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像爬山这样的简单的启发式方法，通过寻找更好的邻居来寻找更好的邻居，当他们达成一个没有更好的邻居的解决方案时就停止，不能保证会导致任何现有的更好的解决方案，他们的结果可能很容易地只是一个局部的最佳解决方案，而实际的最佳解决方案将是一个可能不。元启发法利用解的邻域作为探索解空间的方法，尽管它们更喜欢更好的邻域，但它们也接受更差的邻域以避免陷入局部最优; 如果运行足够长的时间，它们可以找到全局最优。

像爬山这样的简单的启发式方法，通过寻找更好的邻居来寻找更好的邻居，当他们达成一个没有更好的邻居的解决方案时就停止，不能保证会导致任何现有的更好的解决方案，他们的结果可能很容易地只是一个局部的最佳解决方案，而实际的最佳解决方案将是一个可能不。元启发法利用解的邻域作为探索解空间的方法，尽管它们更喜欢更好的邻域，但它们也接受更差的邻域以避免陷入局部最优; 如果运行足够长的时间，它们可以找到全局最优。

The probability of making the transition from the current state <math>s</math> to a candidate new state <math>s'</math> is specified by an acceptance probability function <math>P(e, e', T)</math>, that depends on the energies <math>e = E(s)</math> and <math>e' = E(s')</math> of the two states, and on a global time-varying parameter <math>T</math> called the temperature.  States with a smaller energy are better than those with a greater energy.  The probability function <math>P</math> must be positive even when <math>e'</math> is greater than <math>e</math>.  This feature prevents the method from becoming stuck at a local minimum that is worse than the global one.

The probability of making the transition from the current state <math>s</math> to a candidate new state <math>s'</math> is specified by an acceptance probability function <math>P(e, e', T)</math>, that depends on the energies <math>e = E(s)</math> and <math>e' = E(s')</math> of the two states, and on a global time-varying parameter <math>T</math> called the temperature.  States with a smaller energy are better than those with a greater energy.  The probability function <math>P</math> must be positive even when <math>e'</math> is greater than <math>e</math>.  This feature prevents the method from becoming stuck at a local minimum that is worse than the global one.

从当前状态<math>s</math>到候选状态 <math>s'</math>的转变概率是由接受概率密度函数<math>P(e, e', T)</math>确定的，这取决于两个状态的能量数学 e (s) / 数学 e (s’) / 数学，以及全球时变参数数学 t / 数学，称为温度。能量较小的状态比能量较大的状态要好。概率密度函数的数学 p / 数学必须是积极的，即使数学 e / 数学大于数学 e / 数学。这个特性可以防止方法陷入比全局最小值更糟糕的局部最小值。

从当前状态<math>s</math>到候选状态 <math>s'</math>的转变概率是由接受概率密度函数<math>P(e, e', T)</math>确定的，这取决于两个状态的能量<math>e = E(s)</math> 和<math>e' = E(s')</math>，以及称为温度的全局时变参数<math>T</math>。能量较小的状态比能量较大的状态好。概率密度函数<math>P</math>必须是正数，即使<math>e'</math>大于<math>e</math>。这个特性可以防止陷入比全局最小值糟糕的局部最小值。

When <math>T</math> tends to zero, the probability <math>P(e, e', T)</math> must tend to zero if <math>e' > e</math> and to a positive value otherwise. For sufficiently small values of <math>T</math>, the system will then increasingly favor moves that go "downhill" (i.e., to lower energy values), and avoid those that go "uphill."  With <math>T=0</math> the procedure reduces to the greedy algorithm, which makes only the downhill transitions.

When <math>T</math> tends to zero, the probability <math>P(e, e', T)</math> must tend to zero if <math>e' > e</math> and to a positive value otherwise. For sufficiently small values of <math>T</math>, the system will then increasingly favor moves that go "downhill" (i.e., to lower energy values), and avoid those that go "uphill."  With <math>T=0</math> the procedure reduces to the greedy algorithm, which makes only the downhill transitions.

