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# 数学代写|凸优化代写Convex Optimization代考|Minimization with functional constraints

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## 数学代写|凸优化代写Convex Optimization代考|Minimization with functional constraints

Let us apply a subgradient method to a constrained minimization problem with functional constraints. Consider the problem
$$\min \left{f(x) \mid x \in Q, f_j(x) \leq 0, i=1 \ldots m\right}$$
with convex $f$ and $f_j$, and a simple bounded closed convex set $Q$ :
$$|x-y| \leq R, \quad \forall x, y \in Q .$$
Let us form an aggregate constraint $\bar{f}(x)=\left(\max {1 \leq j \leq m} f_j(x)\right){+}$. Then our problem can be written as follows:
$$\min {f(x) \mid x \in Q, \bar{f}(x) \leq 0} .$$
Note that we can easily compute a subgradient $\bar{g}(x)$ of function $\bar{f}$, provided that we can do so for functions $f_j$ (see Lemma 3.1.10).

Let us fix some $x^$, a solution to (3.2.11). Note that $\bar{f}\left(x^\right)=0$ and $v_{\bar{f}}\left(x^* ; x\right) \geq 0$ for all $x \in R^n$. Therefore, in view of Lemma 3.2.1 we have
$$\bar{f}(x) \leq \omega_{\bar{f}}\left(x^* ; v_{\bar{f}}\left(x^* ; x\right)\right) .$$
If $f_j$ are Lipschitz continuous on $Q$ with constant $M$, then for any $x$ from $R^n$ we have the estimate
$$\bar{f}(x) \leq M \cdot v_{\bar{f}}\left(x^* ; x\right) .$$
Let us write down a subgradient minimization scheme for constrained minimization problem (3.2.12). We assume that $R$ is known.

## 数学代写|凸优化代写Convex Optimization代考|Complexity bounds in finite dimension

Let us look at the unconstrained minimization problem again, assuming that its dimension is relatively small. This means that our computational resources allow us to perform the number of iterations of a minimization method, proportional to the dimension of the space of variables. What will be the lower complexity bounds in this case?
In this section we obtain a finite-dimensional lower complexity bound for a problem, which is closely related to minimization problem. This is the feasibility problem:
Find $x^* \in Q$,
where $Q$ is a convex set. We assume that this problem is endowed with an oracle, which answers our request at point $\bar{x} \in R^n$ in the following way:

• Either it reports that $\bar{x} \in Q$.
• Or, it returns a vector $\bar{g}$, separating $\bar{x}$ from $Q$ :
$$\langle\bar{g}, \bar{x}-x\rangle \geq 0 \quad \forall x \in Q .$$
To estimate the complexity of this problem, we introduce the following assumption.

Assumption 3.2.1 There exists a point $x^* \in Q$ such that for some $\epsilon>0$ the ball $B_2\left(x^*, \epsilon\right)$ belongs to $Q$.

For example, if we know an optimal value $f^$ for problem (3.2.3), we can treat this problem as a feasibility problem with $$\bar{Q}=\left{(t, x) \in R^{n+1} \mid t \geq f(x), t \leq f^+\bar{\epsilon}, x \in Q\right} .$$

## 学代写|凸优化代写Convex Optimization代考|Minimization with functional constraints

$$\min \left{f(x) \mid x \in Q, f_j(x) \leq 0, i=1 \ldots m\right}$$

$$|x-y| \leq R, \quad \forall x, y \in Q .$$

$$\min {f(x) \mid x \in Q, \bar{f}(x) \leq 0} .$$

$$\bar{f}(x) \leq \omega_{\bar{f}}\left(x^* ; v_{\bar{f}}\left(x^* ; x\right)\right) .$$

$$\bar{f}(x) \leq M \cdot v_{\bar{f}}\left(x^* ; x\right) .$$

## 数学代写|凸优化代写Convex Optimization代考|Complexity bounds in finite dimension

$$\langle\bar{g}, \bar{x}-x\rangle \geq 0 \quad \forall x \in Q .$$

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