论文解读：自适应参数控制方案介绍

[算法 1] A novel approach in parameter adaptive and diversity maintenance for genetic algorithms (APGAIN)

1. The techniques of PRAM

A probabilistic rule-driven adaptive model (PRAM) is used to adapt the crossover rate and the mutation rate automatically, where three different parameter values are used on the population for the adaptation. These values will be adjusted automatically during the search by a set of rules according to the fitness improvement gained.

PRAM can be applied to adapting both pc and pm by running two PRAMs concurrently, each PRAM is responsible for adapting a parameter.

Step 1: PRAM uses three parameter values ($v_0$; $v_1$; $v_2$ with $v_0<v_1<v_2$) to adapt a control parameter (pm or pc). The run of a GA is divided into epochs.
Step 2: Each epoch is further divided into two periods. During the first period, the corresponding genetic operations (crossover or mutation) are applied on the chromosomes using any one of the three values randomly. In the second period, the probabilities of applying any one of the three values are proportional to their fitness improvement gained in the first period.
Step 3: At the end of each epoch, a set of ‘‘greedy rules’’ are applied to alter the values of $v_0$; $v_1$; $v_2$ according to three fitness improvement rules: $Expandrule$, $Stayrule$, $Leftmoverule$, and $Rightmoverule$;

2. The repelling algorithm

The repelling algorithm samples a population to a ‘‘representative’’. Population diversity is maintained by driving the population away from the representative. The fitness of a chromosome is modified to include the ‘‘diversity fitness’’, which is inversely proportional to the similarity between the chromosome and the representative. In other wards, the population diversity is maintained by giving a higher survival probability to chromosomes located in a sparsely populated region.

Step 1: The algorithm modifies the fitness evaluation function to increase the survival opportunity of chromosomes with almost lost alleles. The modified fitness of a chromosome ($f_E$) is given as:
$$f_E(\omega ,r_{obj},r_{div})=(1-\omega )\times f_{rankOjb}(r_{obj})+\omega \times f_{rankDiv}(r_{div}),$$ where $\omega$ is the weight of diversity, $r_{ojb}$ ($r_{div}$) is the ranking of the objective fitness (diversity fitness) of the chromosome, $f_{rankOjb}$ ($f_{rankDiv}$) returns a score directly proportional to the ranking of the objective fitness (diversity fitness).

Step 2: The diversity fitness is evaluated by comparing the chromosome with the population’s representative. The representative is represented by an array of real numbers in the range of [0, 1]. Assuming binary chromosomes, the probability of finding a ‘0’ at each allele of the chromosome is assigned to the corresponding array element of the representative. The algorithm to create the representative of a population in the repelling model is as follows.

Step 3: Get the diversity fitness of a chromosome by rewarding the chromosome that are dissimilar to the representative.

Step 4: The replacement scheme after a genetic operation (crossover or mutation) is done by tournament between the offspring and the parents.

3. The lazy repelling algorithm

A lazy repelling algorithm is proposed to further reduce the computational overheads. The diversity fitness is evaluated only if the following condition holds:
$$\frac{{\Sigma }_{i=1}^{\beta }|R^{\prime }[i{]}^2-R_{old}[i{]}^2|}{\beta }\ge \phi$$
where $\phi$ is lazy threshold, $R^{\prime }[i]$ is the $i$-th allele of the latest representative, $R_{old}[i]$ is the $i$-th allele of the $old$ representative computed last time when the diversity fitness is evaluated, $\beta$ is the bit-length of a chromosome.

[算法 2] A Compass to guide genetic algorithms

1. Steps of the compass

Step 1: Let $d_{it}$ and $q_{it}$ be the population’s mean diversity variation and quality (fitness) variation, respectively, Define a vector $o_{it}=(d_{it},q_{it})$ to characterize the effects of the operator over the population in terms of variation of quality and diversity (axis $\Delta D$ and $\Delta Q$);
Step 2: then normalize d i t n = d i t m a x i { ∣ d i t ∣ } d^n_{it} = \frac{d_{it}}{max_i\{|d_{it}|\}} ditn​=maxi​{ ∣dit​∣}dit​​ and q i t n = q i t m a x i { ∣ q i t ∣ } q^n_{it} = \frac{q_{it}}{max_i\{|q_{it}|\}} qitn​=maxi​{ ∣qit​∣}qit​​, we have $o_{it}^n=(d_{it}^n,q_{it}^n)$

Algorithms that just consider the fitness improvement to adjust the operator probabilities would only use the projection of $o_{it}$ over the y-axis (dotted lines in (Fig. b)). On the other hand, if diversity is solely taken in account, measures would be considered as the projection over the x-axis (Fig. c).

Step 3: To control the fitness improvement and the diversity together, a vector $c$ (defined by its angle $\theta \in [0,\frac{\pi }{2}]$) that characterizes also its orthogonal plane P (see Fig. d).

Rewards are then based on the projection of vectors $o_{it}^n$ over $c$, i.e., $|o_{it}^n|cos({\alpha }_{it})$, ${\alpha }_{it}$ being the angle between $o_{it}^n$ and $c$. A value of $\theta$ close to $0$ encourage exploration, while a value close to $\pi /2$ will favor exploitation. In this way, the management of application rates is abstracted by the angle $\theta$, that guides the direction of the search as the needle of a compass shows the north.

Step 4: Projections are turned into positive values by subtracting the smallest one and dividing them by execution time, in order to award faster operators (Fig. e).
δ i t = ∣ o i t n ∣ c o s ( α i t ) − m i n i { ∣ o i t n ∣ c o s ( α i t ) } T i t \delta_{it} = \frac{|o^n_{it}|cos(\alpha_{it}) - min_i\{|o^n_{it}|cos(\alpha_{it})\}}{T_{it}} δit​=Tit​∣oitn​∣cos(αit​)−mini​{ ∣o