Duality in Optimisation
Pair every optimisation problem with a dual.
Key idea
Pair every optimisation problem with a dual.
In depth
Every LP (primal) has a dual whose optimum equals the primal’s (strong duality). Dual variables measure the value of relaxing each constraint (shadow prices). Duality gives bounds and optimality certificates.
Worked example
Primal: max cᵀx with Ax ≤ b; dual: min bᵀy with Aᵀy ≥ c, y ≥ 0.
Real-world uses
- Finding price values for scarce resources and bounds on the best achievable profit
Common mistake
Assuming strong duality holds without regularity conditions in nonlinear problems.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Duality in Optimisation?
Pair every optimisation problem with a dual. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
Where is Duality in Optimisation used in real life?
Real-world examples: Finding price values for scarce resources and bounds on the best achievable profit.
What should I learn before Duality in Optimisation?
Make sure you are comfortable with Linear Programming and Simplex, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Duality in Optimisation?
Assuming strong duality holds without regularity conditions in nonlinear problems.
How can I practise Duality in Optimisation?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Duality in Optimisation?
Move on to KKT Conditions next.
Related expert topics in Differential Equations, Abstract Algebra & Applied Math
- Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
- Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
- Runge–Kutta Methods – Higher-accuracy numerical solvers.
- Second-Order Homogeneous Equations – Use the characteristic equation.
- Undetermined Coefficients – Guess the particular solution form.
- Variation of Parameters – General method for particular solutions.