Posted on Categories:Convex optimization, 凸优化, 数学代写

# 数学代写|凸优化代写Convex Optimization代考|Constrained Minimization

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## 数学代写|凸优化代写Convex Optimization代考|Constrained Minimization

Let us show how to use piece-wise linear models to solve constrained minimization problems. Consider the problem
$$\begin{gathered} \qquad \min _{x \in Q} f(x), \ \text { s.t. } f_j(x) \leq 0, j=1 \ldots m, \end{gathered}$$
where $Q$ is a bounded closed convex set, and functions $f(\cdot), f_j(\cdot)$ are Lipschitz continuous on $Q$.

Note that the functions $f(\cdot)$ and $\bar{f}(\cdot)$ are convex and Lipschitz continuous. In this section, we will try to solve (3.3.5) using the models for both of them.

Let us define the corresponding models. Consider a sequence $X=\left{x_k\right}_{k=0}^{\infty}$. Define
\begin{aligned} & \hat{f}k(X ; x)=\max {0 \leq j \leq k}\left[f\left(x_j\right)+\left\langle g\left(x_j\right), x-x_j\right\rangle\right] \leq f(x), \ & \check{f}k(X ; x)=\max {0 \leq j \leq k}\left[\bar{f}\left(x_j\right)+\left\langle\bar{g}\left(x_j\right), x-x_j\right\rangle\right] \leq \bar{f}(x), \end{aligned}
where $g\left(x_j\right) \in \partial f\left(x_j\right)$ and $\bar{g}\left(x_j\right) \in \partial \bar{f}\left(x_j\right)$.
As in Sect. 2.3.4, our scheme is based on the parametric function
\begin{aligned} f(t ; x) & =\max {f(x)-t, \bar{f}(x)}, \ f^*(t) & =\min _{x \in Q} f(t ; x) . \end{aligned}

## 数学代写|凸优化代写Convex Optimization代考|Cubic Regularization of Quadratic Approximation

In this section, we consider the simplest unconstrained minimization problem
$$\min _{x \in \mathbb{R}^n} f(x)$$

with a twice continuously differentiable objective function. The standard secondorder scheme for this problem, Newton’s method, is as follows:
$$x_{k+1}=x_k-\left[\nabla^2 f\left(x_k\right)\right]^{-1} \nabla f\left(x_k\right)$$
We have already looked at this method in Sect. 1.2.
Despite its very natural motivation, this scheme has several hidden drawbacks. First of all, it may happen that at the current test point the Hessian is degenerate; in this case the method is not well-defined. Secondly, it may happen that this scheme diverges or converges to a saddle point or even to a point of local maximum. In order to overcome these difficulties, there are three standard recipes.

• Levenberg-Marquardt regularization. If $\nabla^2 f\left(x_k\right)$ is indefinite, let us regularize it with a unit matrix. Namely, use the matrix $G_k=\nabla^2 f\left(x_k\right)+\gamma I_n \succ 0$ in order to perform the step:
$$x_{k+1}=x_k-G_k^{-1} \nabla f\left(x_k\right)$$
This strategy is sometimes considered as a way of mixing Newton’s method with the gradient method.
• Line search. Since we are interested in minimization, it is reasonable to introduce in method (4.1.1) a certain step size $h_k>0$ :
$$x_{k+1}=x_k-h_k\left[\nabla^2 f\left(x_k\right)\right]^{-1} \nabla f\left(x_k\right)$$

## 数学代写|凸优化代写Convex Optimization代考|Constrained Minimization

$$\begin{gathered} \qquad \min _{x \in Q} f(x), \ \text { s.t. } f_j(x) \leq 0, j=1 \ldots m, \end{gathered}$$

## 数学代写|凸优化代写Convex Optimization代考|Cubic Regularization of Quadratic Approximation

$$\min _{x \in \mathbb{R}^n} f(x)$$

$$x_{k+1}=x_k-\left[\nabla^2 f\left(x_k\right)\right]^{-1} \nabla f\left(x_k\right)$$

Levenberg-Marquardt正则化。如果$\nabla^2 f\left(x_k\right)$是不定的，让我们用单位矩阵正则化它。也就是说，使用矩阵$G_k=\nabla^2 f\left(x_k\right)+\gamma I_n \succ 0$来执行步骤:
$$x_{k+1}=x_k-G_k^{-1} \nabla f\left(x_k\right)$$

$$x_{k+1}=x_k-h_k\left[\nabla^2 f\left(x_k\right)\right]^{-1} \nabla f\left(x_k\right)$$

## MATLAB代写

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