Metaheuristic Techniques

methods. From Section 3 to Section 14 we cover several popular metaheuristic techniques with completepseudocodes. We discuss the methods in alphabetical order in order to avoid preconceptions aboutperformance, generality and applicability, thus respecting the No Free Lunch theorem [53] of optimization.In Section 15, we enumerate several less popular methods with a brief description of their origins. Thesemethodologies are ordered by their number of Google Scholar citations. We conclude this paper inSection 16 with a few pointers to future research directions.2. Concepts of Metaheuristic TechniquesThe word ‘heuristic’ is defined in the context of computing as a method of denoting a rule-of-thumbfor solving a problem without the exhaustive application of a procedure. In other words, a heuristicmethod is one that (i) looks for an approximate solution, (ii) need not particularly have a mathematicalconvergence proof, and (iii) does not explore each and every possible solution in the search space beforearriving at the final solution, hence is computationally efficient.A metaheuristic method is particularly relevant in the context of solving search and optimizationproblems. It describes a method that uses one or more heuristics and therefore inherits all the threeproperties mentioned above. Thus, a metaheuristic method (i) seeks to find a near-optimal solution,instead of specifically trying to find the exact optimal solution, (ii) usually has no rigorous proof ofconvergence to the optimal solution, and (iii) is usually computationally faster than exhaustive search.These methods are iterative in nature and often use stochastic operations in their search process tomodify one or more initial candidate solutions (usually generated by random sampling of the searchspace). Since many real-world optimization problems are complex due to their inherent practicalities,classical optimization algorithms may not always be applicable and may not fair well in solving suchproblems in a pragmatic manner. Realizing this fact and without disregarding the importance of classicalalgorithms in the development of the field of search and optimization, researchers and practitioners soughtfor metaheuristic methods so that a near-optimal solution can be obtained in a computationally tractablemanner, instead of waiting for a provable optimization algorithm to be developed before attempting tosolve such problems. The ability of the metaheuristic methods to handle different complexities associatedwith practical problems and arrive at a reasonably acceptable solution is the main reason for the popularityof metaheuristic methods in the recent past.Most metaheuristic methods are motivated by natural, physical or biological principles and try tomimic them at a fundamental level through various operators. A common theme seen across all metaheuristics is the balance between exploration and exploitation. Exploration refers to how well the operatorsdiversify solutions in the search space. This aspect gives the metaheuristic a global search behavior. Exploitation refers to how well the operators are able to utilize the information available from solutionsfrom previous iterations to intensify search. Such intensification gives the metaheuristic a local searchcharacteristic. Some metaheuristics tend to be more explorative than exploitative, while some others dothe opposite. For example, the primitive method of randomly picking solutions for a certain number ofiterations, represents a completely explorative search. On the other hand, hill climbing where the currentsolution is incrementally modified until it improves, is an example of completely exploitative search. Morecommonly, metaheuristics allow this balance between diversification and intensification to be adjusted bythe user through operator parameters.The characteristics described above give metaheuristics certain advantages over the classical optimization methods, namely,1. Metaheuristics can lead to good enough solutions for computationally easy (technically, P class)problems with large input complexity, which can be a hurdle for classical methods.2. Metaheuristics can lead to good enough solutions for the NP-hard problems, i.e. problems for whichno known exact algorithm exists that can solve them in a reasonable amount of time.3. Unlike most classical methods, metaheuristics require no gradient information and therefore can beused with non-analytic, black-box or simulation-based objective functions.4

