Equality constraints origin

Lines of constant annual revenue are shown as dotted lines in Figure 3.12, with revenue increasing with increasing distance from the origin. It is clear from Figure 3.12 that the optimum point corresponds with the extreme point at the intersection of the two equality constraints at Point C. 

The vector x can contain slack variables, so the equality constraints (8.33) may contain some constraints that were originally inequalities but have been converted to equalities by inserting slacks. Codes for quadratic programming allow arbitrary upper and lower bounds on x we assume x>0 only for simplicity. 

The geometry of this problem is shown in Figure 8.11. The linear equality constraint is a straight line, and the contours of constant objective function values are circles centered at the origin. From a geometric point of view, the problem is to find the point on the line that is closest to the origin at x = 0, y = 0. The solution to the problem is at x = 2, y = 2, where the objective function value is 8. 

The above problem will be referred to as Problem A in the following. The 7 inequality constraints in equations 1.4e to 1.4k are the bounds on the 7 variables (X4, xs, X2, Xe, Xio, Xg and X3) in the original problem, and they arise from the elimination of these variables from the 7 equality constraints in the model thus making them dependent variables. The cost coefficients in the profit are alkylate product value ( 0.063/octane-barrel), olefin feed cost ( 5.04 arrel), isobutane recycle cost ( 0.035/barrel), fresh acid cost ( 10.0/thousand pounds) and isobutane feed cost ( 3.36/barrel). The optimal solution for this SOO problem is also presented in Table 1.2. The reader can verify this using the Excel file Alkylation.xls in the folder Chapter 1 on the compact disk (CD) provided with the book. 

Mitra et al. (1998) employed NSGA (Srinivas and Deb, 1994) to optimize the operation of an industrial nylon 6 semibatch reactor. The two objectives considered in this study were the minimization of the total reaction time and the concentration of the undesirable cyclic dimer in the polymer produced. The problem involves two equality constraints one to ensure a desired degree of polymerization in the product and the other, to ensure a desired value of the monomer conversion. The former was handled using a penalty function approach whereas the latter was used as a stopping criterion for the integration of the model equations. The decision variables were the vapor release rate history from the semibatch reactor and the jacket fluid temperature. It is important to note that the former variable is a function of time. Therefore, to encode it properly as a sequence of variables, the continuous rate history was discretized into several equally-spaced time points, with the first of these selected randomly between the two (original) bounds, and the rest selected randomly over smaller bounds around the previous generated value (so as... 

In Eq. 4.8, wi is taken to be a large positive number (depending on the value of the original objective function) in case the constraint is violated else it is assigned a value of zero. Equality constraints can be handled in a similar manner. The results for this problem for the 40 generation are shown in Eig. 4.4. The computational parameters used are Istrmg = 10,... 

In an attempt to avoid the ill-conditioning that occurs in the regular pentilty tuid bturier function methods, Hestenes (1969) and PoweU (1969) independently developed a multiplier method for solving nonhnearly constrained problems. This multiplier method was originally developed for equality constraints and involves optimizing a sequence of unconstrained augmented Lagrtuigitui functions. It was later extended to handle inequality constraints by Rockafellar (1973). 

Barrier Junction methods (which can be applied only if no equality constraints are present), which simply modify the original objective function by adding certain special terms, which become progressively smaller when the point is clearly within the feasible region and tend to infinity when approaching the frontier of the feasible region. 

To obtain a valid lower bound on the global solution of the nonconvex problem, the lower bounding problem generated in each domain must have a unique solution. This implies that the formulation includes only convex inequality constraints, linear equality constraints, and an increased feasible region relative to that of the original nonconvex problem. The left-hand side of any nonconvex inequality constraint, g(x) < 0, in the original problem can simply be replaced by its convex underestimator g(x), constructed according to Eq. (9), to yield the relaxed convex inequality g(x) < 0. 

For an equality constraint containing general nonconvex terms, the equation obtained by simple substitution of the appropriate underestimators is also nonlinear. Therefore, the original equality h(x) = 0 must be rewritten as two inequalities of opposite signs ... 

We consider the special biplot in which both rows and columns are represented in a single display of latent variables subjected to the constraint that a + P equals 1. As we have seen above, this constraint allows us to reconstruct the original data X from which the latent variables U, V and the latent values A have been computed (eq. (31.22)). 

In Section 1.5 we briefly discussed the relationships of equality and inequality constraints in the context of independent and dependent variables. Normally in design and control calculations, it is important to eliminate redundant information and equations before any calculations are performed. Modem multivariable optimization software, however, does not require that the user clearly identify independent, dependent, or superfluous variables, or active or redundant constraints. If the number of independent equations is larger than the number of decision variables, the software informs you that no solution exists because the problem is overspecified. Current codes have incorporated diagnostic tools that permit the user to include all possible variables and constraints in the original problem formulation so that you do not necessarily have to eliminate constraints and variables prior to using the software. Keep in mind, however, that the smaller the dimensionality of the problem introduced into the software, the less time it takes to solve the problem. 

