ML+MO
Answered
1. I understand how ML can feed into MO. The ML output (predictions) can be used as a constraint in MO. Example, from historical data, using ML, predict the likely price. Use this likely price as a constraint in a prescriptive model.
2. I want to understand an example of how MO can be an input to ML. ie , MO finds the optimum output for a set of inputs. Then these various input -outputs can be used in ML to find the function which maps these these input-output. Can you give an example of this ?
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Hi Jai,
To illustrate how Mathematical Optimization (MO) can serve as an input to Machine Learning (ML), consider the following example:
Example: Supply Chain Optimization
Step 1: Mathematical Optimization
Imagine a company that needs to optimize its supply chain. The goal is to minimize the total cost of production and transportation while satisfying customer demand. The company has several factories, warehouses, and retail locations, each with different production costs, capacities, and transportation costs.
Inputs for the MO problem:
- Production costs at each factory
- Transportation costs between factories, warehouses, and retail locations
- Capacity limits at factories and warehouses
- Customer demand at each retail location
MO Task: Using these inputs, the MO model determines the optimal production and distribution plan that minimizes the total cost while meeting customer demands. The output of this optimization is the best production levels at each factory and the optimal transportation routes and quantities between locations.
Outputs of MO:
- Optimal production levels at each factory
- Optimal transportation quantities between locations
- Minimum cost associated with the optimal plan
Step 2: Machine Learning
Now, let’s use the results from the optimization model as training data for a Machine Learning model.
Objective of ML: The goal of the ML model is to learn a function that maps the inputs (production costs, transportation costs, capacities, and demands) to the outputs (optimal production levels and transportation quantities). This function can then be used to predict the optimal decisions without having to solve the optimization problem repeatedly, which can be computationally expensive.
ML Training Data:
- Inputs: The original inputs to the MO problem (costs, capacities, demands, etc.)
- Outputs (Labels): The optimal decisions derived from the MO model (production levels, transportation quantities).
ML Task: Using the dataset generated from multiple instances of the MO problem (where inputs vary), train a regression model (like a neural network or a decision tree) to predict the optimal production levels and transportation quantities directly from the input data.
Application: Once the ML model is trained, it can predict the optimal production and transportation decisions for new sets of input parameters quickly, bypassing the need to solve the MO problem from scratch every time. This approach is particularly useful when real-time decision-making is required, or when the optimization problem is too complex to solve in a timely manner.
Benefits:
- Speed: The trained ML model can provide near-instantaneous predictions of optimal decisions.
- Scalability: The ML model can generalize to new, unseen situations, provided they are similar to the training data.
This example shows how optimization outputs can be used to train machine learning models, creating a powerful hybrid approach that leverages the strengths of both MO and ML.
- Bot
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Thanks,
1. From the above answer, it seems that instead of using constraints to form the polytope and intelligently selecting the optimal vertex, it might be quicker to use historic data of constraints as features and optimized results as labels and do supervised learning. Once trained model is available , do prediction of any set of constraints ! In that case why Optimize ?
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Hi Jai,
Why Use Optimization?
Mathematical Optimization is fundamentally about finding the best possible solution to a problem within a defined set of constraints and objectives. Here are some key reasons to still use optimization:
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Exact Solutions: Optimization algorithms are designed to find the exact optimal solution, not just an approximation. For problems where precision is critical, optimization ensures that the solution is mathematically guaranteed to be the best under the given constraints.
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Generalizability to New Situations: Optimization models can be applied to any new set of constraints and objectives, even if the situation is significantly different from past data. This is because the optimization process inherently understands the structure of the problem, not just the patterns in historical data.
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Adaptability to Changes: If the problem constraints or the objective function changes (e.g., a sudden change in resource availability or a shift in market demand), optimization models can accommodate these changes on the fly, recalculating the best solution under the new conditions.
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Handling Unseen Scenarios: Optimization can handle new scenarios that were not present in the historical data. Since ML models learn from historical data, they may struggle with predictions when faced with entirely new or rare constraints.
Why Use Machine Learning?
Machine Learning, particularly supervised learning as you described, can be beneficial in several scenarios:
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Speed: Once trained, an ML model can predict the optimal solution much faster than solving an optimization problem from scratch. This is particularly useful in real-time applications where decisions need to be made quickly.
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Approximation in Complex Scenarios: For very complex optimization problems, solving the optimization might be computationally expensive or slow. An ML model can provide a near-optimal solution much faster, which might be "good enough" in many practical applications.
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Scalability: ML models can easily scale to handle large amounts of data, making them useful when the optimization problem involves many variables and constraints that could slow down traditional optimization methods.
Why Not Just Use ML?
However, relying solely on ML has some potential drawbacks:
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Accuracy: ML models provide predictions based on patterns in historical data. If the situation changes, or if the model encounters scenarios that are significantly different from the training data, the predictions might not be accurate.
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Lack of Guarantees: ML models do not guarantee optimality. They approximate the optimal solution, but there’s no mathematical proof that the prediction is the best possible solution under the current constraints.
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Dependence on Quality Data: The effectiveness of the ML model depends heavily on the quality and quantity of the training data. If the historical data is sparse, noisy, or not representative of future scenarios, the model’s predictions will be poor.
Hybrid Approach: The Best of Both Worlds
In practice, a hybrid approach that combines both MO and ML might be the most powerful:
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Optimization-Informed ML Models: Use MO to generate high-quality training data for ML models. This approach ensures that the ML model learns from optimal solutions and can approximate them quickly when needed.
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ML-Informed Optimization: Use ML models to provide good initial solutions or insights (like predictions for certain variables) that can then be refined by an optimization model, reducing the overall computation time while maintaining a high level of accuracy.
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Switching Strategies: In scenarios where speed is critical and you can tolerate some approximation, use the ML model. For scenarios where precision is paramount, fall back on MO.
Conclusion
Optimization and Machine Learning serve different purposes and excel in different scenarios. Optimization is essential when exact solutions are needed, and when dealing with new or unseen scenarios. Machine Learning is advantageous for speed and scalability, especially when the problem is well-understood and historical data is plentiful. By intelligently combining both, you can achieve robust, efficient, and scalable solutions that leverage the strengths of each approach.
- Bot
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Hi Jai,
In addition to the previous explanation, I recommend reviewing the following examples from the Gurobi Machine Learning documentation:
https://gurobi-machinelearning.readthedocs.io/en/stable/auto_examples/index.html
One of the key concepts of integrating Machine Learning (ML) with Mathematical Optimization (MO) is the ability to optimize specific input or output variables of a trained ML model. For instance, suppose you have an ML model that predicts product demand (an output feature) based on historical data (input features), with product price being one of those inputs. The question then arises: What is the optimal price to be decided for this product next year?In this scenario, you can construct a mathematical optimization model incorporating your trained ML model as an ML constraint. Your trained ML model helps define the relationship between input and output features, while both product price and demand are treated as decision variables within the optimization model. You can refer to the Price Optimization example for a related explanation.
Regards
-Ahmed
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Thanks
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