数学代写|Generalized inequality constraints 凸优化代考
凸优化代写
4.6 Generalized inequality constraints
One very useful generalization of the standard form convex optimization problem (4.15) is obtained by allowing the inequality constraint functions to be vector valued, and using generalized inequalities in the constraints:
where $f_{0}: \mathbf{R}^{n} \rightarrow \mathbf{R}, K_{i} \subseteq \mathbf{R}^{k_{i}}$ are proper cones, and $f_{i}: \mathbf{R}^{n} \rightarrow \mathbf{R}^{k_{i}}$ are $K_{i}$-convex. We refer to this problem as a (standard form) convex optimization problem with generalized inequality constraints. Problem (4.15) is a special case with $K_{i}=\mathbf{R}_{+}$, $i=1, \ldots, m$.
Wany of the results for ordinary convex optimization problems hold for problems with generalized inequalities. Some examples are:
- The feasible set, any sublevel set, and the optimal set are convex.
- Any point that is locally optimal for the problem (4.48) is globally optimal.
- The optimality condition for differentiable $f_{0}$, given in $\S 4.2 .3$, holds without any change.
We will also see (in chapter 11) that convex optimization problems with generalized inequality constraints can often be solved as easily as ordinary convex optimization problems.
4 Convex optimization problems
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4.6.1 Conic form problems
Among the simplest convex optimization problems with generalized inequalities are the conic form problems (or cone programs), which have a linear objective and one inequality constraint function, which is affine (and therefore $K$-convex):
$\begin{array}{ll}\operatorname{minimize} & c^{T} x \ \text { subject to } & F x+g \preceq_{K} 0 \ & A x=b\end{array}$
When $K$ is the nonnegative orthant, the conic form problem reduces to a linear program. We can view conic form problems as a generalization of linear programs in which componentwise inequality is replaced with a generalized linear inequality. lem
as a conic form problem in standard form. Similarly, the problem
$\begin{array}{ll}\text { minimize } & c^{T} x \ \text { subject to } & F x+g \preceq_{K} 0\end{array}$
is called a conic form problem in inequality form.
4.6.2 Semidefinite programming
When $K$ is $\mathbf{S}{+}^{k}$, the cone of positive semidefinite $k \times k$ matrices, the associated conic form problem is called a semidefinite program (SDP), and has the form where $G, F{1}, \ldots, F_{n} \in \mathbb{S}^{k}$, and $A \in \mathbb{R}^{p \times n}$, The inequality here
where $G, F_{1}, \ldots, F_{n} \in \mathbf{S}^{k}$, and $A \in \mathbf{R}^{p \times n}$. The inequality here is a linear matrix inequality (see example 2.10).
If the matrices $G, F_{1}, \ldots, F_{n}$ are all diagonal, then the LMI in (4.50) is equiva- lent to a set of $n$ linear inequalities, and the SDP $(4.50)$ reduces to a linear program.
Standard and inequality form semidefinite programs
Following the analogy to LP, a standard form SDP has linear equality constraints, and a (matrix) nonnegativity constraint on the variable $X \in \mathbf{S}^{n}$ :
$\begin{array}{ll}\text { minimize } & \operatorname{tr}(C X) \ \text { subject to } & \operatorname{tr}\left(A_{i} X\right)=b_{i}, \quad i=1, \ldots, p \ & X \succeq 0,\end{array}$
凸优化代考
4.6 广义不等式约束
标准形式凸优化问题 (4.15) 的一个非常有用的推广是通过允许不等式约束函数为向量值并在约束中使用广义不等式来获得的：
其中 $f_{0}: \mathbf{R}^{n} \rightarrow \mathbf{R}, K_{i} \subseteq \mathbf{R}^{k_{i}}$ 是真锥，$f_ {i}: \mathbf{R}^{n} \rightarrow \mathbf{R}^{k_{i}}$ 是 $K_{i}$-凸的。我们将此问题称为具有广义不等式约束的（标准形式）凸优化问题。问题 (4.15) 是一个特殊情况，$K_{i}=\mathbf{R}_{+}$, $i=1, \ldots, m$。
普通凸优化问题的任何结果都适用于具有广义不等式的问题。一些例子是：
- 可行集、任何子水平集和最优集是凸的。
- 对问题（4.48）局部最优的任何点都是全局最优的。
- 在 $\S 4.2 .3$ 中给出的可微分 $f_{0}$ 的最优条件保持不变。
我们还将看到（在第 11 章中）具有广义不等式约束的凸优化问题通常可以像普通凸优化问题一样容易解决。
4 凸优化问题
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4.6.1 圆锥形问题
具有广义不等式的最简单凸优化问题是圆锥形式问题（或圆锥规划），它有一个线性目标和一个不等式约束函数，它是仿射的（因此是 $K$-凸）：
$\begin{array}{ll}\operatorname{minimize} & c^{T} x \ \text { 服从 } & F x+g \preceq_{K} 0 \ & A x=b\end{数组}$
当$K$ 是非负正数时，圆锥形式问题简化为线性规划。我们可以将圆锥形式问题视为线性程序的推广，其中分量不等式被广义线性不等式取代。莱姆
作为标准形式的圆锥形式问题。同样，问题
$\begin{array}{ll}\text { 最小化 } & c^{T} x \ \text { 服从 } & F x+g \preceq_{K} 0\end{array}$
被称为不等式形式的圆锥形式问题。
4.6.2 半定规划
当 $K$ 是 $\mathbf{S}{+}^{k}$，正半定 $k \times k$ 矩阵的圆锥，相关的圆锥形式问题称为半定规划（SDP），并且有表格 其中 $G, F{1}, \ldots, F_{n} \in \mathbb{S}^{k}$, and $A \in \mathbb{R}^{p \times n}$, 不等式这里
其中 $G, F_{1}, \ldots, F_{n} \in \mathbf{S}^{k}$, 和 $A \in \mathbf{R}^{p \times n}$。这里的不等式是线性矩阵不等式（参见示例 2.10）。
如果矩阵 $G, F_{1}, \ldots, F_{n}$ 都是对角矩阵，那么 (4.50) 中的 LMI 等价于一组 $n$ 线性不等式，并且 SDP $(4.50 )$ 简化为线性程序。
标准和不等式形成半定规划
类似于 LP，标准形式的 SDP 具有线性等式约束，以及对变量 $X \in \mathbf{S}^{n}$ 的（矩阵）非负性约束：
$\begin{array}{ll}\text { 最小化 } & \operatorname{tr}(CX) \ \text { 服从 } & \operatorname{tr}\left(A_{i} X\right)=b_ {i}, \quad i=1, \ldots, p \ & X \succeq 0,\end{array}$
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