Corner solutions
A Socratic walk-through of corner solutions — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why does the best way to allocate limited resources always sit at a corner of the possibilities?
A workshop makes chairs and tables. It has so many hours of labour and so much timber, and each product consumes both. Asked for the most profitable plan, most people expect something balanced — a sensible mixture, adjusted until the margins line up.
The answer is stranger. The best plan sits at a corner of the feasible region: a point where some constraints are exactly used up. Worse, it is stubborn — change the price of tables a little and the plan does not move at all, then at some point jumps to an entirely different corner. Why should optimisation behave like that?
Reasoning it through
REASONING #Draw the possibilities. Let x be chairs and y tables, with labour 2*x* + y ≤ 100, timber x + 2*y* ≤ 80, and the obvious x, y ≥ 0. Each inequality cuts the plane with a straight edge and keeps one side. Four such cuts leave a polygon — the first structural fact: a linear problem's feasible set is always an intersection of half-planes, hence convex, with flat faces and sharp corners. Nothing curved can appear, because nothing curved went in.
Now the objective. Suppose a chair earns 3 and a table 5, so we maximise 3*x* + 5*y*. The set of plans earning exactly 200 is a straight line; earning 220, a parallel line further out. The objective is a family of parallel lines, and optimising means sliding that family outward until it is about to leave the polygon.
Ask what the last point of contact can be. A straight line leaving a convex polygon touches it last either at a single vertex or along a whole edge — and an edge has vertices at both ends. A sliding straight line cannot end up touching only the middle of a face while missing its corners. So an optimum, if one exists, is achieved at a vertex.
That picture has an exact algebraic version needing no drawing. Minkowski's theorem says every point of a bounded polytope is a weighted average of its vertices: z = Σ λᵢ vᵢ, with λᵢ ≥ 0 summing to 1. Apply the linear objective c and linearity passes straight through the average: c·z = Σ λᵢ (c·vᵢ) ≤ maxᵢ (c·vᵢ). Any interior plan is beaten, or at best matched, by the best vertex among those it averages. That is the fundamental theorem of linear programming, in one line.
So we need only check the corners: (0,0) earning 0, (50,0) earning 150, (0,40) earning 200, and where the two constraints cross. Solve 2*x* + y = 100 with x + 2*y* = 80: doubling the first gives 4*x* + 2*y* = 200, and subtracting leaves 3*x* = 120, so x = 40, y = 20. Timber checks at 40 + 40 = 80. Profit is 3(40) + 5(20) = 220 — best of the four, and a plan using every hour and every board.
Now the stubbornness. Raise the table's margin from 5 to 10 and recompute the same corners: (0,40) now earns 400, while (40,20) earns 320 and (50,0) earns 150. The optimum has leapt to a plan making no chairs at all. Between the two prices there was a moment when the objective line ran exactly parallel to a constraint, both corners tied, and the whole edge between them was optimal. Below it, one corner; above it, the other.
The analogy
THE ANALOGY #Think of tilting a flat tray cut into an irregular polygon, with a marble on it. However you tilt it, the marble runs to a corner — never stopping partway along a straight edge, unless that edge lies exactly across the slope, in which case the whole edge is level and it may rest anywhere along it. The tilt is the price vector; the tray is the feasible region.
A marble finds its corner by rolling, whereas an algorithm must search — and in many dimensions the number of corners grows combinatorially, which is why simplex walks edge to edge rather than enumerating, and why the theorem says where the answer lives without saying how to reach it.
Clarifying the model
THE MODEL #The word "always" in the question does too much, and the conditions are where the understanding sits.
A vertex optimum exists provided the feasible region is non-empty, has at least one vertex, and the objective is bounded in the direction you are pushing. Remove the timber constraint and the sliding line never leaves the region: no maximum, no corner. Both failures are ordinary, not pathologies.
More importantly, drop linearity and the result vanishes. The corner is not a fact about scarcity but about proportionality. Suppose each extra chair earns a little less than the last — a concave objective, the ordinary case in economics. The sliding contour is now a curve, and can touch the interior of a face or a point strictly inside the region. What survives is the convex-optimisation statement: the optimum sits where no feasible direction improves things, which at an interior point means the gradient vanishes and at a boundary point means it is balanced by the active constraints. "Marginal benefit equals marginal cost" is only the interior case. At a corner the first-order condition holds as an inequality — you would gain from more and cannot have it — which is what the multiplier on a binding constraint measures. Economists borrow the name for exactly that: taking none of something because you would have chosen a negative amount if allowed.
That gives a falsification test. Perturb the prices slightly. A genuine corner solution does not move — locally insensitive, then discontinuous. An interior optimum drifts smoothly in proportion. If your model's answer slides continuously as inputs change, the objective is not linear there and the corner story is wrong.
One caveat: real problems are frequently degenerate, with more constraints meeting at a corner than the dimension requires, and ties are common. The theorem promises some optimum is a vertex, not that it is unique.
A picture of it
THE PICTURE #How to readStart at the rounded terminal and read downward as a series of tests rather than steps of a calculation — the first three diamonds are the hypotheses the corner theorem needs, and each red box is a way the corner claim fails rather than an error. The middle diamond is the load-bearing one: answer "no, it curves" and you leave the linear world entirely, which is where most economic optimisation lives. Only when all three pass do you reach the store of vertices; the loop back from the last diamond is the simplex method walking corner to corner, ending at the double circle when no neighbour improves.
What became clearer
WHAT CLEARED #Corners are not where scarcity puts you; they are where linearity puts you. Because a linear objective is a family of parallel flats and a linear constraint set is a convex polytope with no curvature anywhere, the last point of contact can always be taken at a vertex — and any interior plan is an average of vertices, so it can never beat the best of them. This feels counter-intuitive because our sense of "balance" comes from diminishing returns, precisely the curvature a linear model omits. Straight lines give all-or-nothing plans that ignore small price changes and then switch abruptly; curves give the smooth interior compromise we expected. Which behaviour you see tells you which model you are really in.
Where to go next
ONWARD #- What the dual problem is, and why a binding constraint's multiplier is the shadow price of one more hour.
- How integer constraints break the geometry, since the best whole-numbered plan need not sit at a vertex.
Key terms
TERMS #| Term | What it means |
|---|---|
| Feasible region | the set of plans satisfying every constraint; for linear constraints, a convex polytope. |
| Vertex (extreme point) | a feasible point that is not the average of two other feasible points. |
| Fundamental theorem of linear programming | if a bounded linear programme has an optimum, some vertex attains it. |
| Corner solution | an optimum on the boundary, where the first-order condition holds as an inequality rather than an equality. |
Every term the collection defines is gathered in the glossary.