Mathematics · Advanced university
Convexity and constrained optimization · revision notes
Values on a line segment lie below the corresponding chord of function values.
Key ideas and conditions
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.
Explain without looking
- What is the second derivative of f(x)=4x²−x+1?
- Give a convex function that is not strictly convex and has more than one minimizer.
Further reading and review references
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