**3. Algorithm definition**

The **Metaheuristic Algorithm** is defined by the Author [Jonah Lissner] as. An ontological mechanism to generate or activate decision paths [algorithms] and make decision potential to solve [essentially two-state] paradoxes, a computational physical network and topos, for practical effort or application. Therefore can be constructed a theoretical or hypothetical guideway from objects and particles to advance ontological gradations of relevance and value, through a logical progression.

A relevant algorithm to solve for discrete stigmergetics in nonlinear optimization challenges for graphing algorithms of power systems has been demonstrated in Ant Colony Optimization [ACO]:

Here in general formula where

$$p\_{xy}^k = \frac{\left(\tau\_{xy}^a\right)\left(\eta\_{xy}^\beta\right)}{\sum\_{x \in \text{allowed}\_x} \left(\tau\_{xx}^a\right)\left(\eta\_{xx}^\beta\right)}$$

by trail update action *<sup>τ</sup>xy* ð Þ <sup>1</sup> � *<sup>ρ</sup> <sup>τ</sup>xy* <sup>þ</sup> <sup>P</sup> *k* Δ*τ<sup>k</sup> xy*. for given function sets

*f x*ð Þ¼ *λx*, for x ≥0; (1)

$$f(\mathbf{x}) = \lambda \mathbf{x}^2, \text{ for } \mathbf{x} \ge \mathbf{0};\tag{2}$$

*Atomistic Mathematical Theory for Metaheuristic Structures of Global Optimization… DOI: http://dx.doi.org/10.5772/intechopen.96516*

$$f(\mathbf{x}) = \begin{cases} \sin\left(\frac{\pi \mathbf{x}}{2\lambda}\right), & \text{for } 0 \le \mathbf{x} \le \lambda; \\ 0, & \text{else} \end{cases} \tag{3}$$

$$f(\mathbf{x}) = \begin{cases} \pi \mathbf{x} \sin\left(\frac{\pi \mathbf{x}}{2\lambda}\right), & \text{for } 0 \le \mathbf{x} \le \lambda; \\ 0, & \text{else} \end{cases}. \tag{4}$$

Evolutionary Game Theory [EGT] challenges in optimization schedules are therein linked to Ant Colony Optimization [ACO], e.g. Eigenvector centrality formula

**Where for G** ≔ ð Þ **E**, **V with V vertices let A** ¼ ð Þ **av**,**t** , **e***:***g***:***av**,**t** ¼ **1 or av**,**t** ¼ **0***:***Therefore xv** ¼ **1***=***l SigtsetM v**ð Þ**xt** ¼ **1***=***l SigtsetG av**,**t xt**½ � *ibid :*

(5)

It is a basis for optimization schedules that there is an asymmetrical velocity, mass and gravity of said scope of systems. At various times in the computational history, particle optimization on the manifolds evolve at a faster rate [or slower rate] than before. Hence the given incremental and discrete rate of increase, in valleys and peaks accelerates and stabilizes at a higher positive, null or negative value and result in extremal mechanics and nonlinear dynamics. An example can be demonstrated utilizing power faults and extremals on the electrical circuits [3].

These problems of prediction for probability of choice of one object or particle of a set, for pariwise sets and in algorithms, have been demonstrated in Arrow's Impossibility Theorem and for Algorithmic Information Theory [AIT] whence we can replace *voter* for global optimization *particle* and replace *group* with *set*:


Important is Criteria 3, from whence adaptive and efficient algorithms have space to be constructed as particles on the run-time, for a given Global Optimization Algorithm [GOA].

#### **4. Building the algorithm**

In a praexological theory [5] this is proposed because of the inherent general inaccuracy of specific problems, learning rubrics, and Macrodynamic properties of a given performance landscape, and ultimately inefficient of any algorithmic system, given isomorphic [atomistic or non-atomistic] qualities of rulebase, algorithmic structure, weights, and variables [6]. These in turn can be represented as information sets, materiel, work, and symbolic representation and/or power in specific *qualia* of *Historical Rule of Perpetuation of Information Inequalities* set to various scales and models.

**2. Materials and methods**

I.Problem of Demarcation.

requiring a kind of.

tion Algorithm [GOA] challenges:

**3. Algorithm definition**

Ant Colony Optimization [ACO]: Here in general formula where

for given function sets

**152**

1.The Rule of the Continuity of Primaries

Adaptive Evolutionary Systems [CAES].

*pk xy* ¼

by trail update action *<sup>τ</sup>xy* ð Þ <sup>1</sup> � *<sup>ρ</sup> <sup>τ</sup>xy* <sup>þ</sup> <sup>P</sup>

P

*f x*ð Þ¼ *<sup>λ</sup>x*<sup>2</sup>

Lissner exist:

**2.1 Metaheuristic structural rules for the algorithm building**

*Computational Optimization Techniques and Applications*

It is a rule of No Unreasonable Effectiveness of Mathematics in any Science [Wigner] [2], and therefore a notion that No Unreasonable Effectiveness of Axiomation in any Science, that 3 Rules of Information Physics [3IP] by Jonah

II.Rule of Information Dichotomy [*Gestalt-Inverse Gestalt*], and thereby

III.Context-Restricted Deep Structure [CRDS] for the given topology.

