Existence and uniqueness of the minimiser

ProofBeginner⏱ 15 minOptimizationstrong convexityuniquenesscoercivity

Theory: Convex Sets, Convex Functions, Smoothness and Strong Convexity →

Problem

Prove that a $\mu$ -strongly convex differentiable $f$ on $\R^d$ has exactly one minimiser, and that $f(x)-f^\star \ge \frac\mu2\|x-x^\star\|_2^2$ for every $x$ .

Write a complete argument. Every inequality you use should be one you could state precisely.

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

Related problems

All problems

Problem	Field	Type	Difficulty	Time
Every norm is convex, and the operator norm is a supremum of linear functions	Optimization	Proof	Beginner	15 min
First-order characterisation of convexity	Optimization	Proof	Beginner	30 min
Separating convexity, strong convexity, smoothness and Lipschitzness	Optimization	Proof	Intermediate	30 min
Strong convexity implies the Polyak–Lojasiewicz inequality	Optimization	Proof	Intermediate	30 min