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Parabola Function

Introduction

The Parabola function is a simple quadratic benchmark function that provides a basic test case for optimization algorithms. It's similar to the Sphere function but with a different mathematical structure, offering insights into algorithm behavior on quadratic landscapes.

Mathematical Definition

The Parabola function can be defined in various forms. A common implementation is:

\[f(x) = \sum_{i=1}^d (x_i - c_i)^2\]

Where \(c_i\) are offset constants (often set to 0 for centered parabola).

Properties

  • Global minimum: \(f(c_1, c_2, ..., c_d) = 0\) where \(c_i\) are the offsets
  • Search domain: Typically \([-10, 10]^d\) or similar
  • Unimodal: Single global minimum
  • Separable: Variables are independent
  • Convex: Bowl-shaped surface
  • Quadratic: Second-order polynomial
  • Scalable: Works in any dimension

Usage Example

Algorithm Testing

Convergence Speed Comparison

Precision Testing

Scalability Analysis

Custom Parabola Variations

Rotated Parabola

Educational Use Cases

Teaching Optimization Basics

Convergence Visualization

Performance Characteristics

For the Parabola function, typical performance expectations:

  • Hill Climbing: Very efficient, direct path to minimum
  • Random Search: Slow convergence, many evaluations needed
  • Simulated Annealing: Good performance, some random exploration
  • Bayesian Optimization: Excellent efficiency, few evaluations needed
  • Population methods: Overkill for this simple function

Common Applications

  1. Algorithm validation: Quick sanity check for new optimizers
  2. Parameter tuning: Test optimizer parameters on simple function
  3. Educational purposes: Demonstrate optimization principles
  4. Baseline comparison: Simple reference for performance
  5. Convergence analysis: Study optimization dynamics

References

  • Fletcher, R. (2013). Practical methods of optimization.
  • Nocedal, J., & Wright, S. (2006). Numerical optimization.