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MathidiaMathematics Academy
Optimisation
Lesson 51 of 75
Introduction

Convexity

Objective. Use convex sets and functions to guarantee optima.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Convexity studies sets and functions whose shape forbids hidden dips, so local minima become global ones. A set is convex if the line segment between any two points stays inside it; a function is convex if its graph lies below every chord, equivalently f(λx+(1−λ)y) ≤ λf(x)+(1−λ)f(y) for λ∈[0,1]. For twice-differentiable f, convexity holds iff f''(x) ≥ 0 everywhere. Example: f(x)=x² has f''=2>0, so it is convex, and its unique stationary point x=0 is automatically the global minimum. This is why convex problems are so tractable in optimisation.

Worked example

Show that f(x) = x² − 4x + 7 is convex and state its global minimum value.

4 … 7
Illustrated card showing 4 … 7
  1. 1Differentiate twice: f'(x) = 2x − 4, f''(x) = 2
  2. 2Since f''(x) = 2 ≥ 0 for all x, f is convex
  3. 3Convex functions have global minima at stationary points: set f'(x)=0 giving x=2
  4. 4Evaluate f(2) = 4 − 8 + 7 = 3

Answer: 3

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Common mistake. Students assume any function with a single minimum is convex; f(x)=x⁴−x² has one well but a dip in curvature, so checking f''≥0 everywhere is essential, not just counting stationary points.

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