The purpose of optimization

Finding the best solution is given the constraints and needs of the problem.

For a problem, there may be several solutions that are defined as a function called the objective function to compare and select the optimal solution. The choice of this function depends on the nature of the problem. For example, travel time or cost are common goals of optimizing transport networks.

However, selecting the right target function is one of the most important optimization steps. Sometimes, multi-objective optimization is considered simultaneously; such optimization problems, which include multiple objective functions, are called multi-objective problems. The simplest way to deal with such problems is to form a new objective function in linear combination of the original objective functions in which the degree of effectiveness of each function is determined by the weight assigned to it. Each optimization problem has a number of independent variables called design variables that are represented by the next n vector x.

The purpose of optimization is to determine the design variables so that the objective function is minimized or maximized.

The different optimization problems are divided into two categories:

 A) Unlimited optimization problems: In these problems the objective is to maximize or minimize the objective function without any constraint on the design variables.

 B) Optimization Problems with Constraints: Optimization is performed in most practical problems, given the constraints in the behavior and performance of a system and the behavioral constraints and constraints that exist in problem physics and geometry, namely geometrical or lateral constraints. .

 The equations representing constraints may be equal or unequal, which in each case is a different optimization method. However, limitations determine the acceptable design area.

Generally the constraint optimization problems can be illustrated as follows:

Minimize or Maximize: F (X) (1-1)

Subject to: I = 1,2,3,…, p

  j = 1,2,3,…, q

  k = 1,2,3,…, n

Where X = {Design vectors and relationships (1-1) are, respectively, inequality constraints and acceptable limits for the design variables.

Searching for search and optimization methods

 The advancement of the computer over the last fifty years has led to the development of optimization techniques, with numerous orders being developed during this period. This section provides an overview of different optimization methods.

 Figure 1-1 classifies the optimization methods into four broad categories. In the following discussion, each of these methods will be examined.

Column production

 
repeat

 
Neighborhood Search

Fig. 1-1: Classification of various optimization methods

1-1-1- Counting methods

 In the enumerative method, only one point belonging to the target function domain space is examined in each iteration. These methods are easier to implement than other methods, but require considerable computation. In these methods there is no mechanism to reduce the search range and the range of search space with this method is very large. Dynamic Programming is a good example of counting methods. This method is completely unconscious and is therefore rarely used alone today.

Computational Methods (Mathematical Search or -Calculus-Based Method)