**2. Basic concept of optimization problem**

In this field, we will first address the general form of the issue of multi-objective optimization and a sequence of definitions and important issues related to the core of the subject under study. So the general form of the problem is:

$$\left(\text{Minimize}\right)F\left(\mathbf{x}\right) = \left[f\_1\left(\mathbf{x}\right), f\_2\left(\mathbf{x}\right), \dots, f\_k\left(\mathbf{x}\right)\right]^T$$

Subject to:

$$w\_i(\mathbf{x}) \le \mathbf{0}, i = 1, \dots, k;\tag{1}$$

$$w\_j(\mathbf{x}) = \mathbf{0}, j = 1, \dots, p;$$

$$\mathbf{x}\_l \ge \mathbf{0}, l = 1, 2, \dots, n$$

*Using Many Objective Bat Algorithms for Solving Many Objective Nonlinear Functions DOI: http://dx.doi.org/10.5772/intechopen.107078*

where *<sup>x</sup>* <sup>¼</sup> ½ � *<sup>x</sup>*<sup>1</sup> : *<sup>x</sup>*2, … , *xn <sup>T</sup>* is the vector of decision variables *Fi* : *<sup>R</sup><sup>n</sup>* ! *<sup>R</sup>*; *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>k</sup>* are the objective functions and *wi*, *nj* : *<sup>R</sup><sup>n</sup>* ! *<sup>R</sup>*,*<sup>i</sup>* <sup>¼</sup> 1,*::*, *<sup>m</sup>*, *and j* <sup>¼</sup> 1, … , *<sup>p</sup>*, are constraints functions a problem. To describe the objective concept of optimization, we will give some of the following definitions:

**Definition (1)** [9]: (Multi-objective optimization problem (MOP)). A MOP is made up of a number of parameters (decision variables), a number of optimization techniques (m), and a number of constraints (m). The determination variables' functions and constraints are functions of the optimization algorithms and requirements. The purpose of optimization is to:

$$\text{Minimize } y\_i = f\left(\mathbf{x}\_i\right) \\ \text{subj. : } \text{to} \\ e(\mathbf{x}) = (e\_1(\mathbf{x}); e\_2(\mathbf{x}); \dots; e\_k(\mathbf{x})) \le \mathbf{0} \tag{2}$$

where *X* ¼ ð Þ *x*1, *x*2, *xn* and *Y* ¼ *y*1, *y*2, *ym* , and *x* the choice pattern is called the decision vector, the ambition velocity is called the objective vector, the determination space is called the decision sector, and the object space is called the subjective space. The constraints *e x*ð Þ≤0 determine the set of feasible solutions.

**Definition (2)** [9]: (Allocative efficiency optimality). A dimension of choice *x* ∊*Xf* when it comes to a set, it is said to be completely non ⊆*Xf iff* ∄ *a* ∊ *A* : *a*˃*x* . If it is evident from the circumstances whichever set A is wanted, the following will simply be omitted. Furthermore, x is described as allocative efficiency optimal *iffx* is nondominated regarding *Xf* .

**Definition (3)** [9]: A set of controller parameters in a scalar *x*1*ϵX* ⊂*R<sup>n</sup>* is nondominant when it comes to *X*, if no *x*<sup>2</sup> ∈*X* appears in the sense that *f x*ð Þ<sup>2</sup> <*f x*ð Þ<sup>1</sup> *:*

**Definition (4)** [9]: The allocative efficiency optimal set P\* is characterized as follows: P\* = {*x*1*ϵF*: *x*<sup>1</sup> is alocative efficiency optimal}.
