Jul 2026

Helpful limitations

Through my work on power-system optimization, I've been thinking more about constraints and started noticing them more often, in various settings. The first association might be: constraints are restrictions, constraints are bad. But interestingly, this isn't always true.

Binding constraints

Let's dive into constraints. Take as an example

Here $x \ge 3$ is the important one — it defines the optimum, and pushes it away from where we'd like it to be. Without it, $x$ could go lower. The constraint $x \ge 2$ is present, but it does nothing. If we can remove $x \ge 3$, the optimum moves to 2 and the solution gets better.

It's interesting to look for such constraints in the real world. If we try to optimize speed for example — getting somewhere faster, finishing something faster — the binding constraint might be the one traffic light on the route, or the one process at work that always takes days. No matter how much we improve everything else, the whole thing still stops there. Sensibly, this is what we should work on (if it lies within our reach).

The traffic light is obvious. But the same is true for things where the feedback is much quieter. Happiness, a partnership, a career — here too there is often one thing that slows everything else down, even if we don't notice it as clearly as a red light. And there too, it's worth actively searching for the binding constraint, and working on that one. (This is known as bottleneck thinking — Goldratt's Theory of Constraints.)

Constraints as information

So far the story is simple: find the constraint that holds you back, and remove it. But that's only half of it. Because a lot of good work seems to happen not with fewer constraints, but with more.

Many photographers shoot with a single fixed focal length. The old iPod was built around a scroll wheel and almost no buttons. Bach composes a whole piece on a tiny motif like B-A-C-H. Some people work best with a deadline breathing down their neck. Spielberg showed little of the shark in Jaws, (since the mechanical shark kept breaking) - but without showing the shark, the film got even scarier.

Why does this work? Why can such strong limitations produce great work? Why isn't the result simply better with full freedom? Zoom lenses, buttons everywhere, sharks everywhere?

In mathematical optimization, there is the idea of the cutting-plane method. Say we're looking for a solution to an optimization problem that has to be integer-valued, $x \in \mathbb{Z}$. That's hard, so first we solve an easier version where fractional $x$ are allowed. The answer might come back as 5.5. So we add a constraint — a cut — chosen to remove that fractional answer, and ideally a whole region of similarly invalid answers, while keeping every valid one. Then we solve again. If the answer still isn't whole, we add another cut. Step by step we carve the space down until what's left is a real solution.

The added constraint isn't just a restriction. It's information: "not there". If we already know the answer is small, we can safely add $x < 10$ — and never have to search near $x \approx 100$ again.

This shines a new light on the examples from before. It's still very possible to photograph a masterpiece at 50mm, and to write a great article in an evening. Spielberg didn't need to show the shark, and the iPod didn't need more buttons to be better. Now, these constraints are not as clean as the cut above. The fixed lens does throw away good photographs — great wide shots exist. But it throws away far more bad options than good ones. Now the good solutions become easier to find.

In art, design, and life we usually move through spaces with infinitely many options. Full freedom sounds like a gift, but really it's paralysis. Every good option is available — but so is every bad one.

So which is it — do constraints hold us back, or push us forward? The binding constraint stands between us and a goal we can already see. The chosen constraint works where we can't see the goal yet.

This makes clear how we can actively use constraints to help us. A restriction is often much easier to find than the optimum. So the question isn't: how much freedom do I need to reach the goal? It's: under which (strong!) restriction is the goal still reachable?

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