Given an optimization problem the optimal solution is usually denoted by The first question is: for which optimization problems, the optimal solution exist?
In general, the question is hard to answer. We only have the following conclusion for some special objective functions and feasible sets.

We now review some definitions in analysis.

We can define open sets and closed sets.

For closed sets, there is another different but equivalent definition.

Then we define compact sets.

In , there is another definition.

For optimization problems whose feasible sets are not compact, we usually cannot have simple ways to determine whether optimal solutions exist. However, for continuous function and , if , , then is a compact set, and thus has minimum values.

Just like the vs. problem, verifying a solution is believed to be easier. So we first study how to justify a solution is indeed an optimal one.

We first identify global minima and local minima.

Similarly, we can also define strictly global minima and strictly local minima.

Unfortunately, it is too hard to verify global minima in general. It also provides evidence why general optimization problems are difficult to solve. In this course we will study a special type of optimization problem, where local minima are also global minima.

We now give some criteria that can be used to prove local minima.

Suppose is continuous and differentiable. We know that if is a extreme point only if . Can we have similar results in high dimensions?

The generalization of derivative in high dimensions is the directional derivative.

Given , we can use to do a linear approximation of at , where can be seen as a linear mapping. It is natural to define the differential of a function at by a linear mapping if .

Now we give some examples and calculation rules of differentials.

Here is a simple proof of the last example: , so which yields that .

We are ready to give the first-order optimality condition.

An important idea is to restrict a multivariate function to a line.

In particular, if is an open set, any point is an interior point. So .

Unfortunately, the first-order condition is a necessary condition. If , we still do not know whether is a local minimum. An simple example is function and . For multivariate functions, there is another case called the saddle point.

We can compute the high-order derivatives to refute saddle points.

For a multivariate function , is a mapping . We can further compute the Jacobian matrix of :The transpose matrix of the Jacobian is called the Hessian matrix of , and denoted by , or . So .

We are ready to establish the second-order condition. Consider a function . Intuitively, if is a local minimum, then we have , and for sufficiently small . Thus .

Now let be a multivariate function . Fix and consider the restriction of . Let . Using the chain rule, we have In particular, we need .

Another idea is to consider the second-order Taylor series: Hence we can reasonable guess that since .