Single objective function

Searching for Pareto-optimal solutions can be computationally very expensive, especially when too many objectives are to be optimized. Therefore, it is very appealing to convert a multiobjective optimization problem into a much simpler single-objective optimization problem by combining the multiple objectives into a single objective function as follows (53-55) ... 

The NONMEM program implements two alternative estimation methods, the first-order conditional estimation and the Laplacian methods. The first-order conditional estimation (FOCE) method uses a first-order expansion about conditional estimates (empirical Bayes estimates) of interindividual random effects, rather than about zero. In this respect, it is like the conditional first-order method of Lindstrom and Bates.f Unlike the latter, which is iterative, a single objective function is minimized, achieving a similar effect as with iteration. The Laplacian method uses second-order expansions about the conditional estimates of the random effects. ... 

Generically the models considered have a clear, quantitative way to compare feasible solutions. That is, they have single objective functions. In many applications single objectives reahstically model the true decision process. Decisions become much more confused when the problem arises in a complex engineering design, where more than one objective may be relevant. For such cases, as referred above, a multi-objective optimization model is required to capture all the possible perspectives. This is the case of the design of batch plants where two objectives are under consideration - one that maximizes the revenues (that is, production) and the other that minimizes the cost. 

Unlike optimization models with a single objective function, the interest is on finding a set of solutions that describe how the improvement of a single objective function value impacts the value of the other objectives. This set is commonly known as the Pareto-optimal set and each of its elements as a Pareto optimal solution. In this respect, a general formulation of a multi-objective problem is ... 

Having set the model and chosen the training data, the objective is to fit the model predictions to the targets. The simplest and most widely used numerical approach is to formulate a single objective function, [Pg.245]

The mathematical formulation of the model consists of the objective function (Eq. 12.1) and constraints (Eq. 12.7-12.13). Equations (12.2-12.6) are auxiliary measures used to the elements of the objective function. The notations used are defined in Table 12.1. The weights wy and wy are used to combine the physical units selection and web service selection criteria in a single objective function. The physical units selection is performed to maximize e-retailer s profit calculated as a difference between revenues R and sourcing cost Ci, delivery cost C2 and fixed cost C3. The web service selection is performed to maximize infrastructure processing efficiency L. 

Figure 3. Performance of the genetic algorithm with single objective function (a) Maximum displacement of bilinear base isolator (b) Maximum displacement of the isolated building top story...

The solutions of the problem are situated on section BDC of the boundary of the feasible region. These are called non-dominated solutions, Pareto ideal solutions or the compromise set that is, no single objective function can be increased without causing the simultaneous decrease of at least one other function. In effect, the difficulty is that there are a great number of Pareto solutions. It is therefore necessary to apply other procedures for the selection of one solution, called the preferable solution. ... 

These objective functions, usually, conflict or compete with each other. In the case of no conflict between the objectives, any traditional single objective function technique can be used easily to solve the problem since optimising one objective will ensure that all the other objectives are optimised within the same direction of minimisation or maximisation. 

There are several approaches to obtaining such solutions (see Sect. 1.5.1). Basically, they are based on the conversion of the MO problem into one single objective function problem. The next section is focused on the e-constrained method which is... 

Minimum-time joint trajectory is a constrained non-linear optimization problem with a single objective function. The optimization procedure used in this work is the non-linear optimization search method with goal programming based on the Modified Hooke and Jeeves Direct Search Method [13]. 

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