Fundamentals: Extreme Value Theorems—Part 1

In the last post, we proved that the gradient descent algorithm can be used to computationally find a global minimizer of the least-squares cost functional, i.e., it converges to a vector \mathbf{x}^\ast \in \mathbb{R}^N such that

\mathcal{C}(\mathbf{x}^\ast) \leq \mathcal{C}(\mathbf{x}),

for all \mathbf{x} \in \mathbb{R}^N, where \mathcal{C}(\mathbf{x}) \triangleq \frac{1}{2}\|\mathbf{y}-\mathbf{H}\mathbf{x}\|_{\ell_2}^2.

Today, we want to be more general and ask the following question: given an arbitrary continuous function \mathcal{C}(\mathbf{x}), with \mathbf{x} \in \mathbb{R}^N, how to know that the function has a global minimizer?

Specifically, we will discuss two famous theorems that establish existence of global minimizers.

Theorem 1 [Extreme Value Theorem]: If \mathcal{X} \subseteq \mathbb{R}^N is a compact set and \mathcal{C} is a continuous function on \mathcal{X}, then \mathcal{C} has a global minimizer on \mathcal{X}

Theorem 2 [Extreme Value Theorem]: If \mathcal{C} is a continuous coercive function defined on all of \mathbb{R}^N, then \mathcal{C} has a global minimizer.

Those theorems provide sufficient conditions for the existence of global minimizers. The rest of the post will be about defining the terms compact and coercive; in Part 2 of the post we shall prove both theorems.

We start by establishing two fundamental concepts of open and closed sets in \mathbb{R}^N.

Definition 1 [open set]: A subset \mathcal{X} \subset \mathbb{R}^N is open if and only if for every \mathbf{x} \in \mathcal{X}  there exist \epsilon > 0 such that the open ball \mathcal{B}_{\epsilon}(\mathbf{x}) of center \mathbf{x} and radius \epsilon remains in \mathcal{X}, i.e., \mathcal{B}_{\epsilon}(\mathbf{x}) \subset \mathcal{X}.

Remark that the radius \epsilon may depend on \mathbf{x}, and also we remind the definition \mathcal{B}_{\epsilon}(\mathbf{x}) \triangleq \{\mathbf{y} \in \mathbb{R}^N : \|\mathbf{x}-\mathbf{y}\|_{\ell_2} < \epsilon\}.

Definition 2 [closed set]: A subset \mathcal{X} \subset \mathbb{R}^N is closed if and only if its complement \mathcal{X}^\complement = \mathbb{R}^N \setminus \mathcal{X} is open.

More intuitively, a closed set is a set that includes its boundary (if there is one), while an open set does not. To establish the term compact set, we introduce one more intermediate definition.

Definition 3 [bounded set]: A subset \mathcal{X} \subseteq \mathbb{R}^N is bounded if there exists a constant C > 0 such that \|\mathbf{x}\|_{\ell_2} < C for all \mathbf{x} \in \mathcal{X}.

Finally, we now define the compact set by combining Definitions 2 and 3.

Definition 4 [compact set]: A subset \mathcal{X} \subseteq \mathbb{R}^N is compact if it is closed and bounded.

From the perspective of Definition 4, Theorem 1 simply states that if we are optimizing \mathcal{C} over some compact subset of \mathcal{X} \subseteq \mathbb{R}^N, then we are sure that there exists a global minimizer \mathbf{x}^\ast that we may hope to find computationally.

Now, what happens if we are interested in optimizing over the whole \mathbb{R}^N rather than on some subset?

This is where the notion of coercive functions becomes useful.

Definition 5 [coercive function]: A continuous function \mathcal{C} that is defined on all of \mathbb{R}^N is coercive if

\lim_{\|\mathbf{x}\|_{\ell_2} \rightarrow \infty} \left\{\mathcal{C}(\mathbf{x})\right\} = +\infty.

This means that for any constant C > 0 there exists a constant X > 0 (that may depend on C) such that \mathcal{C}(\mathbf{x}) > C whenever \|\mathbf{x}\|_{\ell_2} > X.

Intuitively, for a function to be coercive, it must approach +\infty along any path within \mathbb{R}^N on which \|\mathbf{x}\|_{\ell_2} becomes infinite.

Going back to the example of the least-squares cost functional, we have from the reverse triangular inequality that

 \|\mathbf{y}-\mathbf{H}\mathbf{x}\|_{\ell_2} \geq \|\mathbf{H}\mathbf{x}\|_{\ell_2}-\|\mathbf{y}\|_{\ell_2} \geq C\|\mathbf{x}\|_{\ell_2}-\|\mathbf{y}\|_{\ell_2},

where the constant C = \sqrt{\lambda_{\text{min}}(\mathbf{H}^\mathrm{T}\mathbf{H})} with \lambda_{\text{min}}(\mathbf{H}^\mathrm{T}\mathbf{H}) denoting the smallest eigenvalue of the matrix \mathbf{H}^\mathrm{T}\mathbf{H}.

Thus, when the matrix \mathbf{H} is non-singular, i.e., C > 0, and since

 \lim_{\|\mathbf{x}\|_{\ell_2} \rightarrow \infty} \left\{\|\mathbf{x}\|_{\ell_2}\right\} = +\infty,

we see that least-squares is indeed a coercive function.

5 thoughts on “Fundamentals: Extreme Value Theorems—Part 1

  1. Thank you for posting this well-written note on the existence of a global minimizer . It is really a great pleasure to read. Additionally, I have a comment on your coerciveness argument of the least-square cost function. To prove that it blows up as x goes to infinity, I think you should use a lower bound on ||y - Hx||, rather than a upper bound. More specifically, you may want to use the reverse triangle inequality, i.e., ||y - Hx|| >= ||H||*||x|| - ||y||. Please correct me if I am wrong.

    Thanks,
    Bo Zhao

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