What are the best strategies for debugging complex mathematical models?

In context of debugging LP or MIP model if we want to validate the complex math model we must first be ensured whether the math formulation created is valid. We can do it by validating our model with a small data set and validating the set of constraints. In the second case if we are already ensured after rigorous testing that math model is a valid model and the math model is returning either infeasible results or unbounded solution or no solution then we should have to validate the data. Tracing the exact data set is difficult hence we have to carefully debug the model by playing with the set of hard and soft constraints relaxation and find the root constraints. Some solvers have the capability to identify IID constraints as well.

Data engineering skills are a huge plus. Incorporate methods for investigating data against standards and thresholds. this way, you can ensure your larger models are being developed using underlying data that meets expected standards.

One way is by using the data(even the complex or higher levels of datasets) from past for which the confidence level of its behaviour with the model is high shall be a good way to understand/debugg complex mathematical model and its algorithms. Or even if there is a traceable amout of ambiguity exists, using the simplest form but of highly relevant/perfectly fitting datatsets have higher chances of debugging the mathematical models.

Validated data is also a great output for your projects. Let the stakeholders know where they're starting from. Preferably with a nice dashboard.

Debugging complex mathematical models requires a blend of structured techniques and imaginative problem-solving such as: 1. Starting with known outcomes in simple scenarios provides a foundation for validation. 2. Breaking down the model into simpler and manageable components aids in isolating errors. 3. Scrutinizing underlying assumptions ensures alignment with the problem at hand. 3. Double-checking input data and outputs verifies correctness. Integrating assertions within the code catches errors early. 4. Visualizing data aids in anomaly detection. Comparing results with established models validates accuracy. 5. Logging and debugging tools offer systematic analysis and 6. Peer review provides valuable insights.

Create a problem instance that you can solve without having to use your model. This could be a current state that you already know the answer to, a problem with an obvious solution, or a problem of a very small size. Then solve the problem using your model. If you model finds a worse solution, then you know something is wrong. NOTE: In this case, I am assuming the model is something like a linear integer program and you are using an exact method to solve it. If you are using a heuristic, it's possible that there isn't an issue and it just happens to miss out on the optimal solution on your particular use case. Keep in mind that heuristics don't have a guarantee in finding the optimal solution.

1. Build/develop the model through step-by-step approach 2. Reiterate the model in each step with different datasets/scenarios 3. Understand the difference in model behaviour in terms of output/solutions, recommendations, errors 4. Establish standard check points for each steps of model building (which reduces huge efforts in referring back to the model that gets complex during course of time) 5. Keep the logic of the model briefly explained as documented reference (helps when the model is refferred after a period of time)

1. Begin by thoroughly reviewing the assumptions underlying your model, ensuring they accurately reflect the problem domain and constraints. 2. Break down the model into smaller, more manageable components. 3. Utilize print or logging statements strategically throughout your code to track the flow of data and identify potential points of failure. 4. Verify the consistency and accuracy of inputs and outputs to ensure proper data processing within the model. 5. Leverage debugging tools provided by your programming language or IDE to step through the code and monitor variable values during execution. 6. Collaborate with peers or subject matter experts to review the codebase and logic, benefiting from their perspectives and experiences.

One thing I have found helpful is 1. Start with simple cases can validate the basic logic of the model. 2. Modularize to isolate specific areas of concern. 3. Check assumptions and constraints to ensure the model's validity, while leveraging debugging tools like step-by-step debugging and print statements aids in tracking data flow and pinpointing errors. 4. Compare results against expected outcomes and conduct sensitivity analysis to identify discrepancies and sensitive areas. Peer review provides valuable perspectives, and thorough documentation aids in tracking changes. Regression testing ensures fixes do not introduce new issues while adopting an iterative approach allows for incremental improvements.

A systematic approach to model development and validation is essential to ensure that errors are found effectively and efficiently. It can be tempting to quickly jump to more complex models and formulations that are more difficult to interpret and do not allow for easy output validation. One way to address this is to build core model components and validate these as you develop the more complex model. This will provide a level of assurance that the outputs produced are indeed accurate. When model formulations are extensive, such as with Defender-Attacker Optimisations, imprecise formulations may substantially alter optimal solutions. A coherent approach minimises this risk potential.

