数学代写| Geometric programming 数值分析代考
数值分析代写
4.5 Geometric programming
In this section we describe a family of optimization problems that are not convex in their natural form. These problems can, however, be transformed to convex optimization problems, by a change of variables and a transformation of the objective and constraint functions.
4.5.1 Monomials and posynomials
A function $f: \mathbf{R}^{n} \rightarrow \mathbf{R}$ with $\operatorname{dom} f=\mathbf{R}{++}^{n}$, defined as $f(x)=c x{1}^{a_{1}} x_{2}^{a_{2}} \cdots x_{n}^{a_{n}}, } \{ } &{ } \{ } &{ } \end{array}$
where $c>0$ and $a_{i} \in \mathbf{R}$, is called a monomial function, or simply, a monomial. negative, but the coefficient $c$ can only be positive. (The term ‘monomial’ conflicts with the standard definition from algebra, in which the exponents must be nonnegative integers, but this i.e. a function of the form
i.e., a function of the form $$ f(x)=\sum_{k=1}^{K} c_{k} x_{1}^{a_{1 k}} x_{2}^{a_{2 k}} \ldots x_{n}^{a_{n k}} \text {, } $$
where $c_{k}>0$, is called a posynomial function (with $K$ terms), or simply, a posynomial.
4.5 Geometric programming Posynomials are closed under addition, multiplication, and nonnegative scal-
161 Posynomials are closed under addition, multiplication, and nonnegative scalmultiplied by a monomial, the result is a posynomial; similarly, a posynomial can be divided by a monomial, with the result a posynomial.
4.5.2 Geometric programming
An optimization problem of the form
$\begin{array}{ll}\text { minimize } & f_{0}(x) \ \text { subject to } & f_{i}(x) \leq 1, \quad i=1, \ldots, m \ & h_{i}(x)=1, \quad i=1, \ldots, p\end{array}$
where $f_{0}, \ldots, f_{m}$ are posynomials and $h_{1}, \ldots, h_{p}$ are monomials, is called a geometric program (GP). The domain of this problem is $\mathcal{D}=\mathbf{R}{++}^{n} ;$ the constraint $x \succ 0$ is implicit. Extensions of geometric programming Several extensions are readily handled. If $f$ is a posynomial and $h$ is a monomial, then the constraint $f(x) \leq h(x)$ can be handled by expressing it as $f(x) / h(x) \leq 1$ (since $f / h$ is posynomial). This includes as a special case a constraint of the form $f(x) \leq a$, where $f$ is posynomial and $a>0$. In a similar way if $h{1}$ and $h_{2}$ are both nonzero monomial functions, then we can handle the equality constraint $h_{1}(x)=h_{2}(x)$ by expressing it as $h_{1}(x) / h_{2}(x)=1$ (since $h_{1} / h_{2}$ is monomial). We (which is also a monomial).
For example, consider the problem
数值分析代考
4.5 几何规划
在本节中，我们描述了一系列自然形式不是凸的优化问题。然而，这些问题可以通过变量的变化以及目标和约束函数的变换转化为凸优化问题。
4.5.1 单项式和多项式
一个函数 $f: \mathbf{R}^{n} \rightarrow \mathbf{R}$ 和 $\operatorname{dom} f=\mathbf{R}{++}^{n}$，定义为 $f(x)=c x{1}^{a_{1}} x_{2}^{a_{2}} \cdots x_{n}^{a_{n}}, } \{ } &{ } \{ } &{ } \end{数组}$
其中 $c>0$ 和 $a_{i} \in \mathbf{R}$, 称为单项式函数，或者简称为单项式。负，但系数 $c$ 只能为正。 （术语“单项式”与代数的标准定义相冲突，其中指数必须是非负整数，但这就是形式的函数
即，$$ f(x)=\sum_{k=1}^{K} c_{k} x_{1}^{a_{1 k}} x_{2}^{a_{2 形式的函数k}} \ldots x_{n}^{a_{nk}} \text {, } $$
其中 $c_{k}>0$，称为多项式函数（带有 $K$ 项），或者简称为多项式。
4.5 几何规划 多项式在加法、乘法和非负标量下是封闭的
161 多项式在加法、乘法和非负标度乘以一个单项式时是闭合的，结果是一个多项式；类似地，一个多项式可以除以一个单项式，结果是一个多项式。
4.5.2 几何规划
形式的优化问题
$\begin{array}{ll}\text { 最小化 } & f_{0}(x) \ \text { 服从 } & f_{i}(x) \leq 1, \quad i=1, \ldots , m \ & h_{i}(x)=1, \quad i=1, \ldots, p\end{array}$
其中 $f_{0}, \ldots, f_{m}$ 是多项式，$h_{1}, \ldots, h_{p}$ 是单项式，称为几何规划 (GP)。这个问题的域是 $\mathcal{D}=\mathbf{R}{++}^{n} ;$ 约束 $x \succ 0$ 是隐式的。 几何规划的扩展 几个扩展很容易处理。如果 $f$ 是一个多项式并且 $h$ 是一个单项式，那么约束 $f(x) \leq h(x)$ 可以通过将其表示为 $f(x) / h(x) \leq 1 来处理$（因为 $f / h$ 是多项式）。这包括一个特殊情况，形式为 $f(x) \leq a$ 的约束，其中 $f$ 是正多项式且 $a>0$。类似地，如果 $h{1}$ 和 $h_{2}$ 都是非零单项式函数，那么我们可以通过表达来处理等式约束 $h_{1}(x)=h_{2}(x)$它为 $h_{1}(x) / h_{2}(x)=1$ （因为 $h_{1} / h_{2}$ 是单项式）。我们（也是单项式）。
例如，考虑问题
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随机过程代写
随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。 随机变量是随机现象的数量表现，其取值随着偶然因素的影响而改变。 例如，某商店在从时间t0到时间tK这段时间内接待顾客的人数，就是依赖于时间t的一组随机变量，即随机过程
