Genetic Algorithm for Function Minimization

Genetic Algorithm (GA) is an optimization technique inspired by natural evolution processes, particularly effective for solving complex nonlinear optimization problems. When applied to function minimization, GA mimics biological evolution mechanisms including selection, crossover, and mutation to progressively approach optimal solutions.

During implementation, the first step involves formulating the problem as a fitness function. For minimization problems, the objective function is typically inverted (multiplied by -1) to create a fitness function, allowing the algorithm to maximize fitness while effectively locating the original function's minimum. In code implementation, this transformation is usually handled through a simple wrapper function that converts minimization to maximization.

The core algorithm steps include population initialization, fitness evaluation, selection, crossover, and mutation. Each individual in the population represents a potential solution vector (function input values). Using selection mechanisms like roulette wheel selection, individuals with higher fitness have greater probability of being selected for reproduction. Crossover operations (typically single-point or uniform crossover) combine parent chromosomes to create offspring, while mutation operations (applied with low probability) introduce random changes to maintain population diversity.

As iterations progress, superior genetic material accumulates within the population, eventually converging near the function's minimum point. To prevent premature convergence, mutation probability can be adaptively adjusted to preserve population diversity. Common termination criteria include reaching maximum generations or when solution improvement falls below a specified threshold. Implementation often involves maintaining elitism by preserving the best individuals across generations.

GA's primary advantages include not requiring gradient information of the objective function, handling discontinuous and non-differentiable functions, and avoiding local optima through its global search capabilities. The algorithm typically employs real-valued or binary encoding schemes depending on the problem domain, with parameter tuning crucial for optimal performance.