Although classical debugging tools (breakpoints, watches) of programming IDEs are quite helpful, for LP and MIP modeling there are several other useful techniques to debug complex models: - Make sure that number of constraints and variables are within the expected ranges. Get an estimate on the numbers by looking at the data and model. Then compare this estimation with the numbers reported in the logs. - Run your model by changing the data slightly to observe if it behaves as expected. - Print the LP file and quickly browse through the objective function and a few constraints. Check out if they are constructed as expected. - In the case of infeasibility, run IIS (Irreducible Infeasibility Sets) analysis to understand the root causes.

In debugging complex mathematical models, consider incorporating techniques from software engineering such as version control and code proofing. Version control systems like Git enable tracking changes in the model code, facilitating collaboration and reverting to previous versions if issues arise. Code proofing involves systematically reviewing the model code for syntax errors, logical inconsistencies, and potential vulnerabilities. By integrating these practices into the model development process, you can enhance code reliability, maintainability, and traceability, ultimately expediting the debugging process and ensuring robust model performance.

I used this method for creating a linear programing solver & then 3D rebar detailing s/w. This method served me and my team very well. Assuming that the math model is an algorithmic one, going through various stages, its extremely useful to create a spreadsheet which has : a. An initial data set ( Low, mid , high & edge cases) b. Result set after each data processing method transforming the data set using various mathematical expressions sequentially. Its far more easy to add these matrix transformations in Excel & print what outputs are expected. Use the spreadsheet to inspect the outputs of the program. If the value of program output doesnt match the spreadsheet, 1 of the 2 is wrong & you can figure found what needs to be fixed.

It is quite beneficial to have smaller sized problem/data instances for which the solutions are already known. If such a set up is available, then the model can be run against the test dataset to make sure that the same optimal solutions are obtained. Failure to do so would indicate errors either in the model formulation or in the associated code. In order to achieve maximum coverage, the test data set generated should account for all the constraints in the model.

There are several approaches to testing an optimization model. Create a simple enough / small dataset that you can analyze and/or visualize. You can iteratively add constraints and variables to your model and analyze the output of the model on this dataset - to ensure that the output is reasonable. If you have a access to a complete feasible solution, you can seed that in your model and ensure that your model is feasible and the objective function value is reasonable.

An important step before model implementation to reduce the number of iterations/trial & error is seeking feedback from all relevant stakeholders. This will enhance the first-time-right implementation of the mathematical model to real time scenarios. Depending upon the significance/impact of the model, it is important to identify the audience for feedback - direct & indirect.

Reviewing model and approach with peers who are not involved in the process always provides a fresh perspective. It also helps refine you own thought process and unlock creative ideas for debugging the model.

From my years of building and being frustrated with debugging mathematical models, I recommend starting small and testing often. This approach allows you to isolate issues early on, making them much easier to resolve. Additionally, always check your model against simple cases first. It's a great way to ensure the foundational parts are correct before you move on to more complex scenarios.

Understand the mathematical structure of the model. Further, break it into parts that are simple to code. Code these parts separately and integrate.

According to my experience, testing model with data having the controllable size and observing model's behavior is one efficient way to detect and fix the problem. Another possible action is to observe the change of the outcome when removing or adding constraints. How does the model react to your “experiment”? Does it make sense?

understand model logic, divide and conquer strategy, check inputs and representation tools, logging and monitoring including unit testing, check constraints, bounderies and environment, sensitivity analysis, code proofing, comparing against analytical solutions, version control...

It is also really important for the modeller to understand the business requirements. In my experience, for cost minimisation problems sometimes cost does not reflect qualitative aspects of the problem such as customer prioritization, when to backlog. Modeller should ensure understanding requirements and tweak real parameter values by defining shadow parameters. At the end of the day, all an optimiser doing is comparing numerical value of the objective function until there is no more improvement. That is something to capitalize on so use it to your advantage.