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# 数学代写|凸优化代写Convex Optimization代考|Gradient Methods for Differentiable Unconstrained Minimization

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## 数学代写|凸优化代写Convex Optimization代考|Gradient Methods for Differentiable Unconstrained Minimization

For the case where $f$ is differentiable and $X=\Re^n$, there are many popular descent algorithms of the form (2.6). An important example is the classical gradient method, where we use $d_k=-\nabla f\left(x_k\right)$ in Eq. (2.6):
$$x_{k+1}=x_k-\alpha_k \nabla f\left(x_k\right)$$
Since for differentiable $f$ we have
$$f^{\prime}\left(x_k ; d\right)=\nabla f\left(x_k\right)^{\prime} d$$
it follows that
$$-\frac{\nabla f\left(x_k\right)}{\left|\nabla f\left(x_k\right)\right|}=\arg \min _{|d| \leq 1} f^{\prime}\left(x_k ; d\right)$$
[assuming $\nabla f\left(x_k\right) \neq 0$ ]. Thus the gradient method is the descent algorithm of the form (2.6) that uses the direction that yields the greatest rate of cost improvement. For this reason it is also called the method of steepest descent.
Let us now discuss the convergence rate of the steepest descent method, assuming that $f$ is twice continuously differentiable. With proper stepsize choice, it can be shown that the method has a linear rate, assuming that it generates a sequence $\left{x_k\right}$ that converges to a vector $x^$ such that $\nabla f\left(x^\right)=0$ and $\nabla^2 f\left(x^\right)$ is positive definite. For example, if $\alpha_k$ is a sufficiently small constant $\alpha>0$, the corresponding iteration $$x_{k+1}=x_k-\alpha \nabla f\left(x_k\right)$$ can be shown to be contractive within a sphere centered at $x^$, so it converges linearly.
To get a sense of this, assume for convenience that $f$ is quadratic, $\dagger$ so by adding a suitable constant to $f$, we have
$$f(x)=\frac{1}{2}\left(x-x^\right)^{\prime} Q\left(x-x^\right), \quad \nabla f(x)=Q\left(x-x^*\right),$$
where $Q$ is the positive definite symmetric Hessian of $f$. Then for a constant stepsize $\alpha$, the steepest descent iteration (2.7) can be written as
$$x_{k+1}-x^=(I-\alpha Q)\left(x_k-x^\right) .$$

## 数学代写|凸优化代写Convex Optimization代考|Scaling

To improve the convergence rate of the steepest descent method one may “scale” the gradient $\nabla f\left(x_k\right)$ by multiplication with a positive definite symmetric matrix $D_k$, i.e., use a direction $d_k=-D_k \nabla f\left(x_k\right)$, leading to the algorithm
$$x_{k+1}=x_k-\alpha_k D_k \nabla f\left(x_k\right) ;$$
cf. Fig. 2.1.1. Since for $\nabla f\left(x_k\right) \neq 0$ we have
$$f^{\prime}\left(x_k ; d_k\right)=-\nabla f\left(x_k\right)^{\prime} D_k \nabla f\left(x_k\right)<0,$$
it follows that we still have a cost descent method, as long as the positive stepsize $\alpha_k$ is sufficiently small so that $f\left(x_{k+1}\right)<f\left(x_k\right)$.
$$d_k=-D_k \nabla f\left(x_k\right),$$
where $D_k$ is a positive definite matrix, is a descent direction because $d_k^{\prime} \nabla f\left(x_k\right)=$ $-d_k^{\prime} D_k d_k<0$. In this case $d_k$ makes an angle less than $\pi / 2$ with $-\nabla f\left(x_k\right)$.
Scaling is a major concept in the algorithmic theory of nonlinear programming. It is motivated by the idea of modifying the “effective condition number” of the problem through a linear change of variables of the form $x=D_k^{1 / 2} y$. In particular, the iteration (2.11) may be viewed as a steepest descent iteration
$$y_{k+1}=y_k-\alpha \nabla h_k\left(y_k\right)$$
for the equivalent problem of minimizing the function $h_k(y)=f\left(D_k^{1 / 2} y\right)$. For a quadratic problem, where $f(x)=\frac{1}{2} x^{\prime} Q x-b^{\prime} x$, the condition number of $h_k$ is the ratio of largest to smallest eigenvalue of the matrix $D_k^{1 / 2} Q D_k^{1 / 2}$ (rather than $Q$ ).

## 数学代写|凸优化代写Convex Optimization代考|Gradient Methods for Differentiable Unconstrained Minimization

$$x_{k+1}=x_k-\alpha_k \nabla f\left(x_k\right)$$

$$f^{\prime}\left(x_k ; d\right)=\nabla f\left(x_k\right)^{\prime} d$$

$$-\frac{\nabla f\left(x_k\right)}{\left|\nabla f\left(x_k\right)\right|}=\arg \min {|d| \leq 1} f^{\prime}\left(x_k ; d\right)$$ [假设$\nabla f\left(x_k\right) \neq 0$]。因此，梯度方法是(2.6)式的下降算法，它使用产生最大成本改进率的方向。由于这个原因，它也被称为最陡下降法。 现在让我们讨论最陡下降法的收敛速度，假设$f$是两次连续可微的。通过适当的步长选择，可以证明该方法具有线性速率，假设它生成的序列$\left{x_k\right}$收敛于一个向量$x^$，使得$\nabla f\left(x^\right)=0$和$\nabla^2 f\left(x^\right)$是正定的。例如，如果$\alpha_k$是一个足够小的常数$\alpha>0$，则相应的迭代$$x{k+1}=x_k-\alpha \nabla f\left(x_k\right)$$可以在以$x^$为中心的球体内被证明是收缩的，因此它是线性收敛的。

$$f(x)=\frac{1}{2}\left(x-x^\right)^{\prime} Q\left(x-x^\right), \quad \nabla f(x)=Q\left(x-x^*\right),$$

$$x_{k+1}-x^=(I-\alpha Q)\left(x_k-x^\right) .$$

## 数学代写|凸优化代写Convex Optimization代考|Scaling

$$x_{k+1}=x_k-\alpha_k D_k \nabla f\left(x_k\right) ;$$

$$f^{\prime}\left(x_k ; d_k\right)=-\nabla f\left(x_k\right)^{\prime} D_k \nabla f\left(x_k\right)<0,$$

$$d_k=-D_k \nabla f\left(x_k\right),$$

## MATLAB代写

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