Pseudoconvex function

Key takeaways

Pseudoconvex 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 Pseudoconvex function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pseudoconvex function from memory before moving on to harder problems.

Reference excerpt

In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has a positive directional derivative. The property must hold in all of the function domain, and not only for nearby points.

Formal definition Consider a differentiable function f : X ⊆ R n → R {\displaystyle f:X\subseteq \mathbb {R} ^{n}\rightarrow \mathbb {R} } , defined on a (nonempty) convex open set X {\displaystyle X} of the finite-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . This function is said to be pseudoconvex if the following property holds:

for all x , y ∈ X : ∇ f ( x ) ⋅ ( y − x ) ≥ 0 ⇒ f ( y ) ≥ f ( x ) . {\displaystyle x,y\in X:\quad \nabla f(x)\cdot (y-x)\geq 0\Rightarrow f(y)\geq f(x).}

Equivalently:

for all x , y ∈ X : f ( y ) < f ( x ) ⇒ ∇ f ( x ) ⋅ ( y − x ) < 0. {\displaystyle x,y\in X:\quad f(y)<f(x)\Rightarrow \nabla f(x)\cdot (y-x)<0.}

Here ∇ f {\displaystyle \nabla f} is the gradient of f {\displaystyle f} , defined by: ∇ f = ( ∂ f ∂ x 1 , … , ∂ f ∂ x n ) . {\displaystyle \nabla f=\left({\frac {\partial f}{\partial x_{1}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right).}

Note that the definition may also be stated in terms of the directional derivative of f {\displaystyle f} , in the direction given by the vector v = y − x {\displaystyle v=y-x} . This is because, as f {\displaystyle f} is differentiable, this directional derivative is given by:

∂ f ∂ v ( x ) = ∇ f ( x ) ⋅ v = ∇ f ( x ) ⋅ ( y − x ) . {\displaystyle {\frac {\partial f}{\partial v}}(x)=\nabla f(x)\cdot v=\nabla f(x)\cdot (y-x).}

Properties

Relation to other types of "convexity" Every convex function is pseudoconvex, but the converse is not true. For example, the function f ( x ) = x + x 3 {\displaystyle f(x)=x+x^{3}} is pseudoconvex but not convex. Similarly, any pseudoconvex function is quasiconvex; but the converse is not true, since the function f ( x ) = x 3 {\displaystyle f(x)=x^{3}} is quasiconvex but not pseudoconvex. This can be summarized schematically as:

convex ⇒ {\displaystyle \Rightarrow } pseudoconvex ⇒ {\displaystyle \Rightarrow } quasiconvex

To see that f ( x ) = x 3 {\displaystyle f(x)=x^{3}} is not pseudoconvex, consider its derivative at x = 0 {\displaystyle x=0} : f ′ ( 0 ) = 0 {\displaystyle f^{\prime }(0)=0} . Then, if f ( x ) = x 3 {\displaystyle f(x)=x^{3}} was pseudoconvex, we should have:

f ′ ( 0 ) ( y − 0 ) = 0 ≥ 0 ⇒ f ( y ) ≥ f ( 0 ) , ∀ y ∈ R . {\displaystyle f^{\prime }(0)(y-0)=0\geq 0\Rightarrow f(y)\geq f(0),\quad \forall \,y\in \mathbb {R} .}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Pseudoconvex function

Start with the simplest possible case. Write down what Pseudoconvex 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 Pseudoconvex 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 Pseudoconvex 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 Pseudoconvex function

In research: Pseudoconvex 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 Pseudoconvex 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: Pseudoconvex function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Convex analysis, Convex optimization, Generalized convexity, so understanding it makes those chapters shorter.
In everyday life: Look for Pseudoconvex 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 Pseudoconvex function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pseudoconvex 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 Pseudoconvex function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pseudoconvex function in simple terms?

In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increa…

Why does Pseudoconvex 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 Pseudoconvex 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 Pseudoconvex function.