# Convexity and constrained optimization

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

Course: [Advanced mathematical methods](https://studybrix.com/en/courses/advanced-mathematical-methods/index.md)
Level: advanced
Prerequisites: Gradients, inequalities and matrices.

## Notes

### 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.

## Practice questions

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

Answer: 3

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 minimum
B. Always a strict maximum
C. Always exactly two minima
D. No information ever

Answer: A global minimum

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

Explanation: Credit the midpoint lying in the domain and strict inequality.

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

Answer: 18

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 convex
B. Only for positive coordinates
C. Yes
D. No, because it is unbounded

Answer: Yes

Explanation: A linear inequality is preserved under convex combinations.

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

Sample answer / marking guidance: Stationarity equations alone can produce points outside the allowed set; an infeasible point cannot solve the constrained problem.

Explanation: Credit distinction between algebraic candidates and admissible solutions.

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

Answer: 8

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

Explanation: Credit a valid example and explanation.

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.