实用遗传算法第二版：MATLAB实现与混合优化

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"Practical Genetic Algorithms second edition" 是一本关于遗传算法及其在工程和科学领域应用的书籍，由 Randy L. Haupt 和 Sue Ellen Haupt 合著。本书第二版新增了 MATLAB 代码，深入讨论了混合遗传算法，并提供了比第一版更多的实例。 **遗传算法**（Genetic Algorithms）是一种受生物进化过程启发的优化技术，它模拟了自然选择、遗传和突变等生物学过程来搜索问题空间中的最优解。遗传算法通常用于解决复杂的、多模态的优化问题，它们能够处理非线性、非凸和约束条件下的问题。 **优化**（Optimization）是寻找数学模型中最佳参数的过程，目标是最大化或最小化一个函数。遗传算法在优化领域的应用是通过编码个体（代表可能的解决方案），并使用适应度函数来评估这些个体的优劣，进而通过选择、交叉和变异操作迭代地改进种群。 **MATLAB** 是一种广泛使用的数值计算软件，它提供了一个强大的环境来进行数值分析、符号计算、数据可视化和算法开发。在本书中，MATLAB 代码的提供使得读者可以直接实践和理解遗传算法的实现细节，加深对算法原理的理解。 **混合遗传算法**（Hybrid Genetic Algorithms）是结合了其他优化技术（如局部搜索、模拟退火、粒子群优化等）的遗传算法，旨在克服纯遗传算法可能遇到的早熟收敛和局部最优问题。通过结合多种策略，混合算法通常能更有效地探索解决方案空间，提高找到全局最优解的可能性。书中内容包括但不限于以下几个方面： 1. **遗传算法基础**：介绍遗传算法的基本概念，如编码、适应度函数、选择、交叉和变异操作，以及如何初始化种群。 2. **MATLAB实现**：通过具体的MATLAB代码示例，演示如何编程实现遗传算法，帮助读者掌握实际操作。 3. **问题实例**：提供各种工程和科学问题的应用实例，如电路设计、调度问题、机器学习等，展示遗传算法的实际应用效果。 4. **混合算法**：探讨如何构建和设计混合遗传算法，以及如何选择和调整组合的优化方法。 5. **性能评估**：讨论如何评估遗传算法的性能，包括收敛速度、解决方案质量和稳定性。 6. **应用领域**：涵盖多个学科的应用，如统计计算、仿真、系统科学、智能与模糊系统等，显示遗传算法的广泛适用性。这本书适合统计学家、计算机科学家、工程师以及对优化问题感兴趣的读者。无论是作为课程教材还是自学指南，它都能提供深入而实用的知识，帮助读者理解和应用遗传算法来解决实际问题。

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1.1.2 Root Finding versus Optimization

Approaches to optimization are akin to root or zero ﬁnding methods, only

harder. Bracketing the root or optimum is a major step in hunting it down.

For the one-variable case, ﬁnding one positive point and one negative point

brackets the zero. On the other hand, bracketing a minimum requires three

points, with the middle point having a lower value than either end point. In

the mathematical approach, root ﬁnding searches for zeros of a function, while

optimization ﬁnds zeros of the function derivative. Finding the function deriv-

ative adds one more step to the optimization process. Many times the deriva-

tive does not exist or is very difﬁcult to ﬁnd. We like the simplicity of root

ﬁnding problems, so we teach root ﬁnding techniques to students of engi-

neering, math, and science courses. Many technical problems are formulated

to ﬁnd roots when they might be more naturally posed as optimization

problems; except the toolbox containing the optimization tools is small and

inadequate.

Another difﬁculty with optimization is determining if a given minimum is

the best (global) minimum or a suboptimal (local) minimum. Root ﬁnding

doesn’t have this difﬁculty. One root is as good as another, since all roots force

the function to zero.

Finding the minimum of a nonlinear function is especially difﬁcult. Typical

approaches to highly nonlinear problems involve either linearizing the

problem in a very conﬁned region or restricting the optimization to a small

region. In short, we cheat.

1.1.3 Categories of Optimization

Figure 1.2 divides optimization algorithms into six categories. None of

these six views or their branches are necessarily mutually exclusive. For

instance, a dynamic optimization problem could be either constrained or

FINDING THE BEST SOLUTION 3

Figure 1.2 Six categories of optimization algorithms.

unconstrained. In addition some of the variables may be discrete and others

continuous. Let’s begin at the top left of Figure 1.2 and work our way around

clockwise.

