Also, note that problem (6.13) does not feature any nonlinear equality constraints. Hence, the implicit assumption in the OA algorithm is that... 

Remark 2 Under the aforementioned assumptions (i) and (ii), problem (6.13) satisfies property (P) of Geoffrion (1972), and hence the OA corresponds to a subclass of vl-GBD (see sections 6.3.5.1) Furthermore, as we have seen in section 6.3.5.2, assumptions (i) and (ii) make the assumption imposed in v2-GBD valid (see remark of section 6.3.5.2) and therefore the OA can be considered as equivalent to v2-GBD with separability in jc andy and linearity iny. Note though that the vl-GBD can handle nonlinear equality constraints. 

To handle explicitly nonlinear equality constraints of the form h(x) = 0, Kocis and Grossmann (1987) proposed the outer approximation with equality relaxation OA/ER algorithm for the following class of MINLP problems ... 

Remark 1 The nonlinear equalities h(x) = 0 and the set of linear equalities which are included in h(x) = 0, correspond to mass and energy balances and design equations for chemical process systems, and they can be large. Since the nonlinear equality constraints cannot be treated explicitly by the OA algorithm, some of the possible alternatives would be to perform ... 

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. 

Alternative (ii) involves the numerical elimination of the nonlinear equality constraints at each iteration of the OA algorithm through their linearizations. Note though that these linearizations may cause computational difficulties since they may result in singularities depending on the selection of decision variables. In addition to the potential problem of singularities, the numerical elimination of the nonlinear equality constraints may result in an increase of the nonzero elements, and hence loss of sparsity, as shown by Kocis and Grossmann (1987). 

The basic idea in OA/ER is to relax the nonlinear equality constraints into inequalities and subsequently apply the OA algorithm. The relaxation of the nonlinear equalities is based upon the sign of the Lagrange multipliers associated with them when the primal (problem (6.21) with fixed y) is solved. If a multiplier A is positive then the corresponding nonlinear equality hi(x) = 0 is relaxed as hi x) <0. If a multiplier A, is negative, then the nonlinear equality is relaxed as -h (jc) < 0. If, however, A = 0, then the associated nonlinear equality constraint is written as 0 ht(x) = 0, which implies that we can eliminate from consideration this constraint. Having transformed the nonlinear equalities into inequalities, in the sequel we formulate the master problem based on the principles of the OA approach discussed in section 6.4. 

Since the only difference between the OA and the OA/ER lies on the relaxation of the nonlinear equality constraints into inequalities, we will present in this section the key result of the relaxation and the master problem of the OA/ER. 

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

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