当数学 t / math 趋于零时，如果 math e / math 趋于零，概率数学 p (e，e’ ，t) / math 必须趋于零，否则趋于正值。对于足够小的数学 t / math 值，系统会越来越倾向于走“下坡路”(即降低能量值) ，而避免走“上坡路”使用数学 t0 / math，过程简化为贪婪算法，只有下坡过渡。

当<math>T</math>趋于零时，如果<math>e' > e</math>，概率<math>P(e, e', T)</math>必须趋于零，否则趋于正值。对于足够小的<math>T</math>值，系统会越来越倾向于走“下坡路”(即降低能量值) ，而避免走“上坡路”。使用<math>T=0</math>的过程，只有下坡过渡，简化为贪婪算法。

In the original description of simulated annealing, the probability <math>P(e, e', T)</math> was equal to 1 when <math>e' < e</math>—i.e., the procedure always moved downhill when it found a way to do so, irrespective of the temperature.  Many descriptions and implementations of simulated annealing still take this condition as part of the method's definition.  However, this condition is not essential for the method to work.

In the original description of simulated annealing, the probability <math>P(e, e', T)</math> was equal to 1 when <math>e' < e</math>—i.e., the procedure always moved downhill when it found a way to do so, irrespective of the temperature.  Many descriptions and implementations of simulated annealing still take this condition as part of the method's definition.  However, this condition is not essential for the method to work.

在模拟退火的原始描述中，当数学 e’ e / math 时，概率数学 p (e，e’ ，t) / math 等于1。许多模拟退火的描述和实现仍然把这个条件作为方法定义的一部分。然而，这个条件并不是这个方法工作的必要条件。

在模拟退火的原始描述中，当<math>e' < e</math>时，概率<math>P(e, e', T)</math>等于1。即无论温度如何，程序总会找到一种方法，找到“下坡路”。许多模拟退火的描述和实现仍然把这个条件作为方法定义的一部分。然而，这个条件并不是这个方法工作的必要条件。

The <math>P</math> function is usually chosen so that the probability of accepting a move decreases when the difference

The <math>P</math> function is usually chosen so that the probability of accepting a move decreases when the difference

数学 p / 数学函数的选择通常是为了降低接受移动的概率

The <math>P</math>函数的选择通常是为了降低接受移动的概率

<math>e'-e</math> increases—that is, small uphill moves are more likely than large ones.  However, this requirement is not strictly necessary, provided that the above requirements are met.

<math>e'-e</math> increases—that is, small uphill moves are more likely than large ones.  However, this requirement is not strictly necessary, provided that the above requirements are met.

<math>e'-e</math> increases—that is, small uphill moves are more likely than large ones.  However, this requirement is not strictly necessary, provided that the above requirements are met.

<math>e'-e</math> increases—that is, small uphill moves are more likely than large ones.  However, this requirement is not strictly necessary, provided that the above requirements are met.

数学 e’-e / 数学增加ー也就是说，小的上坡动作比大的动作更有可能。但是，如果符合上述要求，这一要求并非绝对必要。

<math>e'-e</math>增加意味着小的上坡动作比大的动作更有可能发生。如果符合上述要求，这一要求并非绝对必要。

Given these properties, the temperature <math>T</math> plays a crucial role in controlling the evolution of the state <math>s</math> of the system with regard to its sensitivity to the variations of system energies. To be precise, for a large <math>T</math>, the evolution of <math>s</math> is sensitive to coarser energy variations, while it is sensitive to finer energy variations when <math>T</math> is small.

Given these properties, the temperature <math>T</math> plays a crucial role in controlling the evolution of the state <math>s</math> of the system with regard to its sensitivity to the variations of system energies. To be precise, for a large <math>T</math>, the evolution of <math>s</math> is sensitive to coarser energy variations, while it is sensitive to finer energy variations when <math>T</math> is small.

考虑到这些性质，温度数学 t / math 在控制系统状态数学的演化方面起着至关重要的作用，因为它对系统能量变化的敏感性。准确地说，对于一个庞大的数学 t / math 来说，数学的进化对粗糙的能量变化是敏感的，而当数学 t / math 很小时，它对精细的能量变化是敏感的。

考虑到这些性质，温度<math>T</math>在控制系统状态<math>s</math>的演化方面起着至关重要的作用，因为它对系统能量变化很敏感。准确地说，对于一个庞大的数学<math>T</math>来说，<math>s</math>的变化对比粗略的能量变化更为敏感，当<math>T</math>很小时，它对精细能量变化更为敏感。

===The annealing schedule===

===The annealing schedule===