1. Trial-and-error optimization refers to the process of adjusting variables

that affect the output without knowing much about the process that produces

the output. A simple example is adjusting the rabbit ears on a TV to get the

best picture and audio reception. An antenna engineer can only guess at why

certain contortions of the rabbit ears result in a better picture than other con-

tortions. Experimentalists prefer this approach. Many great discoveries, like

the discovery and reﬁnement of penicillin as an antibiotic, resulted from the

trial-and-error approach to optimization. In contrast, a mathematical formula

describes the objective function in function optimization. Various mathemat-

ical manipulations of the function lead to the optimal solution. Theoreticians

love this theoretical approach.

2. If there is only one variable, the optimization is one-dimensional. A

problem having more than one variable requires multidimensional optimiza-

tion. Optimization becomes increasingly difﬁcult as the number of dimensions

increases. Many multidimensional optimization approaches generalize to a

series of one-dimensional approaches.

3. Dynamic optimization means that the output is a function of time, while

static means that the output is independent of time. When living in the suburbs

of Boston, there were several ways to drive back and forth to work. What was

the best route? From a distance point of view, the problem is static, and the

solution can be found using a map or the odometer of a car. In practice, this

problem is not simple because of the myriad of variations in the routes. The

shortest route isn’t necessarily the fastest route. Finding the fastest route is a

dynamic problem whose solution depends on the time of day, the weather,

accidents, and so on. The static problem is difﬁcult to solve for the best solu-

tion, but the added dimension of time increases the challenge of solving the

dynamic problem.

4. Optimization can also be distinguished by either discrete or continuous

variables. Discrete variables have only a ﬁnite number of possible values,

whereas continuous variables have an inﬁnite number of possible values. If we

are deciding in what order to attack a series of tasks on a list, discrete opti-

mization is employed. Discrete variable optimization is also known as com-

binatorial optimization, because the optimum solution consists of a certain

combination of variables from the ﬁnite pool of all possible variables.

However, if we are trying to ﬁnd the minimum value of f(x) on a number line,

it is more appropriate to view the problem as continuous.

5. Variables often have limits or constraints. Constrained optimization

incorporates variable equalities and inequalities into the cost function. Uncon-

strained optimization allows the variables to take any value. A constrained

variable often converts into an unconstrained variable through a transforma-

4 INTRODUCTION TO OPTIMIZATION

tion of variables. Most numerical optimization routines work best with uncon-

strained variables. Consider the simple constrained example of minimizing

f(x) over the interval -1 £ x £ 1. The variable converts x into an unconstrained

variable u by letting x = sin(u) and minimizing f(sin(u)) for any value of u.

When constrained optimization formulates variables in terms of linear

equations and linear constraints, it is called a linear program. When the cost

equations or constraints are nonlinear, the problem becomes a nonlinear

programming problem.

6. Some algorithms try to minimize the cost by starting from an initial

set of variable values. These minimum seekers easily get stuck in local minima

but tend to be fast. They are the traditional optimization algorithms and are

generally based on calculus methods. Moving from one variable set to another

is based on some determinant sequence of steps. On the other hand, random

methods use some probabilistic calculations to ﬁnd variable sets. They tend to

be slower but have greater success at ﬁnding the global minimum.

1.2 MINIMUM-SEEKING ALGORITHMS

Searching the cost surface (all possible function values) for the minimum cost

lies at the heart of all optimization routines. Usually a cost surface has many

peaks, valleys, and ridges. An optimization algorithm works much like a hiker

trying to ﬁnd the minimum altitude in Rocky Mountain National Park. Start-

ing at some random location within the park, the goal is to intelligently

proceed to ﬁnd the minimum altitude. There are many ways to hike or glis-

sade to the bottom from a single random point. Once the bottom is found,

however, there is no guarantee that an even lower point doesn’t lie over the

next ridge. Certain constraints, such as cliffs and bears, inﬂuence the path of

the search as well. Pure downhill approaches usually fail to ﬁnd the global

optimum unless the cost surface is quadratic (bowl-shaped).

There are many good texts that describe optimization methods (e.g.,

Press et al., 1992; Cuthbert, 1987). A history is given by Boyer and Merzbach

(1991). Here we give a very brief review of the development of optimization

strategies.

1.2.1 Exhaustive Search

The brute force approach to optimization looks at a sufﬁciently ﬁne sam-

pling of the cost function to ﬁnd the global minimum. It is equivalent to spend-

ing the time, effort, and resources to thoroughly survey Rocky Mountain

National Park. In effect a topographical map can be generated by connecting

lines of equal elevation from an interpolation of the sampled points.

This exhaustive search requires an extremely large number of cost function

evaluations to ﬁnd the optimum. For example, consider solving the two-

dimensional problem

MINIMUM-SEEKING ALGORITHMS 5

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