# 数学代写|凸优化作业代写Convex Optimization代考|Optimality Condition with Directional Derivatives

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## 数学代写|凸优化作业代写Convex Optimization代考|Optimality Condition with Directional Derivatives

The purpose of this exercise is to express the necessary and sufficient condition for optimality of Prop. 3.1.4 in terms of the directional derivative of the cost function. Consider the minimization of a convex function $f: \Re^n \mapsto \Re$ over a convex set $X \subset \Re^n$. For any $x \in X$, the set of feasible directions of $f$ at $x$ is defined to be the convex cone
$$D(x)={\alpha(\bar{x}-x) \mid \bar{x} \in X, \alpha>0} .$$
Show that a vector $x$ minimizes $f$ over $X$ if and only if $x \in X$ and
$$f^{\prime}(x ; d) \geq 0, \quad \forall d \in D(x) .$$
Note: In words, this condition says that $x$ is optimal if and only if there is no feasible descent direction of $f$ at $x$. Solution: Let $\overline{D(x)}$ denote the closure of $D(x)$. By Prop. 3.1.4, $x$ minimizes $f$ over $X$ if and only if there exists $g \in \partial f(x)$ such that
$$g^{\prime} d \geq 0, \quad \forall d \in D(x)$$
which is equivalent to
$$g^{\prime} d \geq 0, \quad \forall d \in \overline{D(x)}$$
Thus, $x$ minimizes $f$ over $X$ if and only if
$$\max {g \in \partial f(x)} \min {|d| \leq 1, d \in \overline{D(x)}} g^{\prime} d \geq 0$$

## 数学代写|凸优化作业代写Convex Optimization代考|Subdifferential of an Extended Real-Valued Function

Extended real-valued convex functions arising in algorithmic practice are often of the form
$$f(x)= \begin{cases}h(x) & \text { if } x \in X, \ \infty & \text { if } x \notin X,\end{cases}$$
where $h: \Re^n \mapsto \Re$ is a real-valued convex function and $X$ is a nonempty convex set. The purpose of this exercise is to show that the subdifferential of such functions admits a more favorable characterization compared to the case where $h$ is extended real-valued.
(a) Use Props. 3.1.3 and 3.1.4 to show that the subdifferential of such a function is nonempty for all $x \in X$, and has the form
$$\partial f(x)=\partial h(x)+N_X(x), \quad \forall x \in X,$$
where $N_X(x)$ is the normal cone of $X$ at $x \in X$. Note: If $h$ is convex but extended-real valued, this formula requires the assumption $\operatorname{ri}(\operatorname{dom}(h)) \cap$ $\operatorname{ri}(X) \neq \varnothing$ or some polyhedral conditions on $h$ and $X$; see Prop. 5.4.6 of Appendix B. Proof: By the subgradient inequality (3.1), we have $g \in \partial f(x)$ if and only if $x$ minimizes $p(z)=h(z)-g^{\prime} z$ over $z \in X$, or equivalently, some subgradient of $p$ at $x$ [i.e., a vector in $\partial h(x)-{g}$, by Prop. 3.1.3] belongs to $-N_X(x)$ (cf. Prop. 3.1.4).
(b) Let $f(x)=-\sqrt{x}$ if $x \geq 0$ and $f(x)=\infty$ if $x<0$. Verify that $f$ is a closed convex function that cannot be written in the form (3.35) and does not have a subgradient at $x=0$.
(c) Show the following formula for the subdifferential of the sum of functions $f_i$ that have the form (3.35) for some $h_i$ and $X_i$ :
$$\partial\left(f_1+\cdots+f_m\right)(x)=\partial h_1(x)+\cdots+\partial h_m(x)+N_{X_1 \cap \cdots \cap X_m}(x),$$
for all $x \in X_1 \cap \cdots \cap X_m$. Demonstrate by example that in this formula we cannot replace $N_{X_1 \cap \cdots \cap X_m}(x)$ by $N_{X_1}(x)+\cdots+N_{X_m}(x)$. Proof: Write $f_1+\cdots+f_m=h+\delta_X$, where $h=h_1+\cdots+h_m$ and $X=X_1 \cap \cdots \cap X_m$. For a counterexample, let $m=2$, and $X_1$ and $X_2$ be unit spheres in the plane with centers at $(-1,0)$ and $(1,0)$, respectively.

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|Optimality Condition with Directional Derivatives

$$D(x)={\alpha(\bar{x}-x) \mid \bar{x} \in X, \alpha>0} .$$

$$f^{\prime}(x ; d) \geq 0, \quad \forall d \in D(x) .$$

$$g^{\prime} d \geq 0, \quad \forall d \in D(x)$$

$$g^{\prime} d \geq 0, \quad \forall d \in \overline{D(x)}$$

$$\max {g \in \partial f(x)} \min {|d| \leq 1, d \in \overline{D(x)}} g^{\prime} d \geq 0$$

## 数学代写|凸优化作业代写Convex Optimization代考|Subdifferential of an Extended Real-Valued Function

$$f(x)= \begin{cases}h(x) & \text { if } x \in X, \ \infty & \text { if } x \notin X,\end{cases}$$

(a)使用第3.1.3和3.1.4节来证明该函数的子微分对于所有$x \in X$都是非空的，并且具有如下形式
$$\partial f(x)=\partial h(x)+N_X(x), \quad \forall x \in X,$$

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