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Sphere Function

Introduction

The Sphere function is one of the simplest and most fundamental benchmark functions in optimization. It represents a convex, unimodal function with a single global minimum, making it ideal for testing basic convergence properties of optimization algorithms.

Mathematical Definition

The Sphere function is defined as:

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

Where \(d\) is the dimensionality of the problem.

Properties

  • Global minimum: \(f(0, 0, ..., 0) = 0\)
  • Search domain: Typically \([-5.12, 5.12]^d\) or \([-10, 10]^d\)
  • Unimodal: Single global minimum, no local minima
  • Separable: Each variable contributes independently
  • Convex: Bowl-shaped surface
  • Differentiable: Smooth everywhere
  • Scalable: Can be used in any dimension

Usage Example

Characteristics for Algorithm Testing

What the Sphere Function Tests

  1. Basic Convergence: Can the algorithm find the global minimum?
  2. Convergence Rate: How quickly does it converge?
  3. Precision: How close can it get to the exact optimum?
  4. Scalability: Performance vs dimensionality relationship
  5. Consistency: Reliable performance across runs

Expected Algorithm Performance

  • Gradient-based methods: Excellent, direct path to optimum
  • Hill Climbing: Very good, steady improvement
  • Random Search: Slow but eventually successful
  • Population methods: Good, may be overkill for this simple function
  • Bayesian Optimization: Excellent, very sample-efficient

Multi-Dimensional Scaling

Convergence Analysis

Algorithm Comparison

Parameter Space Exploration

Custom Sphere Variations

Theoretical Properties

Gradient Information

The gradient of the sphere function is: \(\(\nabla f(x) = 2x\)\)

This means: - Gradient points directly away from the origin - Magnitude increases linearly with distance from origin - Steepest descent leads directly to the global minimum

Hessian Matrix

The Hessian is: \(\(H = 2I\)\)

Where \(I\) is the identity matrix, indicating: - Constant curvature in all directions - No interaction between variables - Well-conditioned optimization problem

Performance Baselines

For the sphere function, expect these approximate performance characteristics:

  • Random Search: \(O(d)\) evaluations to get within reasonable distance
  • Hill Climbing: \(O(\log(\epsilon^{-1}))\) for precision \(\epsilon\)
  • Gradient Methods: Quadratic convergence
  • Population Methods: May require more evaluations but very reliable

Common Mistakes

  1. Over-engineering: Using complex algorithms on this simple function
  2. Insufficient precision: Stopping optimization too early
  3. Wrong bounds: Using bounds that don't include the global minimum
  4. Ignoring scaling: Not accounting for dimension-dependent difficulty

References

  • De Jong, K. A. (1975). An analysis of the behavior of a class of genetic adaptive systems.
  • Jamil, M., & Yang, X. S. (2013). A literature survey of benchmark functions for global optimization problems.