Classifying objective functions

To be more specific, when classifying objective functions the DM indicates which function values should improve, which ones are acceptable and which are allowed to get worse. In addition, amounts of improvement or impairments are asked from the DM. There exist several classification-based interactive multi-objective optimization methods. They use different numbers of classes and generate new solutions in different ways. 

Chapter 1 presents some examples of the constraints that occur in optimization problems. Constraints are classified as being inequality constraints or equality constraints, and as linear or nonlinear. Chapter 7 described the simplex method for solving problems with linear objective functions subject to linear constraints. This chapter treats more difficult problems involving minimization (or maximization) of a nonlinear objective function subject to linear or nonlinear constraints ... 

Optimization methods calculate one best future state as optimal result. Mathematical algorithms e.g. SIMPLEX or Branch Bound are used to solve optimization problems. Optimization problems have a basic structure with an objective function H(X) to be maximized or minimized varying the decision variable vector X with X subject to a set of defined constraints 0 leading to max(min)//(X),Xe 0 (Tekin/Sabuncuoglu 2004, p. 1067). Optimization can be classified by a set of characteristics ... 

Optimization problems and the computational techniques to tackle them are often classified further depending on the properties of these constraints, the objective function, and the domain itself. Linear Programming deals with cases in which the objective function /(x) is linear and the set A is specified through linear equalities and inequalities. If the variables x can only acquire integer values. 

As found in Section 1.8.7, there were two affin linear dependences among the data, classified as exact ones. Therefore, Box et al. (ref. 29) considered the principal components corresponding to the three largest eigenvalues as response functions when minimizing the objective function (3.66). By virtue of the eigenvectors derived in Section 1.8.7, these principal components are ... 

An optimization problem is a mathematical model which in addition to the aforementioned elements contains one or multiple performance criteria. The performance criterion is denoted as objective function, and it can be the minimization of cost, the maximization of profit or yield of a process for instance. If we have multiple performance criteria then the problem is classified as multi-objective optimization problem. A well defined optimization problem features a number of variables greater than the number of equality constraints, which implies that there exist degrees of freedom upon which we optimize. If the number of variables equals the number of equality constraints, then the optimization problem reduces to a solution of nonlinear systems of equations with additional inequality constraints. 

The general mathematical model of the superstructure presented in step 2 of the outline, and indicated as (7.1), has a mixed set of 0 - 1 and continuous variables and as a result is a mixed-integer optimization model. If any of the objective function and constraints is nonlinear, then (7.1) is classified as mixed- integer nonlinear programming MINLP problem. 

Cultivated calculus bovis samples No.4 and No.7 were misclassified into natural ones by K-means algorithm. Both SAC and SAKMC can get a global optimal solution 0. = 94.3589 ), only sample No. -4 belonging to cultivated calculus bovis was classified into a natural one corresponding to j, = 94.3589. If sample No. 4 is classified into a cultivated one, the corresponding objective function 0 would be 95.2626, this indicates that iiample No.4 is closer to natural calculus bovis. From the above results, one notices that calculus bovis samples can be correctly classified into natural and cultivated ones on the basis of their microelement contents by means of SAC and SAKMC except the sample No. 4. The computation times for SAC and SAKMC were 21 and 12 minutes, respectively. 

A linear discriminant function can be found using a linear programming approach (48,49). The objective function to be optimized consists of the fraction of the training set correctly classified. If two vertices have the same classification ability, then the vertex with the smaller sum of distances to misclassified points is taken as better. 

If it is not possible to use automatic trade-off, classifying the objective functions or setting a reference point are almost the same, as discussed earlier. The only difference is that here the reference point is set such that some objective functions must be allowed to get impaired values. Let us finally mention that STOM has been applied to many engineering problems, e.g., in Nakayama (1995) Nakayama and Furukawa (1985) Nakayama and Sawaragi (1984). 

The NIMBUS method (Miettinen, 1999 Miettinen and Makela, 1995, 1999, 2000, 2006) is an interactive method based on classification of the objective functions into up to five classes. To be more specific, the DM is asked to specify how the current Pareto optimal solution f(x ) should be improved by classifying the objective functions into classes where the functions /j... 