2.The Rule of Perpetuation of Information Inequalities of Primaries

3.The Rule of Unprovable Ideals, Cardinals or Delimitations of Complex

The **Metaheuristic Algorithm** is defined by the Author [Jonah Lissner] as. An ontological mechanism to generate or activate decision paths [algorithms] and make decision potential to solve [essentially two-state] paradoxes, a computational physical network and topos, for practical effort or application. Therefore can be constructed a theoretical or hypothetical guideway from objects and particles to advance ontological gradations of relevance and value, through a logical progression. A relevant algorithm to solve for discrete stigmergetics in nonlinear optimization challenges for graphing algorithms of power systems has been demonstrated in

> *τα xy* � �

*<sup>z</sup>* <sup>∈</sup>allowed*<sup>x</sup> τα*

*ηβ xy* � �

*xz* � � *η β xz* � �

*f x*ð Þ¼ *λx*, for x ≥0; (1)

, for x ≥0; (2)

*k* Δ*τ<sup>k</sup> xy*.

Therefore hypothesized to be commutable terms within this 3IP rulebase, The Three Thermodynamic Rules of Macrodynamics by Jonah Lissner. The 3 General Rules of Macrodynamics [3GRM] which are established to define Global Optimiza-

Clerc has demonstrated a general Metaheuristic algorithm where for *<sup>f</sup>***:** *<sup>n</sup>* ! essentially *f***(a)** ≤ *f***(b)**. S includes the number of particles in the swarm having specific position and velocity in the search—space:

for each particle i = 1, … , S do Initialize the particle's position with a uniformly distributed random vector: xi � U(blo, bup) Initialize the particle's best known position to its initial position: pi xi if f(pi) < f(g) then update the swarm's best known position: g pi Initialize the particle's velocity: vi � U(�|bup-blo|, |bup-blo|) while a termination criterion is not met do: for each particle i = 1, … , S do for each dimension d = 1, … , n do Pick random numbers: rp, rg � U(0,1) Update the particle's velocity: vi,d ω vi,d + φp rp (pi,d-xi,d) + φg rg (gd-xi,d) Update the particle's position: xi xi + vi if f(xi) < f(pi) then Update the particle's best known position: pi xi if f(pi) < f(g) then Update the swarm's best known position:

$$\mathbf{g} \leftarrow \mathbf{p} . [\!\mathbf{\!}] \tag{6}$$

Natural Sets, Natural Kinds, Natural Procedures, Natural Strings, Natural Radicals, Natural Binaries, Natural Radices, in Complex Adaptive Evolutionary System

Time-Complexity, Particle-Value, Particle-Weighting in Fuzzy set theory, Gravity of System, <<> > Nanodynamics of System variables [TC-PV-PW-GS-

*Atomistic Mathematical Theory for Metaheuristic Structures of Global Optimization…*

*f* : **Omega set Rn > > R with the global minima** *f* <sup>∗</sup> **and the set of all global minimizers** *X*<sup>∗</sup> **in**

for system conditions, system boundaries, number and density of particles in the total Information Natural Dynamics [IND] of the Global Optimization Algorithm [GOA]. These are applied to algorithmic manifold for the candidate solution on the given search spaces. It can be argued that given the extremes of information disequilibrium applied to macrodynamic disequilibrium models, there are inevitably generated extremals of various degrees of power, in the incremental Information

These differentiable functions can be further defined c.f. Dense heterarchy in Complex Systems Algorithms of a coupled oscillators, where in general formula.

*dt* <sup>¼</sup> ð Þ *P t*ðÞþ *<sup>μ</sup> Q t*ð Þ , *<sup>μ</sup> <sup>x</sup>* <sup>þ</sup> *f t*ð Þ, (10)

, … , *<sup>u</sup>*ð Þ *<sup>n</sup>*�<sup>1</sup> � �, *<sup>n</sup>*≥2, (11)

(9)

[CAES]-Multiagent System [MAS] for 4D model variables.

**Omega to find the minimum best set in the function series of x**ð Þ

**5.2 Complex adaptive evolutionary system: weighting**

*dx*

Here in a differential equation we can demonstrate.

*<sup>u</sup>*ð Þ *<sup>n</sup>* <sup>¼</sup> *f t*, *<sup>u</sup>*, *<sup>u</sup>*<sup>0</sup>

**5.3 Complex adaptive evolutionary system: thermodynamics**

where in general formula to set the integral.

<sup>H</sup>ð Þ¼ *<sup>X</sup>* <sup>X</sup>

*i*

These can be demonstrated in Particle Swarm Optimization [PSO], and Macrodynamic models of Meta-optimization of Particle Swarm Optimization [PSO]

**<sup>v</sup>***i*ð Þ¼ *<sup>t</sup>* <sup>þ</sup> **<sup>1</sup>** *<sup>w</sup>* � **<sup>v</sup>***i*ð Þþ *<sup>t</sup> <sup>n</sup>***<sup>1</sup>**�*r***<sup>1</sup>**� *pi* � *xi*ð Þ*<sup>t</sup>* � � <sup>þ</sup> *<sup>n</sup>***<sup>2</sup>**�*r***<sup>2</sup>**� *pbest* � *xi*ð Þ*<sup>t</sup>* � � (12)

**for each set of given epoch or evolutionary landscape scenario** prediction in analytical and expectation weighting parameter formula algorithm optimization

Regarding bounding definitions, Chaitin demonstrated in Algorithmic Information Theory [AIT] algorithmic decomposition given Boltzmann-Shannon entropy,

> X *i*

Pð Þ *xi* log *<sup>b</sup>*Pð Þ *xi* ,

Pð Þ *xi* Ið Þ¼� *xi*

These variables should each contain criteria:

*DOI: http://dx.doi.org/10.5772/intechopen.96516*

NDS]

Dynamics.

[7], c.f.

and

**155**

[Meissner, et al., *ibid*].

Set approximate to
