Cost functions

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

Simulated Annealing-based solutions [19] are conceptually the same as Genetic Algorithm-based approaches. However, the SA-based techniques, in our experience, are more sensitive to the initial settings of the parameters. Nevertheless, once the correct ones are found, the method can achieve the efficiency of GA-based solutions. We must point out that SA-based solutions have never outperformed the GA-based ones in our studies. Much of what has been mentioned regarding the GA-based solutions is also relevant for the SA technique, particularly, with respect to the cost functions. 

Lonally, the templates were chosen by trial and error or exhaustive enumeration. A itafional method named ZEBEDDE (ZEolites By Evolutionary De novo DEsign) en developed to try to introduce some rationale into the selection of templates et al. 1996 Willock et al. 1997]. The templates are grown within the zeolite by an iterative inside-out approach, starting from a seed molecule. At each jn an action is randomly selected from a list that includes the addition of new (from a library of fragments), random translation or rotation, random bond rota-ing formation or energy minimisation of the template. A cost function based on erlap of van der Waals spheres is used to control the growth of the template ale ... 

We proceed with an investigation of the contact problem for a plate under creep conditions. We know that for every fixed / G L Q) there exists a unique solution w,M satisfying (2.35)-(2.37). Let G L Q) be a given element and F c (Q) be a closed convex and bounded set. We introduce the cost functional... 

In this subsection we consider the other cost functional which describes a deflection of the moments from given functions M°. G L Q)- Namely, let... 

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. 

The cost functional describing the opening of the crack is as follows ... 

It is assumed that for each -06, the level line x,y) = 0 divides G into two subdomains such that x,y) > 0 for the first subdomain and x,y) < 0 for the second one. We consider the cost functional... 

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

Information on collection, one of the most costly functional elements, is presented in four parts deahng with (1) the types of collection services, (2) the types of collection systems, (3) an analysis of collection systems, and (4) the general methodology involved in setting up col-lec tion routes. 

The wattage is directly proportional to the grounding resistance of the whole anode bed and therefore to the specific soil resistivity. Equation (9-5) gives the grounding resistance of the anode installation which either consists of n horizontal or vertical single anodes or of anodes with a horizontal continuous coke bed of total length I = ns. The total cost function is given by [1] ... 

A most successful paradigm for isolating reaction pathways in complex systems begins with a definition of the reaction pathway as a continuous line l(r) that connects known reactant Tr and product rp configurations. We then define an integrated cost functional... 

What is the best choice of differential cost function A variety of definitions of the cost function have been proposed. One stems form the highly original work of Fiber and Karplus [38] and Czerminski and Fiber [39], where... 

An alternative definition of the cost function was proposed by Huo and Straub [35] based on an earlier suggestion of Berkowitz et al. [42],... 

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. 

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