Note that all the constraints in Equation (7.4) are equalities. It is necessary to place the problem in this form to solve it most easily (equations are easier to work with here than inequalities). If the original system is not of this form, it may easily be transformed by use of so-called slack variables. If a given constraint is an inequality, for example,... 

Let the starting point be (1, 0), at which the objective value is 6.5 and the inequality is satisfied strictly, that is, its slack is positive (s = 1). At this point the bounds are also all satisfied, although y is at its lower bound. Because all of the constraints (except for bounds) are inactive at the starting point, there are no equalities that must be solved for values of dependent variables. Hence we proceed to minimize the objective subject only to the bounds on the nonbasic variables x and y. There are no basic variables. The reduced problem is simply the original problem ignoring the inequality constraint. In solving this reduced problem, we do keep track of the inequality. If it becomes active or violated, then the reduced problem changes. 

The quantification of an NOE amounts to determining the volume of the corresponding cross peak in the NOESY spectrum. Since the linewidths can vary appreciably for different resonances, cross-peak volumes should in principle be determined by integration over the peak area rather than by measuring peak heights. However, one should also keep in mind that, according to Eq. (1), the relative error of the distance estimate is only one sixth of the relative error of the volume determination. Furthermore, Eq. (1) involves factors that have their origin in the complex internal dynamics of the macromolecule and are beyond practical reach such that even a very accurate measurement of peak volumes will not yield equally accurate conformational constraints. 

The dual price of the slack variable sm on this constraint indicates the effect of selling this product at the margin, that is, it indicates the marginal profit on the product. Ifthe constraint is slack, so that the slack variable is positive (basic), the profit at the margin must obviously be zero and this is in line with the zero dual price of all basic variables. Since cost + profit — realization for a product, the sum of the dual prices on its balance and requirement constraints equals its coefficient in the original objective function. 

For the present work, we chose the constrained method described by Jansson (1968) and Jansson et al (1968, 1970). See also Section V.A of Chapter 4 and supporting material in Chapter III. This method has also been applied to ESCA spectra by McLachlan et al (1974). In our adaptation (Jansson and Davies, 1974) the procedure was identical to that used in the original application to infrared spectra except that the data were presmoothed three times instead of once, and the variable relaxation factor was modified to accommodate the lack of an upper bound. Referring to Eqs. (15) and (16) of Section V.A.2 of Chapter 4, we set k = 2o(k)K0 for 6(k) < j and k = Kq exp[3 — for o(k) > This function is seen to apply the positivity constraint in a manner similar to that previously employed but eliminates the upper bound in favor of an exponential falloff. We also experimented with k = k0 for o(k) > j, and found it to be equally effective. As in the infrared application, only 10 iterations were needed. 

The constraint of finite extent applies to data that exist only over a finite interval and have zero values elsewhere. Let N1 and N2 denote the nonzero extent of the original undistorted data. The restored function u(k) + v(k), then, should have no deviations from zero outside the known extent of the data. To find the coefficients in v(k) that best satisfy this constraint, we should minimize the sum of the squared points outside the known extent of the object. Actually, recovering only a band of frequencies in v(k) implies the additional constraint of holding all higher frequencies above this band equal to zero. This is necessary for stability and is an example of one of the smoothing constraints discussed earlier. We minimize the expression... 

The master problem in OA and it variants OA/ER, OA/ER/AP involves linearizations of the nonlinear objective function, the vector of transformed nonlinear equalities and the original nonlinear inequalities around the optimum solution xk of the primal problem at each iteration. As a result, a large number of constraints are added at each iteration to the master problem. Therefore, if convergence has not been reached in a few iterations the effort of solving the master problem, which is a mixed-integer linear programming MILP problem, increases. 

So far the micro-mechanical origin of the Mullins effect is not totally understood [26, 36, 61]. Beside the action of the entropy elastic polymer network that is quite well understood on a molecular-statistical basis [24, 62], the impact of filler particles on stress-strain properties is of high importance. On the one hand the addition of hard filler particles leads to a stiffening of the rubber matrix that can be described by a hydrodynamic strain amplification factor [22, 63-65]. On the other, the constraints introduced into the system by filler-polymer bonds result in a decreased network entropy. Accordingly, the free energy that equals the negative entropy times the temperature increases linear with the effective number of network junctions [64-67]. A further effect is obtained from the formation of filler clusters or a... 

---