Supermodular function

Key takeaways

Supermodular function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Supermodular function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Supermodular function from memory before moving on to harder problems.

Reference excerpt

In mathematics, a supermodular function is a function on a lattice that, informally, has the property of being characterized by "increasing differences." Seen from the point of set functions, this can also be viewed as a relationship of "increasing returns", where adding more elements to a subset increases its valuation. In economics, supermodular functions are often used as a formal expression of complementarity in preferences among goods. Supermodular functions are studied and have applications in game theory, economics, lattice theory, combinatorial optimization, and machine learning.

Definition Let ( X , ⪯ ) {\displaystyle (X,\preceq )} be a lattice. A real-valued function f : X → R {\displaystyle f:X\rightarrow \mathbb {R} } is called supermodular if

f ( x ∨ y ) + f ( x ∧ y ) ≥ f ( x ) + f ( y ) {\displaystyle f(x\vee y)+f(x\wedge y)\geq f(x)+f(y)}

for all x , y ∈ X {\displaystyle x,y\in X} . If the inequality is strict, then f {\displaystyle f} is strictly supermodular on X {\displaystyle X} . If − f {\displaystyle -f} is (strictly) supermodular then f is called (strictly) submodular. A function that is both submodular and supermodular is called modular. This corresponds to the inequality being changed to an equality. We can also define supermodular functions where the underlying lattice is the vector space R n {\displaystyle \mathbb {R} ^{n}} . Then the function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is supermodular if

f ( x ↑ y ) + f ( x ↓ y ) ≥ f ( x ) + f ( y ) {\displaystyle f(x\uparrow y)+f(x\downarrow y)\geq f(x)+f(y)}

for all x {\displaystyle x} , y ∈ R n {\displaystyle y\in \mathbb {R} ^{n}} , where x ↑ y {\displaystyle x\uparrow y} denotes the componentwise maximum and x ↓ y {\displaystyle x\downarrow y} the componentwise minimum of x {\displaystyle x} and y {\displaystyle y} . If f is twice continuously differentiable, then supermodularity is equivalent to the condition

∂ 2 f ∂ z i ∂ z j ≥ 0 for all i ≠ j . {\displaystyle {\frac {\partial ^{2}f}{\partial z_{i}\,\partial z_{j}}}\geq 0{\mbox{ for all }}i\neq j.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Supermodular function

Start with the simplest possible case. Write down what Supermodular function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Supermodular function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Supermodular function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Supermodular function

In research: Supermodular function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Supermodular function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Supermodular function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Generalized convexity, Optimization of ordered sets, Order theory, so understanding it makes those chapters shorter.
In everyday life: Look for Supermodular function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Supermodular function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Supermodular function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Supermodular function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Supermodular function in simple terms?

In mathematics, a supermodular function is a function on a lattice that, informally, has the property of being characterized by "increasing differences." Seen from the point of set functions, this can also be viewed as a relationship of "increasing returns", where adding more elements to a subset i…

Why does Supermodular function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Supermodular function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Supermodular function.