Modified objective function

Figure 9.12 shows the modified objective function for a one-dimensional continuous object parameter with first- and second-stage infeasibilities. 

The unit conversion of t from inches to feet does not affect the optimum LID), nor do the values of p and S, which are multiplicative constants. The modified objective function, substituting Equation (/) in Equation (j), is therefore... 

The formalism used to calculate the pulse shape that maximizes J is optimal control theory. This formalism can be considered to be an extension of the calculus of variations to the case where the constraints include differential equations. In general, the constraints expressed in the form of differential equations express the restriction that the amplitude must always satisfy the Schrodinger equation. In addition, there can be a variety of other constraints, such as a restriction on the total energy in the pulse or on the shape of the pulse. These constraints are accounted for by the method of Lagrange multipliers, which modify the objective functional (4.6) and thereby permit the calculation of the unconstrained maximum of the modified objective functional. When the only constraints are satisfaction of the Schrodinger equation and limitation of the pulse energy, the modified objective functional can be written in the form... 

In the OCT formulation, the TDSE written as a 2 x 2 matrix in a BO basis set, equation tAl. 6.72). is introduced into the objective functional with a Lagrange multiplier, j x, t) [54]. The modified objective functional may now be written as... 

As shown in Section 12.2.1, the quadratic penalty Junction requires high values for the parameters to satisfy the constraints. It implies that the modified objective function has very narrow valleys and the search for the minimum is quite difficult. 

In order to insure that there were as few over and under bakes as possible, the object function was modified so that it increased rapidly when SKjw was less than SK or when SKb. b was greater than SK. ... 

The explicit feasibility constraints of (MASTER) are given by the linear first-stage constraints in (9.4.2). In a classical penalty function approach, the explicit feasibility constraints are relaxed while the violation of these constraints is considered by an additional penalty term in the fitness function. However, this method would waste valuable CPU time since the MILP subproblems (SUB) have to be solved also for the fitness evaluation of infeasible individuals. A similar method which does not require the solution of the MILP subproblems for infeasible individuals is the use of a modified objective Junction that separates the objective and the feasibility... 

Note that both the direction and step length are specified as a result of Equation (6.11). Iffix) is actually quadratic, only one step is required to reach the minimum of fix). For a general nonlinear objective function, however, the minimum of fix) cannot be reached in one step, so that Equation (6.12) can be modified to conform to Equation (6.7) by introducing the parameter for the step length into (6.12). 

Modify the objective function to simulate a larger importance of investment costs by introducing an exponent for PROD larger than 1. How does the value of the exponent influence the value of the objective function OBJ How is the time of the maximum of the objective function changed ... 

Step 2 Formulation of the unconstrained problem. Applying previous results, the (y — x) vector of the objective function is modified as follows ... 

Modified max tsT fsF Objective Function -invf - xft-xft i)-cfixf-xft+ YJPs -y),- price,-cvarf ) (3.107) seS... 

Here n Is the refractive Index of the medium and X Is the wavelength of Incident light In a vacuum. We modified Provencher program to call a subroutine which would supply values of (l (a)/a ) for the kernel of the Integral. The Initial solution Is that with little or no regularization. A chosen solution where the Increase In the objective function over the Initial solution could about 50% of the time be due to experimental noise and about 50% of the time be due to oversmoothing, Is selected by a statistical criterion (4,5). 

A method to overcome the impact of model plant mismatch on optimisation performance was previously investigated by Zhang [8] where model prediction confidence bormds are incorporated as a penalty in the objective function. Therefore, the objective function can be modified as... 

Cyclic adsorption processes Two examples (a) thermal swing adsorption -maximization of total adsorption efficiency and minimization of consumption rate of regeneration energy, and (b) rapid pressure swing adsorption -maximization of both purity and recovery of the desired product for RPSA. Modified Sum of Weighted Objective Function (SWOF) method Modified SWOF method is superior to the conventional SWOF as it was able to find the non-convex part of the Pareto-optimal set. Ko and Moon (2002)... 

Constraints (other than bounds) of the kind, gi(x) < 0 i = 1, 2,..., p, can be taken care of by subtracting these (for a maximization problem) in a weighted form, from the objective function, and maximizing the modified fitness function. These terms act as penalties (Deb, 1995) when any constraint is violated, since they reduce the value of the modified fitness function, thus favoring the ehmination of that chromosome over the next few generations. The following example (Deb, 1995) illustrates the procedure ... 

The SMB model consists of partial differential equations (PDFs) for the concentrations of chemical components, restrictions for the connections between different columns and cyclic steady-state constraints (Kawajiri and Biegler, 2006b). Previously, the SMB processes have been usually optimized with respect to one objective only. Recently, multi-objective optimization has been applied in periodic separation processes (Ko and Moon, 2002), in gas separation and in SMB processes (Subramani et al., 2003). Ko and Moon used a modified sum of weighted objective functions to obtain a representation of the Pareto optimal set. Their approach is valid for two objective functions only. On the other hand, Subramani et al. applied EMO to a problem where they had two or three objective functions. 

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