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Rosenbrock function

As explained in Optimization Algorithms: AI techniques for design, planning, and control problems

The Rosenbrock function is a nonconvex test function with a narrow valley leading to its global minimum, commonly used to evaluate optimization algorithms.

The Rosenbrock function, also called the valley or banana function, is a nonconvex optimization test problem. Its global minimum is f(x*) = 0 at (1, ..., 1), while the search landscape guides optimizers through a difficult valley. section 10.5

Key properties for optimization experiments include:

  • Use n-dimensional inputs, commonly bounded by [-5, 10] or the narrower interval [-2.048, 2.048]. section 10.5
  • Evaluate candidate solutions by their objective value; lower values indicate better solutions. section 10.5
  • Use the function to expose how an optimizer explores a challenging nonconvex landscape. section 10.5

Minimizing the two-dimensional function with ABC

A small artificial bee colony experiment initializes six candidate positions in [-2.048, 2.048]. Their objective values range from 30.62 to 2,097.22. After neighborhood updates, the best candidate reaches 15.15, and fitness-proportionate selection gives it probability 0.49. A Python implementation with 50 bees and 1,000 iterations reports [0.99766117, 0.99542949] with fitness 6.50e-06. section 10.5

How Optimization Algorithms: AI techniques for design, planning, and control problems covers it

How deep it goes
Defines the landscape, bounds, minimum, and ABC walkthrough, then demonstrates implementation results without deriving optimizer theory.
Examples
Demonstrates six hand-worked food sources, fitness probabilities, Python ABC code, and contour plots of candidate solutions.
What it assumes
Assumes basic optimization vocabulary and Python familiarity while supplying the function, bounds, candidate evaluation, and algorithm setup.
What you can do after
Evaluate optimization behavior by implementing the function and minimizing it with a basic ABC configuration.
How it connects
Connects nonconvex optimization with gradient-based optimization, artificial bee colony, and ant colony optimization.
Cover of Optimization Algorithms: AI techniques for design, planning, and control problems
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