Optimization cost function

The synthesis tool performs optimization by minimizing cost functions - one for design rule costs and the other for optimization costs. These cost values are usually displayed during optimization. The optimization cost function consists of four parts in the following order of importance ... 

Optimization of the system can be carried out by minimizing a cost function or maximizing economic potential EP defined by (see App. A)... 

In the sequel we shall study an optimal control problem. Let C (fl) be a convex, bounded and closed set. Assume that ( < 0 on T for each G. In particular, this condition provides nonemptiness for Kf. Denote the solution of (2.131) by % = introduce the cost functional... 

In the next two subsections the parameter c is supposed to be fixed. The convergence of solutions of the optimal control problem (2.134) as —> 0 will be analysed in Section 2.5.4. For this reason the -dependence of the cost functional is indicated. 

An existence theorem to the equilibrium problem of the plate is proved. A complete system of equations and inequalities fulfilled at the crack faces is found. The solvability of the optimal control problem with a cost functional characterizing an opening of the crack is established. The solution is shown to belong to the space C °° near crack points provided the crack opening is equal to zero. The results of this section are published in (Khludnev, 1996c). 

Further, in Section 3.1.4, an optimal control problem is analysed. The external forces u serve as a control. The solution existence of the optimal control problem with a cost functional describing the crack opening is proved. Finally, in Section 3.1.5, we prove C°°-regularity of the solution near crack points having a zero opening. 

In this subsection we analyse an optimal control problem. The exterior forces f,g) are chosen to minimize the cost functional... 

Consider an approximate description of the nonpenetration condition between the crack faces which can be obtained by putting c = 0 in (3.45). Similar to the case c > 0, we can analyse the equilibrium problem of the plates and prove the solution existence of the optimal control problem of the plates with the same cost functional. We aim at the convergence proof of solutions of the optimal control problem as —> 0. In this subsection we assume that T, is a segment of a straight line parallel to the axis x. 

An objec tive function to be optimized (revenue function, cost function, etc.)... 

Alternative algorithms employ global optimization methods such as simulated annealing that can explore the set of all possible reaction pathways [35]. In the MaxFlux method it is helpful to vary the value of [3 (temperamre) that appears in the differential cost function from an initially low [3 (high temperature), where the effective surface is smooth, to a high [3 (the reaction temperature of interest), where the reaction surface is more rugged. 

The data in Table 6.2 illustrate a problem when optimizing a function by making one-at-a-time guesses. The cost at F=50m and ro , = 366K is not the minimum, but is lower than the entries above and below it, on either side of it, or even diagonally above or below it. Great care must be taken to avoid false optimums. This is tedious to do manually, even with only two variables, and quickly becomes unmanageable as the number of variables increases. 

In the formulation above, the discrete optimization on the number of compressors has been transformed into a continuous optimization on suction and delivery pressures. This transformation was made possible by the form of the compressor cost function which vanishes when pd = ps. However, if the compressor costs include a fixed capital outlay, i.e., the cost function is a linear function of horsepower with a nonzero constant term, then a branch and bound procedure must be used in conjunction with the GRG method. 

Step 5. Look at the total cost function, Equation (c). Observe that the cost function includes a constant term, K3Q. If the total cost function is differentiated, the term K3Q vanishes and thus K3 does not enter into the determination of the optimal value for D. K3, however, contributes to the total cost. 

Suppose you wanted to find the configuration that minimizes the capital costs of a cylindrical pressure vessel. To select the best dimensions (length L and diameter D) of the vessel, formulate a suitable objective function for the capital costs and find the optimal (LID) that minimizes the cost function. Let the tank volume be V, which is fixed. Compare your result with the design rule-of-thumb used in practice, (L/D)opt = 3.0. 

In real life, other problems involving discrete variables may not be so nicely posed. For example, if cost is a function of the number of discrete pieces of equipment, such as compressors, the optimization procedure cannot ignore the integer character of the cost function because usually only a small number of pieces of equipment are involved. You cannot install 1.54 compressors, and rounding off to 1 or 2 compressors may be quite unsatisfactory. This subject will be discussed in more detail in Chapter 9. 

The optimal plant operation can be determined by minimizing the total cost function, including steam costs, with respect to P (liquid pumping costs are negligible)... 

Figure 20.9 shows a typical optimisation trajectory found by the model for one load and power price combination. The starting point for the raw and recycle brine is based on current plant operating rules and appears as 100% on the flow axes. Subsequently the model varies the brine flows and runs to a new steady-state solution. The modelling package has a built-in optimisation routine which controls the searching process as dictated by the cost function, which is the price per unit of chlorine. In the case illustrated by Fig. 20.9 it requires eight runs to find the optimal solution. 

Optimization approaches and cost functions 1.3. MSE magnet coil arrangement and 164... 

In contrast to our MSB technique, Kakugawa et al. formulated the cost function to be minimized as a product of coil cross sectional area and its current density. The result is an optimal coil configuration for the coil cross sectional area and coil current density. However, it is not clear that the superconducting volume is also minimized when cross sectional area is reduced, as individual coil volumes are a fimction of the coil radius, which was not considered. 

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