Comparing Optimization Algorithms: Lessons from the Himmelblau Function

Which Optimization Algorithm Wins? Lessons from the Himmelblau Function
Optimization algorithms are the backbone of countless computational tasks, from training machine learning models to solving engineering puzzles. But how do you pick the right one? A recent study, "A Comparative Analysis of Optimization Algorithms: The Himmelblau Function Case Study" (April 22, 2025), puts four algorithms to the test on the Himmelblau function—a notorious benchmark that’s like a treasure map with multiple hidden gems. In this post, we’ll break down the study’s key findings for developers, keeping things clear and practical, with no math overload. Plus, we’ll share a Python snippet to get you experimenting with optimization yourself.

What’s the Himmelblau Function?
Imagine a landscape with four deep valleys (the “treasures” or global minima) and several shallow dips that can trick you into thinking you’ve found the best spot. That’s the Himmelblau function, a classic test for optimization algorithms. Its complexity—four identical optimal solutions and sneaky local minima—makes it perfect for seeing how well an algorithm can find the true valleys without getting stuck. The study uses it to compare how different algorithms navigate this tricky terrain.

The Algorithms Tested
I tested four optimization algorithms, each run 360 times on the Himmelblau function to ensure robust results [1]:

• SA_Noise: Simulated Annealing with added randomness to roam the landscape more freely, like a hiker exploring every path.
• SA_T10: Simulated Annealing with a fixed “temperature” of 10, balancing exploration and focus, like a hiker who’s a bit more methodical.
• Hybrid_SA_Adam: A hybrid of Simulated Annealing and the Adam optimizer, combining bold exploration with precise steps, like a hiker with a GPS.
• Adam_lr0.01: The Adam optimizer with a learning rate of 0.01, a machine learning favorite that takes steady, calculated steps toward the goal.

How They Were Judged
The study measured performance across four metrics:

• Steps to Converge: How many iterations it took to settle on a solution.
• Final Loss: How close the solution’s value was to zero (the ideal value for the Himmelblau function’s valleys).
• Success Rate: The percentage of runs with a final loss below 1.0, meaning the algorithm got close to a valley.
• Consistency: 95% confidence intervals for steps, loss, and distance to the nearest valley, showing how reliable each algorithm was.

Key Results
Across all 360 runs, the algorithms averaged 238.73 steps, a final loss of 0.52, and an 81.11% success rate. But digging deeper, each algorithm showed distinct strengths [1]:

Steps to Converge

• SA_Noise and SA_T10: Lightning-fast, needing just 52–62 steps on average. They’re the sprinters of the group, racing to a solution.
• Hybrid_SA_Adam: More deliberate, taking 194–208 steps, like a runner pacing themselves for accuracy.
• Adam_lr0.01: The slowest, clocking 585–693 steps, as it carefully navigates to the best spot.

Final Loss
Hybrid_SA_Adam and Adam_lr0.01: Absolute champs, hitting near-zero loss (0.00–0.00), meaning they consistently found the true valleys.
SA_Noise and SA_T10: Less accurate, with losses of 0.69–1.38 and 0.87–1.38, often settling in shallow dips instead of the deepest valleys.

Distance to the True Solution
All algorithms ended up about 4.05–4.41 units from the nearest valley, but Hybrid_SA_Adam was the most precise, with a tight range of 4.08–4.28.

Visual Insights
The study’s visuals tell a vivid story [1]:

Boxplots: SA_Noise and SA_T10 are fast but sloppy, while Hybrid_SA_Adam and Adam_lr0.01 are slower but nail the target (see Figure 1 in [1]).
Convergence Plot: Adam_lr0.01 zooms to near-zero loss in just 200 steps, outpacing others in accuracy (Figure 2 in [1]).
Heatmap of Failures: SA_T10 struggled most, racking up high losses when stuck about 4.0 units from a valley (Figure 3 in [1]).

Overall Metrics:
Number of Runs: 360
Average Steps: 238.73
Average Final Loss: 0.52
Success Rate (<1.0): 81.11%

Why This Matters for Devs
The study highlights a classic trade-off: speed vs. accuracy. Here’s what it means for your projects:

Need Speed? SA_Noise and SA_T10 are ideal for real-time systems or tasks where a “good enough” solution is fine, like quick simulations or rapid prototyping.
Need Precision? Hybrid_SA_Adam and Adam_lr0.01 are your go-to for tasks like training neural networks or optimizing designs, where finding the absolute best solution is critical.
Think Hybrid: The Hybrid_SA_Adam approach shows that blending exploration (searching broadly) with exploitation (zeroing in) can strike a great balance.

These insights are a playbook for picking the right algorithm based on your project’s goals.
Why It’s Cool
This study isn’t just academic—it’s a practical guide for developers:

Algorithm Choice Matters: The Himmelblau function’s multiple valleys mirror real-world challenges, like tuning a neural network or optimizing resource allocation.
Experimentation Pays Off: Running 360 tests per algorithm gave clear insights into reliability, a strategy you can use to test your own code.
Visuals Are Your Friend: Boxplots, convergence plots, and heatmaps (like those in [1]) make it easier to debug and compare algorithms, so lean on them in your work.

Try It Out
Ready to play with optimization? Below is a Python snippet using TensorFlow to minimize a simple quadratic function with Adam. It’s not the Himmelblau function, but it’s a great starting point. To test Himmelblau, swap the loss_function with

(x[0]**2 + x[1] - 11)**2 + (x[0] + x[1]**2 - 7)**2

.

import tensorflow as tf
import numpy as np
import matplotlib.pyplot as plt

# Simple loss function (replace with Himmelblau for the real deal)
def loss_function(x):
    return tf.reduce_sum(x**2)  # Minimizes x^2 + y^2

# Initialize variables
x = tf.Variable([1.0, 1.0], dtype=tf.float32)
optimizer = tf.keras.optimizers.Adam(learning_rate=0.01)
steps = 200
losses = []

# Optimization loop
for step in range(steps):
    with tf.GradientTape() as tape:
        loss = loss_function(x)
    gradients = tape.gradient(loss, [x])
    optimizer.apply_gradients(zip(gradients, [x]))
    losses.append(loss.numpy())

# Plot convergence
plt.plot(losses)
plt.xlabel('Step')
plt.ylabel('Loss')
plt.title('Adam Optimization')
plt.savefig('adam_convergence.png')
plt.show()

Try tweaking the learning rate or swapping in the Himmelblau function to see how Adam behaves. Compare it with Simulated Annealing by implementing a basic version (hint: add random perturbations and a cooling schedule).
Wrapping Up there is still more iteration that needs to be done stay tuned

The Himmelblau study shows that no single optimization algorithm is “the best”—it depends on your needs. SA_Noise and SA_T10 are speedy but less precise, while Hybrid_SA_Adam and Adam_lr0.01 take their time to nail the target. For developers, this is a reminder to test, visualize, and choose algorithms wisely. Whether you’re training a model or solving an engineering problem, the right optimizer can make all the difference.
Want to dive deeper? Check out the full study [1] for detailed metrics and visuals, or experiment with the code above to find your own optimization sweet spot.

References:[1] Allan, "A Comparative Analysis of Optimization Algorithms: The Himmelblau Function Case Study", April 22, 2025.
The Full paper: https://www.kaggle.com/code/allanwandia/non-convex-research/url
The notebook: https://www.kaggle.com/datasets/allanwandia/himmelblau