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- The following topics will be covered in this lecture:
- Concepts in optimization
- Local versus global optimum
- Convexity
- Gradient descent versus Newton descent

The maximization and minimization of functions, or

**optimization problems**, contain two components:- an
**objective function \( f(\mathbf{x}) \)**; and **constraints \( g(\mathbf{x}) \)**.

- an
E.g., we may wish to

**optimize factory output \( f(x) \)**as a function of hours \( x \) in a week, with a measure of our**active machine-hours \( g(x) \) not exceeding a pre-specified limitation \( g(x)\leq C \)**.Optimization problems can thus be classified into two categories:

- If there are
**constraints \( g(\mathbf{x}) \)**affiliated with the objective function \( f(\mathbf{x}) \), then it is a**constrained optimization problem**,**otherwise, it is a unconstrained optimization problem**.

- If there are
We will focus on

**unconstrained optimization as often arises in MLE**; this is formulated as the following problem,\[ \begin{align} f: \mathbb{R}^n &\rightarrow \mathbb{R}\\ \mathbf{x} &\rightarrow f(\mathbf{x})\\ f(\mathbf{x}^\ast) &= \mathrm{max}_{\mathbf{x} \in \mathcal{D}} f \end{align} \]

We note that the above problem is equivalent to a minimization problem by a subsitution of \( \tilde{f} = -f \), i.e.,

\[ \begin{align} \tilde{f}: \mathbb{R}^n &\rightarrow \mathbb{R}\\ \mathbf{x} &\rightarrow -f(\mathbf{x})\\ f(\mathbf{x}^\ast) &= \mathrm{max}_{\mathbf{x} \in \mathcal{D}} \tilde{f} = \mathrm{min}_{\mathbf{x}\in \mathcal{D}} f \end{align} \]

- Because these problems are equivalent, we
**focus on the minimization of functions**at the moment as they are traditionally phrased in optimization literature. - The same techniques will apply for maximization by a simple change of variables.
- We will need to identify a few key concepts:
**global**and**local**minimizers. - Suppose we are trying to minimize an objective function \( f \).
- We would ideally find a
**global minimizer of \( f \)**, a point where the function attains its least value over all possible values under consideration:A point \( \mathbf{x}^\ast \) is a global minimizer if \( f(\mathbf{x}^\ast) \leq f(\mathbf{x}) \) for all other possible \( \mathbf{x} \) in the domain of consideration \( D\subset \mathbb{R}^n \).

- A
**global minimizer can be difficult to find**, because our**knowledge of \( f \) is usually only local**;

Courtesy of: J. Nocedal and S. Wright. *Numerical optimization*. Springer Science & Business Media, 2006.

- i.e., we only can approximate the behavior of the function \( f \) within
**small perturbations \( \boldsymbol{\delta}_{x} \) of values \( \mathbf{x} \)**where we already know \( f(\mathbf{x}) \). - Since our algorithm hopefully does not need to compute \( f \) over many points, we usually do not have a good picture of the overall shape of \( f \),
- generally, we can never be sure that the function does not take a sharp dip in some region that has not been sampled by the algorithm.

**Most algorithms are able to find only a local minimizer**, which is a point that achieves the smallest value of \( f \) in its neighborhood, i.e.,

Let \( \mathcal{N}\subset \mathcal{D} \subset\mathbb{R}^n \) be a neighborhood of the point \( \mathbf{x}^\ast \) in the domain of consideration. We say \( \mathbf{x}^\ast \) is a local minimizer in the neighborhood of \( \mathcal{N} \) if \[ \begin{align} f(\mathbf{x}^\ast) \leq f(\mathbf{x}) & & \text{ for each other }\mathbf{x}\in \mathcal{N} \end{align} \]

For finding a local minimizer, the main tools will be derived directly from the

**second order approximation of the objective function \( f \)**, defined by\[ \begin{align} f(\mathbf{x}_1) \approx f(\mathbf{x}_0) + \left(\nabla f(\mathbf{x}_0)\right)^\mathrm{T} \boldsymbol{\delta}_{x_1}+\frac{1}{2} \boldsymbol{\delta}_{x_1}^\mathrm{T}\mathbf{H}_f (\mathbf{x}_0) \boldsymbol{\delta}_{x_1} \end{align} \]

We will consider how this is related to the notion of

**convexity**as follows.

Courtesy of: Oleg Alexandrov. Public domain, via Wikimedia Commons.

- Throughout mathematics, the notion of convexity is a powerful tool, often used in optimization.
- Particularly, a
**function is convex**if and only if**the region above its graph is a convex set**. - The
**convexity of the full epigraph**set means that the function attains a**global minimum over its entire domain**. - In
**non-convex functions**, we can have regions that are also**locally convex**in the graph of the function. - For such regions, we can find
**local minimizers**as defined in the last slide. - In a single variable, this is phrased in terms of the
**second derivative test**, i.e.,For the function of one variable \( f(x) \) we say that \( x^\ast \) is a local minimizer if \( f'(x^\ast)=0 \) and \( f''(x^\ast)> 0 \).

- There is a direct analogy for a function of multiple variables, but this need to be rephrased slightly.

- For a real-valued function of multiple variables,
\[ f:\mathbb{R}^n \rightarrow \mathbb{R}, \]
we will instead phrase the
**second derivative test in terms of the Hessian of \( f \)**, \[ \begin{align} \mathbf{H}_{f} = \begin{pmatrix} \partial_{x_1}^2 f & \cdots & \partial_{x_1}\partial_{x_n}f \\ \vdots &\ddots & \vdots \\ \partial_{x_n}\partial_{x_1} f & \cdots & \partial_{x_n}^2 f \end{pmatrix} \end{align} \] - Particularly, the spectral theorem says the Hessian has an eigen decomposition such that by a change of coordinates, \( \mathbf{H}_f \) will be diagonal.