Objective function of the optimization

To gain further understanding of the applications of optimal control, let us study some examples of optimal control problems. In each problem, there is a system changing with time or some other independent variable. The system is mathematically described or modeled with the help of differential equations. At least one such differential equation is needed in order to have an optimal control problem. Appearing in the model is a set of undetermined control functions, which determines the dependent variables. The set of controls and dependent variables in turn determine the objective functional of the optimal control problem. 

All these choices lead to capital and operating costs that become the objective function of the optimization, shown in the lower RHS of Fig. 2.1. 

Simultaneous hybrid modeling—integration between simulation and optimization is implemented by evaluation of objective function. In this case, objective function of the optimization model is not available as a closed form expression (or its analytical evaluation is too complex). An optimization model sets values of decision variables. Simulation modeling results obtained using these decision variables as input parameters are used to find a value of the optimization objective function. The value found is passed back to the optimization model. One can say that simulation is called on each optimization trial. Simulation... 

The objective functional for the optimal control of photodissociation may be defined as ... 

Node 1. The first step is to set up and solve the relaxation of the binary IP via LP. The optimal solution has one fractional (noninteger) variable (y2) and an objective function value of 129.1. Because the feasible region of the relaxed problem includes the feasible region of the initial IP problem, 129.1 is an upper bound on the value of the objective function of the IP. If we knew a feasible binary solution, its objective value would be a lower bound on the value of the objective function, but none is assumed here, so the lower bound is set to -< >. There is as yet no incumbent, which is the best feasible integer solution found thus far. 

In sub-problem 5M the process model constraints are considered along with the objective function and the optimal mixture is identified by solving a smaller MINLP problem (if the number of feasible solutions is large) or a set of NLP problems (if the number of feasible solutions is small). 

In general, an objective function in the optimization problem can be chosen, depending on the nature of the problem. Here, two practical optimization problems related to batch operation maximization of product concentration in a fixed batch time and minimization of batch operation time given amount of desired product, are considered to determine an optimal reactor temperature profile. The first problem formulation is applied to a situation where we need to increase the amount of desired product while batch operation time is fixed. This is due to the limitation of complete production line in a sequential processing. However, in some circumstances, we need to reduce the duration of batch run to allow the operation of more runs per day. This requirement leads to the minimum time optimization problem. These problems can be described in details as follows. 

A number of numerical experiments were carried out to investigate the sensitivity of the objective function around the optimal solution. The values of the objective function in the environment of optimal point are also shown in Table 1, and these very values are shown graphically in Fig. 3. 

The overall objective function of the entire process is to optimize for a maximum profit over a five-year period. Assume equipment life period is 3 years. 

The base case simulation can be optimized by introducing an objective function and by adding constraints to the model. The objective function of the dynamic optimization is the crystal mean size (mfmf). The following constraints are subject to the simulation in addition to those indicating limits of the physical possible domain ... 

The basic simulated annealing algorithm does not require much more than a dozen lines of code (not including evaluation of the objective function) and the optimization constraints are easily handled by rejecting candidate points that violate them. 

Figure 18.18 illustrates the shift of the position of the optimum experimental conditions when Pr is replaced by the Pr X Y objective function for the optimization. The maximum production rate is at point A, while Pr x Y reaches its maximum at point B. The contour lines clearly show that the production rate is hardly lower at the new optimum. On the other hand, the recovery yield is improved when the experimental conditions are shifted from point A to point B. The surface determined by Pr x Y exhibits a well defined maximum, which makes the numerical path toward optimization stable. 

Pintaric, Z. N. and Kravanja, Z. (2006). Selection of the Economic Objective Function for the Optimization of Process Flow Sheets, Industrial and Engineering Chemistry Research, 45, pp. 4222-4232. 

When dealing with difficult discrete optimization problems, it is natural to search for related, but easier optimization models that can aid in the analysis. Relaxations are auxiliary optimization problems of this sort formed by weakening either the constraints or the objective function of the main problem. Specifically, an optimization problem P) is said to be a constraint relaxation of another optimization problem (P) if every solution feasible to (P) is also feasible for (P). Similarly, maximize problem (respectively minimize problem) P) is an objective relaxation of another maximize (respectively minimize) problem (P) if the two problems have the same feasible solutions and the objective function value in P) of any feasible solution is (respectively <) the objective function value of the same solution in (P). 

The objective functional of an optimal control problem depends on functions as well as their derivatives. Under what circumstances would the variation of such a functional become the differential ... 

If Ao is zero, then the Hamiltonian would be independent of the objective functional. Consequently, the optimal control problem would have nothing to do with the objective functional. Since an objective functional is essentially... 

In Figure 18.2b, an upper bound is placed on x, and that bound is below the value at the maximum value of the objective function. Therefore, the optimal value of x is its upper bound, and the constraint is referred to as a binding constraint. 

This is the variational principle for the variance of the local energy. The evaluation of the variance is easy with the MC method. Thus, the variance is often adopted as the objective function in the optimization of the wave function within the QMC method. The efficient optimization of the wave function is an important problem in the QMC method and several works can be found in this line [23-30]. 

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