Mathematics · lesson 5 of 5

Convexity and constrained optimization

Values on a line segment lie below the corresponding chord of function values.

Advanced university8 practice questionsNotes + worked examples

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Learn the ideas

Prerequisites: Gradients, inequalities and matrices.

Convex function

Values on a line segment lie below the corresponding chord of function values.

First-order criterion

For differentiable convex f, f(y)≥f(x)+∇f(x)·(y−x).

Global minima

A stationary point of a differentiable convex function is a global minimizer; strict convexity gives at most one.

Constraints

Lagrange multipliers and KKT conditions require their own regularity and feasibility checks.

Worked examples

Example 1

Find the x-coordinate minimizing f(x,y)=x²+2y²−6x.

The x derivative is 2x−6; setting it to zero gives 3, and the Hessian is positive definite.

Example 2

For a differentiable convex function, a zero gradient implies what?

The supporting-hyperplane inequality gives f(y)≥f(x).

A global minimum

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 Find the x-coordinate minimizing f(x,y)=x²+2y²−6x.

Show answer and explanation

The x derivative is 2x−6; setting it to zero gives 3, and the Hessian is positive definite.

2 For a differentiable convex function, a zero gradient implies what?

A. A global minimumB. Always a strict maximumC. Always exactly two minimaD. No information ever

Show answer and explanation

A global minimum

The supporting-hyperplane inequality gives f(y)≥f(x).

3 Prove that a strictly convex function has at most one minimizer on a convex domain.

Sample answer and marking guidance

If two distinct minimizers existed, strict convexity at their midpoint would give a value strictly below their common minimum, a contradiction.

Credit the midpoint lying in the domain and strict inequality.

4 Minimize x²+y² subject to x+y=6. Find the minimum value.

Show answer and explanation

The minimum occurs at x=y=3 by completing a square or Lagrange multipliers; value 9+9=18.

5 Is the feasible set x+y≤1 in R² convex?

A. Only its boundary is convexB. Only for positive coordinatesC. YesD. No, because it is unbounded

Show answer and explanation

Yes

A linear inequality is preserved under convex combinations.

6 Why must a proposed constrained optimum be checked for feasibility?

Sample answer and marking guidance

Stationarity equations alone can produce points outside the allowed set; an infeasible point cannot solve the constrained problem.

Credit distinction between algebraic candidates and admissible solutions.

7 What is the second derivative of f(x)=4x²−x+1?

Show answer and explanation

Differentiate twice: f′=8x−1 and f″=8.

8 Give a convex function that is not strictly convex and has more than one minimizer.

Sample answer and marking guidance

f(x)=0 on R is convex but not strictly convex, and every x is a minimizer.

Credit a valid example and explanation.

Learn the ideas

Convex function

First-order criterion

Global minima

Constraints

Worked examples

Find the x-coordinate minimizing f(x,y)=x²+2y²−6x.

For a differentiable convex function, a zero gradient implies what?

Check your understanding

Further reading and review references