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Mathematics · Advanced university

Convexity and constrained optimization

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

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Prerequisites

Gradients, inequalities and matrices.

Study 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.

Worked examples

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.

3

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

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

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

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

    1. Only its boundary is convex
    2. Only for positive coordinates
    3. Yes
    4. No, because it is unbounded

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

Answers and marking guidance

Answers and marking notes
  1. 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.

  2. 18

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

  3. Yes

    A linear inequality is preserved under convex combinations.

  4. 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.

Further reading and review references

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