Weak duality

Key takeaways

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

Reference excerpt

In applied mathematics, weak duality is a concept in optimization which states that the duality gap is always greater than or equal to 0. This means that for any minimization problem, called the primal problem, the solution to the primal problem is always greater than or equal to the solution to the dual maximization problem. Alternatively, the solution to a primal maximization problem is always less than or equal to the solution to the dual minimization problem. So, in short: weak duality states that any solution feasible for the dual problem is a lower bound to the solution of the primal problem. Weak duality is in contrast to strong duality, which states that the primal optimal objective and the dual optimal objective are equal. Strong duality only holds in certain cases.

Uses Many primal-dual approximation algorithms are based on the principle of weak duality.

Weak duality theorem Consider a linear programming problem,

where A {\displaystyle A} is m × n {\displaystyle m\times n} and b {\displaystyle b} is m × 1 {\displaystyle m\times 1} . The dual problem of (1) is

The weak duality theorem states that c ⊤ x ∗ ≤ b ⊤ y ∗ {\displaystyle c^{\top }x^{*}\leq b^{\top }y^{*}} for every solution x ∗ {\displaystyle x^{*}} to the primal problem (1) and every solution y ∗ {\displaystyle y^{*}} to the dual problem (2). Namely, if ( x 1 , x 2 , . . . . , x n ) {\displaystyle (x_{1},x_{2},....,x_{n})} is a feasible solution for the primal maximization linear program and ( y 1 , y 2 , . . . . , y m ) {\displaystyle (y_{1},y_{2},....,y_{m})} is a feasible solution for the dual minimization linear program, then the weak duality theorem can be stated as

∑ j = 1 n c j x j ≤ ∑ i = 1 m b i y i {\displaystyle \sum _{j=1}^{n}c_{j}x_{j}\leq \sum _{i=1}^{m}b_{i}y_{i}} , where c j {\displaystyle c_{j}} and b i {\displaystyle b_{i}} are the coefficients of the respective objective functions. Proof:

cTx = xTc ≤ xTATy ≤ bTy

Generalizations More generally, if x {\displaystyle x} is a feasible solution for the primal maximization problem and y {\displaystyle y} is a feasible solution for the dual minimization problem, then weak duality implies f ( x ) ≤ g ( y ) {\displaystyle f(x)\leq g(y)} where f {\displaystyle f} and g {\displaystyle g} are the objective functions for the primal and dual problems respectively.

See also Convex optimization Max–min inequality

References

Worked examples

Example 1 — a first encounter with Weak duality

Start with the simplest possible case. Write down what Weak duality claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Weak duality 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 Weak duality 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 Weak duality

In research: Weak duality appears in science 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 Weak duality 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: Weak duality 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 Weak duality 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 Weak duality in 20 minutes

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

Frequently asked questions

What is Weak duality in simple terms?

In applied mathematics, weak duality is a concept in optimization which states that the duality gap is always greater than or equal to 0. This means that for any minimization problem, called the primal problem, the solution to the primal problem is always greater than or equal to the solution to th…

Why does Weak duality matter?

Because it connects several science 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 Weak duality?

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 Weak duality.