A problem that once took hours can now take minutes. Another that resisted every available method can become routine. Nothing about the mathematical problem has changed: the same variables, the same constraints, the same definition of a good answer. What has changed is the machinery of search.
We tend to recognize progress in computing by looking at its hardware. Chips become faster, memory becomes more capable, and entire new architectures emerge. Yet some of the most consequential advances happen inside the software, in the mathematical ideas that decide which possibilities to explore and which can safely be ignored.
Optimization offers a remarkable way to measure the difference.
A twenty-year experiment
In 2022, Thorsten Koch, Timo Berthold, Jaap Pedersen and Charlie Vanaret published a study comparing mathematical programming solvers from 2001 with their counterparts from 2020. Crucially, they separated advances in computer hardware from advances in solver algorithms.
The results are striking. For linear programming, their measurements suggested that hardware had become roughly 20 times faster and algorithms about nine times faster, combining to approximately 180 times the solving power. For mixed-integer linear programming, which includes indivisible decisions such as whether to assign a job, open a facility or use a vehicle, hardware again contributed about 20 times, while solver software contributed approximately 50 times. Together, that amounted to around 1,000 times the performance on the problems used for the comparison.
Study-wide estimates on the benchmark problems used for the comparison; individual models vary widely. Combined figures are approximate.
These are study-wide estimates, not a promise about any particular model. Some problems benefited far more than others. Even more revealing were the instances that older software could not solve within a reasonable time but modern methods could finish in seconds. A single average speedup cannot fully describe that change.
What the experiment establishes is more interesting than a contest between hardware and software. They reinforce each other. Faster machines make more ambitious search practical; more intelligent algorithms extract more value from every second of computation.
How a solver learns to do less
Optimization is often pictured as searching through an enormous collection of possible decisions. The obvious approach is to try more possibilities, faster. The more profound approach is to discover why most of them need never be tried.
A modern mixed-integer solver can simplify a model before search begins, derive mathematical bounds that eliminate unpromising regions, find feasible answers early, and revisit promising parts of the search with different strategies. Branching rules decide which decisions to examine first. Cutting planes strengthen the mathematical description of what is possible. Primal heuristics look for good solutions while exact methods work toward proof.
Each component has evolved. More importantly, their interactions have improved. A good feasible answer can help a mathematical search discard entire branches. A stronger bound can redirect a heuristic toward opportunities that matter. A useful reformulation can make a difficult model easier for several algorithms at once.
The most powerful improvement is sometimes a better way to use the computation we already have.
These gains are not merely the result of adding code. They come from deeper mathematical understanding, painstaking experimentation and the willingness to reconsider assumptions about how an algorithm should search.
Progress does not arrive in one leap
The history of commercial solvers illustrates how advances accumulate. Robert Bixby's account of computational linear programming traced improvements in the implementation of familiar mathematical methods, showing that seemingly mature algorithms still had substantial room to advance.
New ideas continue to enter established products. In 2025, the integration of Feasibility Jump, a method developed in research at SINTEF, into a commercial solver was announced. The technique focuses on finding feasible solutions to demanding mixed-integer models. It is a useful example of the broader pattern: progress can emerge from one improved part of a sophisticated solver, then become valuable as part of a larger system.
The development of another commercial solver tells a related story. Its original design was built to exploit multicore processors, but better parallelism was accompanied by improvements to heuristics, interfaces and other parts of the solving process. Over successive releases, the software kept changing as well as the machines running it.
Meanwhile, hardware changes the kinds of algorithms worth exploring. Parallel processors enable different search strategies to run together. GPUs can accelerate certain numerical kernels, though their benefit depends on a problem's structure. Memory performance and data movement may matter as much as arithmetic speed.
The future will not be built by hardware improvements alone or by a single miraculous algorithm. It will emerge from better combinations of mathematics, engineering and computation.
What faster search makes possible
A thousandfold speedup is easy to admire and surprisingly hard to translate into business value without knowing the problem.
For one organization, it might mean a weekly production plan can be improved before the shift begins. For another, it could make scenario analysis practical: several alternatives can be compared while a decision is still open. Sometimes the gain is not speed at all, but the ability to consider a larger, more faithful model with additional operational requirements.
The mathematical achievement becomes economically meaningful when it expands the quality, scale or timeliness of the decisions a team can make.
This is also why public benchmarks and customer evaluations serve different purposes. Benchmarks allow researchers to see whether search methods are genuinely advancing. Real operational instances reveal how those advances behave with the organization's data, constraints, interfaces and decision deadlines.
At 4colors, the same principle motivates our work on optimization algorithms: the search for better methods is valuable because it changes the solutions that software can deliver.
There is a temptation to describe optimization as a mature technology, its important discoveries already made. The history suggests otherwise. Problems familiar for decades continue to yield to better ideas. As computers become more capable, the algorithms deciding how to use them remain one of the most powerful sources of progress.
Further reading
- Koch, Berthold, Pedersen and Vanaret, Progress in mathematical programming solvers from 2001 to 2020 (2022)
- Robert E. Bixby, Solving Real-World Linear Programs: A Decade and More of Progress (2002)