Mastering Non-Convex Function Minimization with Linear Constraints Using Mystic

Discover how to effectively minimize non-convex functions with linear constraints and bounds in Python using the Mystic library.
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Mastering Non-Convex Function Minimization with Linear Constraints Using Mystic

Non-convex optimization problems often present a challenge due to their complexity, characterized by numerous local minima that can hinder the search for a global solution. When combined with linear constraints, it can be even more daunting. In this guide, we will explore a practical approach to minimizing a non-convex function in Python using the Mystic library. By providing a clear overview of the problem and how to solve it, we aim to equip you with the skills needed to tackle similar challenges.

Understanding the Problem

Imagine that we have a non-convex objective function called loss. This function takes an np.ndarray named X, which represents our variables of interest, and yields a float value as the output. The function may exhibit multiple local minima due to its dependence on a rounded transformation of X multiplied by a constant vector c. Here’s a simplified representation of how it may look:

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We also need to account for linear constraints defined with two constant matrices, A and b, where the optimization should satisfy the relation A.dot(X) == b. Additionally, each element of X must fall within specified bounds (0 to 2), and we have an initial guess defined as X0 = [1, 1, ..., 1].

To summarize the requirements:

Objective: Minimize loss(X).

Constraints: A.dot(X) == b.

Bounds: 0 <= X_i <= 2.

Termination conditions:

Stop when loss(X) <= 1.

Stop after ~200 evaluations or after a set duration of 5 minutes.

Utilizing Mystic for Solution

To tackle the problem using Mystic, we start by setting up the loss function and defining our constraints. Below is the step-by-step implementation.

Step 1: Setup

We need to import the required libraries and set up our matrices:

[[See Video to Reveal this Text or Code Snippet]]

Step 2: Define Constraints and Bounds

Next, we construct the constraints and bounds for our variables:

[[See Video to Reveal this Text or Code Snippet]]

Step 3: Optimization Using Differential Evolution

We can now apply a differential evolution solver to find a solution that satisfies the conditions:

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The output will show the progression of the optimization process, with each generation’s chi-square values.

Step 4: Local Solver for Faster Convergence

For faster convergence, you can use the fmin_powell method, which is particularly suitable when you have a decent starting point:

[[See Video to Reveal this Text or Code Snippet]]

With this code, you can control the maximum number of function evaluations (maxfun) and the function tolerance (ftol), ensuring that the optimization process meets your criteria.

Conclusion

While minimizing non-convex functions under linear constraints can be complex, the Mystic library offers powerful tools to navigate these challenges. By defining your loss function, constraints, and bounds clearly, and using appropriate solvers, you can achieve satisfactory solutions efficiently. Through careful iterations and condition handling, as demonstrated in this post, you can unlock the potential of your optimization problems.

Now, you are better equipped to handle non-convex optimization in Python—embrace the journey, and may your solutions converge smoothly!

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