# 数学代写|凸优化作业代写Convex Optimization代考|Incremental Newton Method with Diagonal Approximation

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## 数学代写|凸优化作业代写Convex Optimization代考|Incremental Newton Method with Diagonal Approximation

Generally, with proper implementation, the incremental Newton method is often substantially faster than the incremental gradient method, in terms of numbers of iterations (there are theoretical results suggesting this property for stochastic versions of the two methods; see the end-of-chapter references). However, in addition to computation of second derivatives, the incremental Newton method involves greater overhead per iteration due to matrix-vector calculations in Eqs. (2.44), (2.46), and (2.47), so it is suitable only for problems where $n$, the dimension of $x$, is relatively small.

One way to remedy in part this difficulty is to approximate $\nabla^2 f_i\left(\psi_{i, k}\right)$ by a diagonal matrix, and recursively update a diagonal approximation of $D_{i, k}$ using Eqs. (2.46) or (2.47). One possibility, inspired by similar diagonal scaling schemes for nonincremental gradient methods, is to set to 0 the off-diagonal components of $\nabla^2 f_i\left(\psi_{i, k}\right)$. In this case, the iteration (2.44) becomes a diagonally scaled version of the incremental gradient method, and involves comparable overhead per iteration (assuming the required diagonal second derivatives are easily computed). As an additional option, one may multiply the diagonal components with a stepsize parameter that is close to 1 and add a small positive constant (to bound them away from 0 ). Ordinarily, for the convex problems considered here, this method should require little experimentation with stepsize selection.

## 数学代写|凸优化作业代写Convex Optimization代考|Incremental Newton Methods with Constraints

The incremental Newton method can also be adapted to constrained problems of the form
\begin{aligned} & \operatorname{minimize} \sum_{i=1}^m f_i(x) \ & \text { subject to } x \in X \text {, } \ & \end{aligned}
where $f_i: \Re^n \mapsto \Re$ are convex, twice continuously differentiable convex functions. If $X$ has a relatively simple form, such as upper and lower bounds on the variables, one may use a two-metric implementation, such as the ones discussed earlier, whereby the matrix $D_{i, k}$ is partially diagonalized before it is applied to the iteration
$$\psi_{i, k}=P_X\left(\psi_{i-1, k}-D_{i, k} \nabla f_i\left(\psi_{i-1, k}\right)\right)$$
[cf. Eqs. (2.21) and (2.44)].
For more complicated constraint sets of the form
$$X=\cap_{i=1}^m X_i$$
where each $X_i$ is a relatively simple component constraint set (such as a halfspace), there is another possibility. This is to apply an incremental projected Newton iteration, with projection on a single individual component $X_i$, i.e., an iteration of the form
$$\psi_{i, k} \in \arg \min {\psi \in X_i}\left{\nabla f_i\left(\psi{i-1, k}\right)^{\prime}\left(\psi-\psi_{i-1, k}\right)+\frac{1}{2}\left(\psi-\psi_{i-1, k}\right)^{\prime} H_{i, k}\left(\psi-\psi_{i-1, k}\right)\right},$$
where
$$H_{i, k}=\sum_{\ell=1}^i \nabla^2 f_{\ell}\left(\psi_{\ell-1, k}\right)$$

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|Incremental Newton Methods with Constraints

\begin{aligned} & \operatorname{minimize} \sum_{i=1}^m f_i(x) \ & \text { subject to } x \in X \text {, } \ & \end{aligned}

$$\psi_{i, k}=P_X\left(\psi_{i-1, k}-D_{i, k} \nabla f_i\left(\psi_{i-1, k}\right)\right)$$

$$X=\cap_{i=1}^m X_i$$

$$\psi_{i, k} \in \arg \min {\psi \in X_i}\left{\nabla f_i\left(\psi{i-1, k}\right)^{\prime}\left(\psi-\psi_{i-1, k}\right)+\frac{1}{2}\left(\psi-\psi_{i-1, k}\right)^{\prime} H_{i, k}\left(\psi-\psi_{i-1, k}\right)\right},$$

$$H_{i, k}=\sum_{\ell=1}^i \nabla^2 f_{\ell}\left(\psi_{\ell-1, k}\right)$$

## 有限元方法代写

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## MATLAB代写

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