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What is optimization? Roughly speaking, optimization is to minimize or maximize a function (which is called the *objective function*) under some constraints.

For example, we some several ways to return the campus from Hongqiao Station: by taxi (Didi / Gaode), by metro or by bus (Hongqiao 4 Line / Min-Hong 2 Line), etc. We would like to minimize the time, but our money is limited. This is an optimization problem.

Formally, an optimization problem can be defined by *objective function* and *feasible set*, usually specified by *constraint functions* *optimal solution* is usually denoted by *continuous* optimization problem, where the *objective function* and the *constraints* are continuous functions. We now give some more examples.

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Example

Suppose there are

For each

Question

- What if we require integer
? - What if there are more types?

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Example

Consider the *free falling motion*. The height and the time of a free fall follow the law

Suppose we have the following data and we would like to use

10 | 20 | 30 | 40 | |
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1.011 | 2.019 | 3.032 | 4.041 |

However, before solving this problem, we should first ask the following question: if we choose a certain value of

Question

Generally, we have the following question. Let

If we only have two numbers *absolute value*

We need to extend the concept of the absolute value to measure the distances between vectors in

Definition (*Norm*)

Given a vector space

1. (*Nonnegativity*)

2. (*Positive definiteness*)

3. (*Absolute homogeneity*)

4. (*Triangle inequality*)