# 数学代写|凸优化作业代写Convex Optimization代考|SUBGRADIENTS OF CONVEX REAL-VALUED FUNCTIONS

#### Doug I. Jones

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## 数学代写|凸优化作业代写Convex Optimization代考|SUBGRADIENTS OF CONVEX REAL-VALUED FUNCTIONS

Given a proper convex function $f: \Re^n \mapsto(-\infty, \infty)$, we say that a vector $g \in \Re^n$ is a subgradient of $f$ at a point $x \in \operatorname{dom}(f)$ if
$$f(z) \geq f(x)+g^{\prime}(z-x), \quad \forall z \in \Re^n$$
see Fig. 3.1.1. The set of all subgradients of $f$ at $x \in \Re^n$ is called the subdifferential of $f$ at $x$, and is denoted by $\partial f(x)$. For $x \notin \operatorname{dom}(f)$ we use the convention $\partial f(x)=\varnothing$. Figure 3.1.2 provides some examples of subdifferentials. Note that $\partial f(x)$ is a closed convex set, since based on Eq. $(3.1)$, it is the intersection of a collection of closed halfspaces (one for each $\left.z \in \Re^n\right)$.

It is generally true that $\partial f(x)$ is nonempty for all $x \in \operatorname{ri}(\operatorname{dom}(f))$, the relative interior of the domain of $f$, but it is possible that $\partial f(x)=\varnothing$ at some points in the relative boundary of $\operatorname{dom}(f)$. The properties of subgradients of extended real-valued functions are summarized in Section 5.4 of Appendix B. When $f$ is real-valued, however, stronger results can be shown: $\partial f(x)$ is not only closed and convex, but also nonempty and compact for all $x \in \Re^n$. Moreover the proofs of this and other related results are generally simpler than for the extended real-valued case. For this reason, we will provide an independent development of the results that we need for the case where $f$ is real-valued (which is the primary case of interest in algorithms).
To this end, we recall the definition of the directional derivative of $f$ at a point $x$ in a direction $d$ :
$$f^{\prime}(x ; d)=\lim _{\alpha \downarrow 0} \frac{f(x+\alpha d)-f(x)}{\alpha}$$
(cf. Section 5.4.4 of Appendix B). The ratio on the right-hand side is monotonically nonincreasing to $f^{\prime}(x ; d)$, as shown in Section 5.4.4 of Appendix B; also see Fig. 3.1.3.
Our first result shows some basic properties, and provides the connection between $\partial f(x)$ and $f^{\prime}(x ; d)$ for real-valued $f$. A related and more refined result is given in Prop. 5.4.8 in Appendix B for extended real-valued $f$. Its proof, however, is more intricate and includes some conditions that are unnecessary for the case where $f$ is real-valued.

## 数学代写|凸优化作业代写Convex Optimization代考|Characterization of the Subdifferential

The characterization and computation of $\partial f(x)$ may not be convenient in general. It is, however, possible in some special cases. Principal among these is when
$$f(x)=\sup {z \in Z} \phi(x, z)$$ where $x \in \Re^n, z \in \Re^m, \phi: \Re^n \times \Re^m \mapsto \Re$ is a function, $Z$ is a compact subset of $\Re^m, \phi(\cdot, z)$ is convex and differentiable for each $z \in Z$, and $\nabla_x \phi(x, \cdot)$ is continuous on $Z$ for each $x$. Then the form of $\partial f(x)$ is given by Danskin’s Theorem [Dan67], which states that $$\partial f(x)=\operatorname{conv}\left{\nabla_x \phi(x, z) \mid z \in Z(x)\right}, \quad x \in \Re^n,$$ where $Z(x)$ is the set of maximizing points in Eq. (3.9), $$Z(x)=\left{\bar{z} \mid \phi(x, \bar{z})=\max {z \in Z} \phi(x, z)\right} .$$
The proof is somewhat long, so it is relegated to the exercises.
An important special case of Eq. (3.10) is when $Z$ is a finite set, so $f$ is the maximum of $m$ differentiable convex functions $\phi_1, \ldots, \phi_m$ :
$$f(x)=\max \left{\phi_1(x), \ldots, \phi_m(x)\right}, \quad x \in \Re^n .$$
Then we have
$$\partial f(x)=\operatorname{conv}\left{\nabla \phi_i(x) \mid i \in I(x)\right}$$
where $I(x)$ is the set of indexes $i$ for which the maximum is attained, i.e., $\phi_i(x)=f(x)$. Another important special case is when $\phi(\cdot, z)$ is differentiable for all $z \in Z$, and the supremum in Eq. (3.9) is attained at a unique point, so $Z(x)$ consists of a single point $z(x)$. Then $f$ is differentiable at $x$ and
$$\nabla f(x)=\nabla \phi(x, z(x))$$

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|SUBGRADIENTS OF CONVEX REAL-VALUED FUNCTIONS

$$f(z) \geq f(x)+g^{\prime}(z-x), \quad \forall z \in \Re^n$$

$$f^{\prime}(x ; d)=\lim _{\alpha \downarrow 0} \frac{f(x+\alpha d)-f(x)}{\alpha}$$
(参见附录B第5.4.4节)。右侧的比率单调不增加到$f^{\prime}(x ; d)$，如附录B第5.4.4节所示;见图3.1.3。

## 数学代写|凸优化作业代写Convex Optimization代考|Characterization of the Subdifferential

$$f(x)=\sup {z \in Z} \phi(x, z)$$其中$x \in \Re^n, z \in \Re^m, \phi: \Re^n \times \Re^m \mapsto \Re$是一个函数，$Z$是一个紧凑的子集$\Re^m, \phi(\cdot, z)$对于每个$z \in Z$是凸的和可微的，$\nabla_x \phi(x, \cdot)$对于每个$x$在$Z$上是连续的。然后，$\partial f(x)$的形式由Danskin定理[Dan67]给出，该定理指出$$\partial f(x)=\operatorname{conv}\left{\nabla_x \phi(x, z) \mid z \in Z(x)\right}, \quad x \in \Re^n,$$，其中$Z(x)$是公式(3.9)中最大化点的集合，$$Z(x)=\left{\bar{z} \mid \phi(x, \bar{z})=\max {z \in Z} \phi(x, z)\right} .$$

(3.10)式的一个重要特例是$Z$是有限集，因此$f$是$m$可微凸函数$\phi_1, \ldots, \phi_m$的最大值:
$$f(x)=\max \left{\phi_1(x), \ldots, \phi_m(x)\right}, \quad x \in \Re^n .$$

$$\partial f(x)=\operatorname{conv}\left{\nabla \phi_i(x) \mid i \in I(x)\right}$$

$$\nabla f(x)=\nabla \phi(x, z(x))$$

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