Once the DM has classified the objective functions, (s)he can decide how many Pareto optimal solutions (between one and four) based on this information (s)he wants to see and compare. Then, as many scalarized problems are formed and solved and the new solutions are shown to the DM together with the current solution. If the DM has found the most preferred solution, the solution process stops. Otherwise, the DM can select a solution as a starting point of a new classification or ask for a desired number of intermediate (Pareto optimal) solutions between any two solutions generated so far. The DM can also save any interesting solutions to a database and return to them later. All the solutions considered are Pareto optimal. For details of the algorithm, see Miettinen and Makela (2006). 

The correct classification rate (CCR) or misclassification rate (MCR) are perhaps the most favoured assessment criteria in discriminant analysis. Their widespread popularity is obviously due to their ease in interpretation and implementation. Other assessment criteria are based on probability measures. Unlike correct classification rates which provide a discrete measure of assignment accuracy, probability based criteria provide a more continuous measure and reflect the degree of certainty with which assignments have been made. In this chapter we present results in terms of correct classification rates, for their ease in interpretation, but use a probability based criterion function in the construction of the filter coefficients (see Section 2.3). Whilst we speak of correct classification rates, misclassification rates (MCR == 1 - CCR) would equally suffice. The correct classification rate is typically formulated as the ratio of correctly classified objects with the total... 

If all of the functions /, g, hj are linear functions of x, then (P) is called a linear program. Otherwise (P) is a nonlinear program. Note that Example 1 is a nonlinear program since the objective function (1) is a nonlinear function. Actually, as will be seen later, this problem can be classified as a quadratic program since the objective function is a quadratic function and the constraints are all linear functions. 

Optimization problems can be classified as unconstrained, where no limitations are imposed on the range of possible values of independent factors, and constrained, where additional conditions (constraints) define the range of admissible values of the factors. The objective function and the... 

Problems with the general form of Eq. (15.1) can be classified according to the nature of the objective function and constraints (linear, nonlinear, convex), the number of variables, large or small, the smoothness of the functions, differentiable or non-differentiable, and so on. An important distinction is between problems that have constraints on the variables and those that do not. Unconstrained optimization problems, for which we have = n, = 0 in Eq. (15.1), arise in many practical... 

Nonlinear problems— NLP and MINLP—can be further classified as convex or nonconvex, depending on the convexity of the objective function and feasible region. The understanding of the type of problem in terms of classification and convexity is very important in the utilization of modeling systems, since there are specific solvers and solution techniques for each type of problem, and depending on the problem, there may be local and global solutions. A more comprehensive study on mathanati-cal programming topics and continuous nonlinear optimization is out of the scope of this chapter. The interested reader is directed to references [4,6,7]. 

Rather than fitting a predetermined function and reporting the coefficients of such a fit, neural networks process input information to produce a hidden model of the relationships. Depending on the structure of the network, they can be used to classify objects, to map complex relationships to fewer dimensions, and to model the relationship between input properties—molecular descriptors in the case of QSAR—and the output property, which for this discussion is biological potency. " > -> For QSAR the back-propagation network method, described below, is used. However, PLS neural networks are also available. ... 

In fact, the success of any optimization technique critically depends on the degree to which the model represents and accurately predicts the investigated system. For this reason, the model must capture the complex dynamics in the system and predict with acceptable accuracy the proper elements of reality. Moreover, it is important to be able to recognize the characteristics of a problem and identify appropriate solution techniques within each class of problems there are different optimization methods which vary in computational requirements and convergence properties. These problems are generally classified according to the mathematical characteristics of the objective function, the constraints, and the controllable decision variables. 

The optimal control problem encountered in biodiesel production in a batch reactor can be classified into three categories depending on the objective function as given below. The decision variable or the control variable is the temperature profile for the reactor. 

Much of the recent research on optimal control problems can be classified into this problem. [53] were the first to use the profit function for maximization in batch distillation, and they solved the optimal control problem. The following simple objective function is given by [53] ... 

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