FOLA capitalises on the merits of the structure of multiple automata, the di- mensional search, the dividing of the dimensional search domain into cells, and the memories of the performance evaluation of the dimensional states in a form of cell values. By these approaches, FOLA is able to undertake search in continuous states
and achieve accurate solutions efficiently. There are two key parameters, wc and
Nfemax, to tune in FOLA when it is applied to resolve a specific application prob-
lem. The two parameters can be determined with the basic knowledge of the range of variables involved and the solution accuracy required in the application problem. The simulation studies have been undertaken on 13 widely used 30-dimensional benchmark functions which include uni-modal and multi-modal problems. The ex- perimental results have shown that FOLA is able to achieve more accurate results
2.5 Conclusions 66
than the other eight EAs in finding a global minimum solution, and is more reli- able, for its standard deviation of the results over different independent runs is much smaller than that of other algorithms. To investigate whether FOLA can be scaled up to handle the optimisation problems which are highly dimensional, FOLA has been applied to solve the 300-dimensional multi-modal functions which are extremely difficult to solve due to a large number of local minima. In comparison with the other four EAs, FOLA presents its great superiority, as it finds much more accurate solutions, and significantly improves the efficiency and convergence rate.
In addition, FOLA has also been applied to solve nine challenging multi-modal benchmark functions, which are rotated and shifted. In this case, FOLA is compared with two popularly used EAs and four newly-proposed EAs. The experimental re- sults have shown that FOLA offers better performance for most of the benchmark functions, in terms of the accuracy of the obtained optimal solutions and the con- vergence rate. FOLA is able to reduce the computation time greatly, especially for high-dimensional functions.
Multi-objective Optimisation by
Learning Automata
3.1
Introduction
Many (perhaps most) real-world problems are, in fact, multi-objective optimisa- tion problems. Unlike single-objective optimisation, whose goal is to find the global maximum or minimum subject to an objective function, a multi-objective optimisa- tion problem has usually no unique, perfect solution, but a series of non-inferior al- ternative solutions, known as Pareto optimal solutions, which represent the possible trade-off among conflicting objectives. The multi-objective optimisation problems can be formulated as follows:
Minimise F(X) = [f1(X), f2(X),· · · , fmf(X)] (3.1.1)
s.t. gi(x)≤0, i= 1,2,· · · , mg
whereX = [x1, x2,· · · , xN]T ∈RN is the vector of variables to be optimised, func-
tions gi (i = 1,2,· · · , mg)are constraint functions of the problem, and functions
fi (i = 1,2,· · · , mf) are mf objective functions. mf fitness values, obtained by
applying solutionX in themf objective functions, compose an objective function
3.1 Introduction 68 !"#$%"&'#()*% +"&'#()*% ,-. */)'0* 1 2 3 45 6 7 89 5 : ; (<*)" =<"#)
Figure 3.1: Dominance relation in multi-objective problems
The optimal solutions of a multi-objective optimisation problem can be de- scribed by the concept of Pareto dominance and Pareto optimality, which are math- ematically defined as follows [101][102]:
Definition 1 (Pareto Dominance): A vector U = [u1, u2,· · · , umf] is said to dominateV = [v1, v2,· · · , vmf] if and only ifU is partially less than V, i.e. ∀i ∈
{1,2,· · · , mf}, ui ≤vi ∧ ∃i∈ {1,2,· · · , mf}:ui < vi.
Definition 2 (Pareto Optimality): A solutionXU is said to be Pareto optimal if and only if there is no XV for which V = F(XV) = (v1, v2,· · · , vmf)dominates
U =F(XU) = (u1, u2,· · · , umf).
As an example to explain this concept, assume that a multi-objective problem is to minimise F(Xa), a ∈ {U, V}, where U = F(XU) = (3.25,1.76,4.67) and
V = F(XV) = (3.15,1.76,4.22). In this example, objective function vectorU is
dominated byV, andV is non-dominated in the objective space. XV is the Pareto
optimal solution (also called non-dominated solution) of set {XU, XV}, and V is Pareto optimal objective function vector (also called non-dominated objective func- tion vector). A set of all the Pareto optimal solutions is called the Pareto set, which can be used to form a Pareto front in the objective space, as shown in Fig. 3.1.
It can be seen that the aim of multi-objective optimisation is to gain the Pareto
set whose Pareto optimal objective function vectors are evenly distributed on the Pareto front [2]. The Pareto front is evaluated through two aspects: 1) convergence to the Pareto-optimal set and 2) maintenance of diversity among the solutions of the Pareto-optimal set.
In order to obtain an accurate Pareto front, there are two standard methods for treating multi-objective problems. One is to convert multi-objective problems into a single objective problem using the weighted-sum method or weighted Tchebycheff method; and the other one is Pareto front-based method, which applies a population of individuals, and each of them represents one Pareto optimal solution. Based on this concept, various algorithms have been proposed to solve multi-objective op- timisation problems in the past few decades, such as multi-objective evolutionary algorithms [103][104], multi-objective genetic algorithms [29][7] and group search optimiser [60]. These multi-objective algorithms propose mathematical improve- ments to meet the demand of solving problems. They have been comprehensively investigated in various application areas, such as power plant [43], wireless sensor networks [30], structural mechanics problems [105], and other engineering prob- lem [31], and so on. The first two applications adopt the weighted-sum methods, which are only capable of solving convex Pareto front problems but have a diffi- culty in solving the multi-objective problems whose Pareto fronts are non-convex [7]. Nonetheless, there is no way to predetermine if a problem is convex or concave in many applications. The latter is an NSGA II-based method, which suffers from the drawback of high computational complexity caused by non-dominated sorting.
This chapter presents a novel method for multi-objective optimisation by learn- ing automata (MOLA). MOLA adopts the strategy of multiple automata, dimen- sional search and action selection, which are similar to those in FOLA. Unlike FOLA, the reinforcement signal adopted here is generated through comparing the state with all the non-dominated solutions found so far. MOLA mainly comprises two processes: the process of searching and the process of learning from neigh- borhood. The process of searching is carried out through a tournament that is held between Pareto global search and Pareto local search. This tournament can lead to a better trade-off between exploitation and exploration, which is a critical factor in