On the Theoretical Analysis of the Plant Propagation Algorithms

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.


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][2][3][4][5].The majority of the algorithms considered use random search one way or another to find the optimum solution [6][7][8][9][10][11][12][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][10][11][14][15][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.

Terminology and Notation
We consider single objective minimization problems [17]. optimal ∈  such that ( optimal ) ≤ () 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 pop  = { 1, ,  2, , . . .,  , }, where  is the population size.The coordinates of runners, or more precisely their endpoints, are denoted by   = ( 1  ,  2  , . . .,    )  , where  is the space dimension of the given problem.

Search Equations and Evaluation of New Plants.
Variants of PPA can be found in [14][15][16].In this paper we analyse PPA-C as Algorithm 1 of [14].

A Case Study.
Let  be the class of runners sent by the th parent plant and stored in .Each runner in class  is decomposed into two vectors   and   , where   denotes the vector of indices which are perturbed with respect to the current position of the plant, while   represents the vector of corresponding indices of the unperturbed coordinates with respect to the current position of plants.This can be represented as To clarify this idea, let us take an example [17] of a newly generated runner by the th plant as such that then we can write where The dot product of these vectors is zero, which shows that they are mutually orthogonal.Mathematically, this can be written as Let    and    denote two vector spaces such that    = containing vectors having dimensions as in   ,    = containing vectors having dimensions as in   .
and    are subspaces of   .This implies that     ∈    and     ∈    .A scalar objective function  defined over    can be represented as where where (11) represents an objective value corresponding to a new runner in position    .Similarly, different runners are produced to correspond to different classes  and evaluated by the same procedure.This procedure can be generalized for -dimensional problems [1][2][3].Figure 1: Overall performance of (1a), (1b), (1c), and (2) on a design optimisation problem given in Appendix [16].

Graphical and Theoretical Analysis of a Population of Plants
unchanged with probability 1 −   .Thus there are 2  possible runners to be generated for each th parent plant using (1a), (1b), and (1c), where  is the space dimension of the given problem.
Let, at any generation , the random population be represented as pop = { 1, ,  2, , . . .,  , }, where  denotes the population size.To create a runner by using (1a), (1b), and (1c), for next generation +1, PPA uses a population of parent plants at generation  for this purpose.It is not required to know about any other runner in generation  + 1.
This shows that all runners created at generation  + 1 are statistically mutually independent.Furthermore, the initial population is random and all parent plants do not depend on each other.Thus, by induction, the runners at any further generations are mutually independent.
From (1a), (1b), and (1c), a runner    may be formed by itself or by choosing three different coordinates from current population.In case of using (1a) [19], there are  possibilities to send (long or short) runners as in Figure 1.On the other hand, by using (1b) or (1c),  − 1 vectors are used to send a new runner.Thus, in later cases there are −1  3 possibilities to send new runners.In (1a)-(1c), different possibilities are of the form where  , ,  , , and  , are calculated according to (1a), (1b), and (1c), in which  denotes the current generation and  is an -dimensional random vector within interval The probability density functions (PDFs) [17,23] (15).Definition 1 (convolution operation ⋆ [24]).Let () and () be Laplace transformable piecewise continuous functions defined on [0 ∞].The convolution product of these two functions is again a function of  defined as Let  +1 be any plant in generation  + Note that a runner is selected for the next population only if its rank is less than or equal to , the population size.Its objective value is less than the maximum objective value (in case of minimization problem) in the current population.Note also that instead of greedy selection we sort the population and eliminate those plants whose rank is higher than .The model for this selection mechanism can be represented as follows: 3.1.Convergence Analysis of PPA-C.For illustration purposes, we have implemented a combined version of (1a), ( . This version of PPA-C called H-PPA-SbPPA is a hybridisation of PPA and SbPPA [10].We have plotted the position of plants in populations through solving the Branin and Matyas test functions (see Figures 2,3,4,and 5).It is obvious from Figures 2 and 4 that (1a), (1b), and (1c) have generated short runners which exploit the search space locally.On the other hand, in Figures 3 and 5, (2) has generated a diverse range of solutions which are spread over the whole search space.This equation helps the algorithm escape from local minima and to explore the solution space better and hence the global search qualities of this algorithm.Mathematically this can be shown as follows.
Let  denote the search space containing the solution of a given optimisation problem defined as where () is the objective function.Then the optimal solution set [25] can be represented as where  optimal is the optimum solution.The region of attraction [25] of the solution set  * is defined as where  is a small positive real number and  best is the current best solution.
In PPA-C, each parent plant produces  ℎ  runners (solutions), where ℎ = 1, 2, 3.The probability that at generation  a subset of the temporary population   ⊂ , containing solutions which are not good enough to be retained in the next generation by the selection model as in (17), is given as where   is a small positive real number.Obviously, in all previous generations  − 1, some of the solutions died and some succeeded to survive into the next generation.This shows that in previous generations we have some solutions  ℎ  which do not belong to the region of attraction  *  . −1 At the end of each generation, the temporary population  is appended to the main population pop  .Then all individuals are sorted with respect to their objective values.The individuals with higher rank than size of population are   omitted.Thus the probability that a generation  does not contain an optimum is given below After sorting the final population at the end of each generation, the probability that the optimum may exist in a subpopulation   (population of weak or dead runners/solutions) is less than that of the population pop  .This can be represented as Following [25][26][27], the right hand side term of inequality ( 24) is zero if the series ∑ +∞ =1 (  ) diverges.The convergence of PPA follows and can be summarised in the theorem below.[1][2][3]

Conclusion
The Plant Propagation Algorithm (PPA) and its variants for continuous optimisation problems are getting notoriety as flexible and powerful solvers.PPA is a heuristic inspired by the way plants and in particular the strawberry plant propagate.It is also referred to as the Strawberry Algorithm.While there is a growing body of experimental and computational works that show its good behaviour and performance against well-established algorithms and heuristics, there is very little if any in terms of theoretical investigation.This gap, therefore, needs to be filled.The aim of course, in analysing the convergence of any algorithm (in this case PPA-C), is to give confidence to the potential users that the solutions that it returns are of good quality.The convergence analysis put forward in this paper relies on the exploitation and exploration characteristics of the algorithm.Since it does not get stuck in local optima and explores thoroughly the search space it is only a matter of time before the global optimum is discovered.The approach is probabilistic in nature and ascertains that the global optimum will be found with probability 1 provided the algorithm is run reasonably long enough.The argument for this to hold is that at each iteration new and better solutions are generated which means that, in the limit, the global optimum is reached.Questions still remain concerning what is considered a reasonable amount of time.Bounds on the time it will take to converge are being developed and results will be presented in a followup paper.

Figure 2 :Figure 3 :
Figure 2: The exploitation capability of PPA while solving Branin test function.

Figure 4 :
Figure 4: A scatter plot of plants produced by PPA when optimising Matyas function [15].
Remark 3 implies that  best is always improving or changing its position until the optimum is reached.Remark 6.  best converges approximately to  optimal , as generation  grows.