Perturbation function

Key takeaways

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

Reference excerpt

In mathematical optimization, the perturbation function is any function which relates to primal and dual problems. The name comes from the fact that any such function defines a perturbation of the initial problem. In many cases this takes the form of shifting the constraints. In some texts the value function is called the perturbation function, and the perturbation function is called the bifunction.

Definition Given two dual pairs of separated locally convex spaces ( X , X ∗ ) {\displaystyle \left(X,X^{*}\right)} and ( Y , Y ∗ ) {\displaystyle \left(Y,Y^{*}\right)} . Then given the function f : X → R ∪ { + ∞ } {\displaystyle f:X\to \mathbb {R} \cup \{+\infty \}} , we can define the primal problem by

inf x ∈ X f ( x ) . {\displaystyle \inf _{x\in X}f(x).\,}

If there are constraint conditions, these can be built into the function f {\displaystyle f} by letting f ← f + I c o n s t r a i n t s {\displaystyle f\leftarrow f+I_{\mathrm {constraints} }} where I {\displaystyle I} is the characteristic function. Then F : X × Y → R ∪ { + ∞ } {\displaystyle F:X\times Y\to \mathbb {R} \cup \{+\infty \}} is a perturbation function if and only if F ( x , 0 ) = f ( x ) {\displaystyle F(x,0)=f(x)} .

Use in duality The duality gap is the difference of the right and left hand side of the inequality

sup y ∗ ∈ Y ∗ − F ∗ ( 0 , y ∗ ) ≤ inf x ∈ X F ( x , 0 ) , {\displaystyle \sup _{y^{*}\in Y^{*}}-F^{*}(0,y^{*})\leq \inf _{x\in X}F(x,0),}

where F ∗ {\displaystyle F^{*}} is the convex conjugate in both variables. For any choice of perturbation function F weak duality holds. There are a number of conditions which if satisfied imply strong duality. For instance, if F is proper, jointly convex, lower semi-continuous with 0 ∈ core ⁡ ( Pr Y ( dom ⁡ F ) ) {\displaystyle 0\in \operatorname {core} ({\Pr }_{Y}(\operatorname {dom} F))} (where core {\displaystyle \operatorname {core} } is the algebraic interior and Pr Y {\displaystyle {\Pr }_{Y}} is the projection onto Y defined by Pr Y ( x , y ) = y {\displaystyle {\Pr }_{Y}(x,y)=y} ) and X, Y are Fréchet spaces then strong duality holds.

Examples

Lagrangian Let ( X , X ∗ ) {\displaystyle (X,X^{*})} and ( Y , Y ∗ ) {\displaystyle (Y,Y^{*})} be dual pairs. Given a primal problem (minimize f(x)) and a related perturbation function (F(x,y)) then the Lagrangian L : X × Y ∗ → R ∪ { + ∞ } {\displaystyle L:X\times Y^{*}\to \mathbb {R} \cup \{+\infty \}} is the negative conjugate of F with respect to y (i.e. the concave conjugate). That is the Lagrangian is defined by

L ( x , y ∗ ) = inf y ∈ Y { F ( x , y ) − y ∗ ( y ) } . {\displaystyle L(x,y^{*})=\inf _{y\in Y}\left\{F(x,y)-y^{*}(y)\right\}.}

In particular the weak duality minmax equation can be shown to be

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Perturbation function

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

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

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

Frequently asked questions

What is Perturbation function in simple terms?

In mathematical optimization, the perturbation function is any function which relates to primal and dual problems. The name comes from the fact that any such function defines a perturbation of the initial problem.

Why does Perturbation 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 Perturbation 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 Perturbation function.