4. Most metaheuristics have the ability to recover from local optima due to inherent stochasticity ordeterministic heuristics specifically meant for this purpose.5. Because of the ability to recover from local optima, metaheuristics can better handle uncertaintiesin objectives.6. Most metaheuristics can handle multiple objectives with only a few algorithmic changes.2.1. Optimization ProblemsA non-linear programming (NLP) problem involving n real or discrete (integer, boolean or otherwise)variables or a combination thereof is stated as follows:Minimize f (x),subject to gj (x) 0,j 1, 2, . . . , J,hk (x) 0,k 1, 2, . . . , K,(L)(U )xi xi xi , i 1, 2, . . . , n,(1)where f (x) is the objective function to be optimized, gj (x) represent J inequality constraints and hk (x)represent K equality constraints. Typically, equality constraints are handled by converting them intosoft inequality constraints. A survey of constraint handling methods can be found in [54, 55]. A solution(L)(U )x is feasible if all J K constraints and variable bounds [xi , xi ] are satisfied.The nature of the optimization problem plays an important role in determining the optimizationmethodology to be used. Classical optimization methods should always be the first choice for solvingconvex problems. An NLP problem is said to be convex if and only if, (i) f is convex, (ii) all gj (x)are concave, and (iii) all hk (x) are linear. Unfortunately, most real-world problems are non-convex andNP-hard, which makes metaheuristic techniques a popular choice.Metaheuristics are especially popular for solving combinatorial optimization problems because notmany classical methods can handle the kind of variables that they involve. A typical combinatorialoptimization problem involves an n-dimensional permutation p in which every entity appears only once:Minimize f (p),subject to gj (p) 0,hk (p) 0,j 1, 2, . . . , Jk 1, 2, . . . , K.(2)Again, a candidate permutation p is feasible only when all J constraints are satisfied. Many practicalproblems involve combinatorial optimization. Examples include knapsack problems, bin-packing, networkdesign, traveling salesman, vehicle routing, facility location and scheduling.When multiple objectives are involved that conflict with each other, no single solution exists that cansimultaneously optimize all the objectives. Rather, a multitude of optimal solutions are possible whichprovide a trade-off between the objectives. These solutions are known as the Pareto-optimal solutionsand they lie on what is known as the Pareto-optimal front. A candidate solution x1 is said to dominatex2 and denoted as x1 x2 if and only if the following conditions are satisfied:1. fi (x1 ) fi (x2 ) i {1, 2, . . . , m},2. and j {1, 2, . . . , m} such that fj (x1 ) fj (x2 ).If only the first of these conditions is satisfied then x1 is said to weakly dominate x2 and is denoted asx1 x2 . If neither x1 x2 nor x2 x1 , then x1 and x2 are said to be non-dominated with respect toeach other and denoted as x1 x2 . A feasible solution x is said to be Pareto-optimal if there does notexist any other feasible x such that x x . The set of all such x (which are non-dominated with respectto each other) is referred to as the Pareto-optimal set. Multi-objective optimization poses additionalchallenges to metaheuristics due to this concept of non-dominance.In this paper, we will restrict ourselves to describing metaheuristics in the context of single-objectiveoptimization. Most metaheuristics can be readily used for multi-objective optimization by simply considering Pareto-dominance or other suitable selection mechanisms when comparing different solutions5

and by taking extra measures for preserving solution diversity. However, it is important to note thatsince multi-objective optimization problems with conflicting objectives have multiple optimal solutions,metaheuristics that use a ‘population’ of solutions are preferred so that the entire Pareto-optimal frontcan be represented simultaneously.2.2. Classification of Metaheuristic TechniquesThe most common way of classifying metaheuristic techniques is based on the number of initialsolutions that are modified in subsequent iterations. Single-solution metaheuristics start with one initialsolution which gets modified iteratively. Note that the modification process itself may involve more thanone solution, but only a single solution is used in each following iteration. Population-based metaheuristicsuse more than one initial solution to start optimization. In each iteration, multiple solutions get modified,and some of them make it to the next iteration. Modification of solutions is done through operatorsthat often use special statistical properties of the population. The additional parameter for size of thepopulation is set by the user and usually remains constant across iterations.Another way of classifying metaheuristics is through the domain that they mimic. Umbrella terms suchas bio-inspired and nature-inspired are often used for metaheuristics. However, they can be further subcategorized as evolutionary algorithms, swarm-intelligence based algorithms and physical phenomenonbased algorithms. Evolutionary algorithms (like genetic algorithms, evolution strategies, differential evolution, genetic programming, evolutionary programming, etc.) mimic various aspects of evolution innature such as survival of the fittest, reproduction and genetic mutation. Swarm-intelligence algorithmsmimic the group behavior and/or interactions of living organisms (like ants, bees, birds, fireflies, glowworms, fishes, white-blood cells, bacteria, etc.) and non-living things (like water drops, river systems,masses under gravity, etc.). The rest of the metaheuristics mimic various physical phenomena like annealing of metals, musical aesthetics (harmony), etc. A fourth sub-category may be used to classifymetaheuristics whose source of inspiration is unclear (like tabu search and scatter search) or those thatare too few in number to have a category for themselves.Other popular ways of classifying metaheuristics are [1, 35]:1. Deterministic versus stochastic methods: Deterministic methods follow a definite trajectory fromthe random initial solution(s). Therefore, they are sometimes referred to as trajectory methods.Stochastic methods (also discontinuous methods) allow probabilistic jumps from the current solution(s) to the next.2. Greedy versus non-greedy methods: Greedy algorithms usually search in the neighborhood of thecurrent solution and immediately move to a better solution when its found. This behavior oftenleads to a local optimum. Non-greedy methods either hold ou

13. Ant Colony Optimization and Swarm Intelligence (ANTS Conference) 14. Swarm, Evolutionary and Memetic Computing Conference (SEMCCO) 15. Bio-Inspired Computing: Theories and Applications (BICTA) 16. Nature and Biologically Inspired Computing (NaBIC) 17. International Conference on Soft Computing for Problem Solving (SocProS) 18.