Six pictures of the same idea: a problem is a shape, and solving it is finding a point on that shape. Every number you see is computed in your browser, and checked against a second method.
The Feasible Region course draws most of its shapes flat. Some of them are not flat: a three-variable linear program is a solid with corners, an integer program is a cloud of points inside it, Max-Cut’s relaxation lives on a sphere. This page lets you turn them over. Nothing here is a claim about quantum speed-ups. It is the geometry that the quantum optimisation papers start from.
The 3D view loads only when you press Start 3D. On a weak phone it draws with a lighter setting, and the tables under each scene say the same thing in numbers if you would rather not load it.
Drag to turn. Scroll, pinch or the + and − keys to zoom. Arrow keys turn it when the picture has focus.
Maximise 5x + 4y + 3z subject to three limits and x, y, z ≥ 0. The allowed points form a solid with flat faces. The best point is always a corner, so the simplex method starts at the origin and slides from corner to corner, each step along an edge and each step improving the objective, until no neighbour is better.
| Corner | x | y | z | 5x+4y+3z |
|---|
The amber path is the simplex method’s own pivots (the textbook tableau, Dantzig’s rule). The check script confirms the objective never falls along it, every step lands on a corner, and the end is the same corner an exhaustive search finds.
Maximise 5x + 8y with x + y ≤ 6 and 5x + 9y ≤ 45, x, y whole numbers. Relax the whole-number rule and the best point is a fractional corner worth 41.25. The real answer, found by trying every whole point, is 40. The smallest shape that contains exactly the whole points is the integer hull (teal), and finding it is the hard part. A cutting plane (red) is a cheap approximation: an extra limit that removes the fractional corner and no whole point.
The cut is a Chvátal–Gomory cut: take a non-negative mix of the limits, round the left-hand coefficients down, then round the right-hand side down. The check script confirms it removes the fractional optimum, removes no whole point, and that the bound strictly falls after adding it. It is flat here because two variables are all a page can show honestly.
Max-Cut asks for a split of a graph’s nodes into two sides that cuts as many edges as possible. Replace each node’s yes/no choice with an arrow on a sphere, and push arrows at the ends of an edge to point away from each other. That relaxed problem is a semidefinite program. Then round it: throw a random plane through the centre and put the nodes on one side or the other. The graph here is a five-sided ring with a hub joined to every corner: ten edges, and the best split cuts 7. The relaxation says 7.368, and a random plane cuts 5, 6 or 7 depending on the throw, averaging 6.616, which is 0.898 of the relaxation. The proven worst-case guarantee is 0.878 (Goemans–Williamson, 1995).
The arrows here come from a small projected-gradient solver, not a library, and the check script compares it with the closed form 5(1 − cos(4π/5))/2 on the plain 5-ring and confirms five random starts reach the same value on the wheel. The teal disc is the random plane; press Throw another plane to see the cut change. The 0.878 guarantee is a worst case over all graphs. It does not say anything about a quantum computer.
Heights are costs. Four wells, one of them deeper than the rest. A walker that only ever steps downhill gets stuck in whichever well it started near. Simulated annealing lets it climb out early (when it is hot) and settles it late (when it is cold). The red dot replays one run that starts in a shallow well; the teal dot marks the true minimum.
The check script runs sixty cooling schedules against sixty cold starts from the same shallow well and confirms slow cooling finds the deep well far more often. One run is an illustration, not a proof, and the page does not claim a quantum annealer does better on landscapes like this.
Each point is a design, scored on three things to minimise. A design is dominated if another is at least as good on every score and better on one. The designs nothing dominates form the Pareto surface (teal and red). A weighted sum of the three scores picks only some of them. The red ones sit in a dent in the surface: no choice of weights ever selects them, though they are perfectly good trade-offs.
The counts are checked against a brute-force pairwise comparison of every design. The designs are generated by a formula so the example is reproducible, and they are not data from any real product.
A network of pipes with capacities. The most you can push from the left node to the right is the max flow. Max-flow equals the capacity of the cheapest set of pipes whose removal separates the two ends, the min cut (red). The amber blob is the source side. Pushing more through any pipe that is not in the cut changes nothing, which is why the cut is the answer to “what is the bottleneck”.
The check script finds the same number by trying all 64 ways of splitting the six middle nodes.
Every figure above comes from region3d.js, and tools/verify_region3d.mjs recomputes each scene by a different method: a grid search for the linear program, exhaustive enumeration for the integer program, a closed form for the sphere, a dense scan and many seeded runs for the landscape, pairwise comparison for the Pareto surface, and enumeration of every cut for the network. The picture is drawing, not evidence; the numbers are the evidence.
Low-end phones: the page draws no shadows, caps the pixel ratio, drops antialiasing on weak devices, pauses when scrolled out of view or the tab is hidden, and does not animate on its own if your device asks for reduced motion.