Mathematical Problems in Engineering

Volume 2018, Article ID 6357935, 8 pages

https://doi.org/10.1155/2018/6357935

## On the Theoretical Analysis of the Plant Propagation Algorithms

^{1}Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, Pakistan^{2}Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK^{3}Department of Mathematics, Kohat University of Science & Technology (KUST), Khyber Pakhtunkhwa, Pakistan

Correspondence should be addressed to Muhammad Sulaiman; ku.oc.oohay@315namialus and Abdellah Salhi; ku.ca.xesse@sa

Received 10 September 2017; Accepted 21 January 2018; Published 21 March 2018

Academic Editor: Haranath Kar

Copyright © 2018 Muhammad Sulaiman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Plant Propagation Algorithms (PPA) are powerful and flexible solvers for optimisation problems. They are nature-inspired heuristics which can be applied to any optimisation/search problem. There is a growing body of research, mainly experimental, on PPA in the literature. Little, however, has been done on the theoretical front. Given the prominence this algorithm is gaining in terms of performance on benchmark problems as well as practical ones, some theoretical insight into its convergence is needed. The current paper is aimed at fulfilling this by providing a sketch for a global convergence analysis.

#### 1. Introduction

The theoretical analysis of stochastic algorithms for global optimisation is not new and can be found in a number of sources such as [1–5]. The majority of the algorithms considered use random search one way or another to find the optimum solution [6–13]. Here, we consider the algorithmic scheme of the Plant Propagation Algorithm for continuous optimisation or PPA-C [14] and theoretically investigate its global convergence to the optimum solution. The optimisation problems of concern are continuous and defined in finite -dimensional domains.

The basic version of PPA [15] models the propagation of strawberry plants. The scheme uses short runners for exploitation or local search refinement while long runners are used for diversification and exploration of the search space. Since the propagation of strawberries is due to seeds as well as runners, a Seed-based Plant Propagation Algorithm (SbPPA) has also been introduced in [16]. Both PPA-C and SbPPA have been shown to be efficient on continuous unconstrained and constrained optimisation problems; statistical convergence analyses of PPA-C and SbPPA can be found in [9–11, 14–16].

PPA-C [14, 15] consists of two steps:(1)Initialization: a population of parent plants is generated randomly.(2)Propagation: a new population is created from persistent parents (strawberry plants) and their children (new strawberry plants at the end of runners, i.e., a distance away from parent plants).

Let denote the search space such that , where is its dimension. By an iteration of PPA-C we mean a new generation of child plants produced by parent plants. These child plants are the result of either short or long runners [14, 16]. This is the basic setup that we consider to sketch a proof of convergence to the global optimum of a given continuous optimisation problem.

The paper is organised as follows. Section 2 presents the terminology used in the analysis of PPA-C. Section 3 analyses a population of plants. Section 3.1 describes the convergence analysis of PPA-C. Section 4 is the conclusion.

#### 2. Terminology and Notation

We consider single objective minimization problems [17]. such that for all , where the objective function is defined as , denotes the best spot for a plant in the search space. is an -dimensional position vector.

The population at the th iteration is denoted by , where is the population size. The coordinates of runners, or more precisely their endpoints, are denoted by , where is the space dimension of the given problem.

##### 2.1. Search Equations and Evaluation of New Plants

Variants of PPA can be found in [14–16]. In this paper we analyse PPA-C as Algorithm 1 of [14].

In order to send a short or long runner, is generated [14, 19–21], as in (1a), (1b), and (1c)where is the population size, is the Monte Carlo trial run counter, is the modification probability, and is a randomly generated number for each th entry, . The indices are mutually exclusive; that is, . Another version of PPA called SbPPA [16] which is inspired by propagation via seeds implements the following search equation instead: where is a step drawn from the Lévy distribution [22] and is a random coordinate within the search space. Equations (1a), (1b), (1c), and (2) perturb the current solution, the results of which can be seen in Figures 1(a) and 1(b), respectively.