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# 计算机代写|计算机视觉代写Computer Vision代考|EECS498-007 Iterative Multidimensional Optimization: General Proceeding

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## 计算机代写|计算机视觉代写Computer Vision代考|Iterative Multidimensional Optimization: General Proceeding

The techniques presented in this chapter are universally applicable, because they:

Directly operate on the energy function $f(x)$.

Do not rely on a special structure of $f(x)$.
The broadest possible applicability can be achieved if only information about the values of $f(x)$ themselves is required.

The other side of the coin is that these methods are usually rather slow. Therefore, some of the methods of the following sections additionally utilize knowledge about the first- and second-order derivatives $f^{\prime}(x)$ and $f^{\prime \prime}(x)$. This restricts their applicability a bit in exchange for an acceleration of the solution calculation. As was already stated earlier, it usually is best to make use of as much specific knowledge about the optimization problem at hand as possible. The usage of this knowledge will typically lead to faster algorithms.

Furthermore, as the methods of this chapter typically perform a local search, there is no guarantee that the global optimum is found. In order to avoid to get stuck in a local minimum, a reasonably good first estimate $x_0$ of the minimal position $x^*$ is required for those methods.

The general proceeding of most of the methods presented in this chapter is a two-stage iterative approach. Usually, the function $f(\mathbf{x})$ is vector valued, i.e., $\mathbf{x}$ consists of multiple ( say $N$ ) elements. Hence, the optimization procedure has to find $N$ values simultaneously. Starting at an initial solution $\mathbf{x}^k$, this can be done by an iterated application of the following two steps:

Calculation of a so-called search direction $\mathbf{s}^k$ along which the minimal position is to be searched.

Update the solution by finding a $\mathbf{x}^{k+1}$ which reduces $f\left(\mathbf{x}^{k+1}\right)$ (compared to $f\left(\mathbf{x}^k\right)$ ) by performing a one-dimensional search along the direction $\mathbf{s}^k$. Because $\mathbf{s}^k$ remains fixed during one iteration, this step is also called a line search. Mathematically, this can be written as
$$\mathbf{x}^{k+1}=\mathbf{x}^k+\alpha^k \cdot \mathbf{s}^k$$
where, in the most simple case, $\alpha^k$ is a fixed step size or – more sophisticated – is estimated during the one-dimensional search such that $\mathbf{x}^{k+1}$ minimizes the objective function along the search direction $\mathbf{s}^k$. The repeated procedure stops either if convergence is achieved, i.e., $f\left(\mathbf{x}^{k+1}\right)$ is sufficiently close to $f\left(\mathbf{x}^k\right)$ and hence we assume that no more progress is possible, or if the number of iterations exceeds an upper threshold $K_{\max }$. This iterative process is visualized in Fig. $2.3$ with the help of a rather simple example objective function.

## 计算机代写|计算机视觉代写Computer Vision代考|One-Dimensional Optimization Along a Search Direction

In this section, step two of the general iterative procedure is discussed in more detail. The outline of the proceeding of this step is to first bound the solution and subsequently iteratively narrow down the bounds until it is possible to interpolate the solution with good accuracy. In more detail, the one-dimensional optimization comprises the following steps (see also flowchart of Fig. 2.4):

1. Determine upper and lower bounds $\mathbf{x}_u^k=\mathbf{x}^k+\alpha_u^0 \cdot \mathbf{s}^k$ and $\mathbf{x}_l^k=\mathbf{x}^k+\alpha_l^0 \cdot \mathbf{s}^k$ which bound the current solution $\mathbf{x}^k$ ( $\alpha_u^0$ and $\alpha_l^0$ have opposite signs).
2. Recalculate the upper and lower bounds $\mathbf{x}_u^i$ and $\mathbf{x}_l^i$ in an iterative scheme, i.e., decrease the distance between $\mathbf{x}_u^i$ and $\mathbf{x}_l^i$ as the iteration proceeds. Please note that, for a better understanding, the superscript $\boldsymbol{k}$ indicating the index of the multidimensional iteration is dropped and replaced by the index $i$ of the current one-dimensional iteration. One possibility to update the bounds is the so-called golden section method, which will be presented below. As the distance between the bounds is reduced by a fixed fraction at each iteration there, we can set the number of iterations necessary to a fixed value $N$.
3. Refine the solution by applying a polynomial interpolation to the data found in step 2, where the refined bounds $\mathbf{x}_u^i$ and $\mathbf{x}_l^i$ as well as two intermediate points were found (see below). Hence, we have four data points and can perform a cubic interpolation based on these four points.

## 计算机代写计算机视觉代写COMPUTER VISION代 考|ITERATIVE MULTIDIMENSIONAL OPTIMIZATION: GENERAL PROCEEDING

. Because $\mid$ mathbf []$^{\wedge} \mathrm{kremains}$ fixedduringoneiteration, thisstepisalsocalledalinesearch. Mathematically, thiscanbewrittenas $\mathrm{x}^{k+1}=\mathrm{x}^k+\alpha^k \cdot \mathrm{s}^k$
where, inthemostsimplecase, $\backslash$ alpha^kisafixedstepsizeor-moresophisticated-isestimatedduringtheone-dimensionalsearchsuchthat
andhenceweassumethatnomoreprogressispossible, orifthenumberofiterationsexceedsanupperthresholdK_{\最大}

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