Function Extremum Optimization Using Real-Coded Genetic Algorithms

Real-coded genetic algorithm (RCGA) is an efficient method for solving function extremum optimization problems. Unlike traditional binary encoding, RCGA directly uses real numbers to represent candidate solutions in the search space, which provides higher precision and efficiency when handling continuous variable problems.

Implementing real-coded genetic algorithms for function extremum optimization in MATLAB environment involves the following key steps with corresponding code approaches:

Population Initialization: Randomly generate a set of real-valued vectors as the initial population, where each vector represents a potential solution to the function. The population size should be reasonably set according to problem complexity. In MATLAB, this can be implemented using: population = lb + (ub-lb).*rand(popSize,dim); where lb and ub represent lower and upper bounds.

Fitness Evaluation: Calculate the fitness value for each individual using the objective function. For maximization problems, the function value is typically used directly; for minimization problems, conversion through reciprocals or negative values is required. MATLAB implementation: fitness = objectiveFunction(population);

Selection Operation: Apply methods like roulette wheel selection or tournament selection to choose superior individuals for the next generation based on fitness values. Individuals with higher fitness have greater selection probability. Tournament selection implementation: selectedIndices = tournamentSelect(fitness, tournamentSize);

Crossover Operation: Common crossover operators for real-coded GA include arithmetic crossover and heuristic crossover. The crossover probability determines the algorithm's ability to explore new regions. Arithmetic crossover example: offspring = parent1 + alpha*(parent2-parent1); where alpha is a random factor.

Mutation Operation: Apply Gaussian mutation or uniform mutation to introduce small perturbations to individuals, increasing population diversity. Mutation probability is typically set to small values. Gaussian mutation implementation: mutated = individual + sigma*randn(1,dim); where sigma controls mutation magnitude.

Termination Condition: Stop the algorithm when maximum iterations are reached or the optimal solution converges, then output the current best solution. MATLAB implementation: while iteration < maxIter && convergenceCriterion > tolerance

Real-coded genetic algorithms eliminate the encoding/decoding process required in binary coding, offering better local search capability and convergence speed for continuous function optimization problems. MATLAB's matrix computation capabilities are particularly suitable for implementing such algorithms, enabling efficient handling of genetic operations on populations through vectorized operations.