单变量优化与多变量优化

维基百科将优化定义为通过系统地从允许的集合中选择输入值并计算函数值来最大化或最小化实际函数的问题。这意味着当我们谈论优化时，我们总是对找到最佳解决方案感兴趣。因此，假设一个人有某种函数形式（例如以 f(x) 的形式），并且他正在尝试为这种函数形式找到最佳解决方案。现在，最好的意思是什么？可以说他对最小化这种函数形式或最大化这种函数形式感兴趣。
通常，优化问题具有三个组成部分。

最小化 f(x)，
x,
服从 a < x < b

其中， f(x) ：目标函数
x : 决策变量
a < x < b ：约束

根据决策变量的数量，优化可以分为两部分，

1. 单变量优化问题：单变量优化可以定义为一种没有约束的非线性优化，并且在这个优化中只有一个我们试图为其找到值的决策变量。
   

   最小 f(x)
   写
   x∈R

2. 多元优化问题：在多元优化问题中，在这个优化中必须有多个决策变量，我们试图为其找到一个值。
   

   min f(x ₁ , x ₂ , x ₃ .....x _n )
   

下表列出了单变量优化和多变量优化之间的差异：

Uni-variate Optimization	Multivariate Optimization
Uni-variate optimization may be defined as a non-linear optimization with no constraint and there is only one decision variable in this optimization that we are trying to find a value for. min f(x) w.r.t x x ∈ R So, when you look at this optimization problem you typically write it in this above form where you say you are going to minimize f(x), and this function is called the objective function. And the variable that you can use to minimize this function which is called the decision variable is written below like this w.r.t x here and you also say x is continuous that is it could take any value in the real number line.	In a multivariate optimization problem, multiple variables act as decision variables in the optimization problem. z = f(x1, x2, x3…..xn) So, when you look at these types of problems a general function z could be some non-linear function of decision variables x1, x2, x3 to xn. So, there are n variables that one could manipulate or choose to optimize this function z.
In case of uni-variate optimization problem there is only one decision variable.	In case of multivariate optimization problem there is more than one decision variables.
In a uni-variate optimization problem x is a scalar variable and not a vector variable.	In a multivariate optimization problem x may be a scalar variable or a vector variable.
One could explain univariate optimization using pictures in two dimensions that is because in the x-direction we had the decision variable value and in the y-direction, we had the value of the function.	However, if it is multivariate optimization then we have to use pictures in three dimensions
In a uni-variate optimization problem there is no constraint.	In a multivariate optimization problem there may be no constraint case or equality constraint case or inequality constraint case.
In case of uni-variate optimization the first order necessary conditions for x to be the minimizer of the function f(x) is f'(x) = 0	In case of unconstrained multivariate optimization the first order necessary conditions for x̄* to be the minimizer of the function f(x̄) is ∇ f(x̄*) = 0
In case of uni-variate optimization the second-order sufficiency condition for x to be the minimizer of the function f(x) is f”(x) > 0	In case of unconstrained multivariate optimization the second-order sufficiency condition for x̄* to be the minimizer of the function f(x̄) is ∇ 2 f(x̄*) > 0