~Russian Bear’Z Blog~

The question of the utilization of optimization for the strategies is disputable but I don t want to make you change your mind. I just want to mention that if you have invented some strategy without any parameters it will work only for having some historical data. And its impossible to draw a conclusion concerning its utilization. And having an assumption that some connection exists the dynamics of the alteration of the quotations with the past you can empirically select the best solutions (it can be either a single strategy or a set of it, reliant from the parameters). But there may be a situation when there are a lot of parameters or there are a little data under test, on which the resolution may behave aloof. It is examined by the carrying out the additional tests on other data.

Let’s examine in details the search of the optimum conclusion. It is obvious that the main objective of the quality of the strategy is a future profit that is why if you simplify the terms of the task by varying out the tests only for the historical data you must not be guided only by this single parameter as it does not reflect the main essence of the task. Thus, it is necessary to analyze such a criteria which can be regarded as a stable in respect of data. Such objectives are: Profit Factor, Drawdown, Percent Profitable, Ratio Avg. Win/Avg. Loss etc. But at the same time having obtained a conclusion which is optimum in respect of one objective it does not necessarily mean that it will be optimum for another one. Nevertheless you would like to solve the following task: to minimize the risk and maximize the profit at the same time.

Continuation

Such a target setting is proper and may have a solution. It may look like the number of solutions or just a multitude. For example, the solution of the dual tasks of the Markovits method concerning investment portfolio will be the special case of solving multiobjective problem of minimizing the risk and maximizing the profit. The solution of this problem is a Efficient Frontier. I will try to explain this concept in a word for those who are not acquainted with the multiobjective analysis.

Let’s assume that we have n criterias that can be conceived as a space Rⁿ.

Element x sorts with y as:

1. “better”, ∀i∈[1,n]: x_i≥y_i ,∃k∈[1,n]: x_k>y_k.
2. “not better”, ∃k∈[1,n]: x_kk.
3. “worse”, ∀i∈[1,n]: x_i≤ y_i ,∃k∈[1,n]: x_kk.
4. “not worse”, ∃k∈[1,n]: x_k>y_k.

Element x*∈X - is called an Pareto-optimal point if doesn’t exist such x∈X that can be better than x*.
Pareto Set – is a host of elements which are Pareto-optimal point.

Pareto Set is also known as efficient, non-dominated, or non-inferior points.

Example 1. Let’s assume we have a set of 2 criteria: (1,2), (2,2), (1,3), (2, 1), (1,1), (2,2). Pareto Set for this assortment will be (2,2), (1,3), (2,2), and if we add (3,2), will remain (3,2) and (1,3).
Observation: Pareto Set consists of subset of all elements and each of them is not worse than the others.
Example 2. Circumference in R²:

Fig. 1: Pareto Set for circumference.

Pareto Set for this circumference will be the arch pointed out in the picture. (Fig. 1).

Example 3. The result of optimization of the strategy for Sberbank by means of Genetic Algorithm using the criteria Profit Factor and Drawdown (thay can be risk and value). (Fig. 2):

Fig. 2: Pareto Set for 2 criteria.

Thus, Pareto Set may either consist of solutions that are optimal according to 1 of the guidelines or of the intermediate versions that are not so interesting for consideration. It is apparent from the example, where an optimum solution ( 2.91, -657), is that marked of by the left note. At the same time it is intuitively obvious that the other solution (3.87, -658) has the better exponents and both of them belongs to Pareto Set.

Example 4. The result of optimization having the same data and using 3 criteria: Profit Factor, Drawdown and Net Profit (Fig. 3):

Fig. 3: Pareto Set for 3 criteria.
(Axis: X – Profit Factor, Y – Drawdown, Z – Net Profit).

In this case Pareto Set forms the accumulation in some areas, but on the whole it is presented by the more chaotic multitude (red balls) according to which it is difficult to draw some conclusions. The result is more of experimental character and is interesting if you want to determine the structure of this multitude and the self-descriptiveness of the criteria in this combination.

If you want to find Pareto Set in the final aggregates you may use the enumeration of possibilities as a universal method of a Genetic Algorithm. In the second case you may carry out a search without any restrictions on the cardinal number. In order to show how you can apply the genetic algorithm for the solution of multiobjectives problems you must first describe the way of its realization.

For the understanding of Genetic Algorithm you may be unaware of the objectives of the optimization but if you have a clear idea of GA it will help you to denote the aims and to look for the solutions. For example, some tasks can be solved using the method of gradient descent.
As a rule they are the unimodal tasks of optimization (we assume that they have only 1 extremum). The variety of tasks that can be solved with the help of Genetic Algorithm is not limited to the objectives of optimization. They can be the tasks of fragmentation on classes, identifications and the search of the scenery of the surfaces. But from the practical point of view they have proved to be very effective in solving the absolute tasks aimed at the search of the global extremum with the host of local magnitudes.

In this case any restrictions are not imposed on the criterion function except its existence in the range of definition.There are some other methods of solving such tasks but for all of them except the simple excess, exist the probability of the direct hit in the local extremum and any of these methods can not guarantee that the right solution has been found.

For the Genetic Algorithm this probability is smaller in comparison with other methods. That is you have to sacrifice the convergence in favor of the universality of the problem put by. For example, the function of the variable is defined on the whole straight (except 0) and amounts to 1. The task with such a criterion function is possible to solve only with the help of the analytical procedure. And if some method of optimization, including the fingering, coincides, Genetic Algorithm also will coincide. With the help of Genetic Algorithm you may solve the task of the constrained optimization, but you must take into account that this task must be solved by stages using the Lagrange Formula.

Let’s assume we have the task of the unconstrained optimization: on the definitional domain x∈X the criterion vector-function F(x) for the n criteria, is defined X equals R^m. Than let’s convert the area of phenotypes in R^m by the hyperplanes, so that we will have the final quantity of blocks; from this fragmentation depends the adequacy of solution. Each block can be confronted with the m-dimention vector that is formed by the number of the block along the axis. This vector is called a chromosome, and it’s position data - gen. A great amount of chromosomes form the discrete space of the genotypes, and genofond. In practice this procedure is executed not necessarily at the beginning as it may be executed in any moment when there is the necessity to change the adequacy of the calculations, but in this case you will have to define the fitness between the previous and new gene pools. The basic population is formed using the arbitrary set of chromosomes and afterwards its fitness is valued. The fitness of the chromosome is valued using the function F for the fitness phenotypes. The subsequent activities reflect the essence of the reproductive cycle that can be represented in this way:

1. 1. Selection of parents for the continuation of the population.
2. 2. Formation of couples from the extracts.
3. 3. Operator of crossover for the couples organized.
4. 4. Mutation.
5. 5. Estimation of fitness of all chromosomes.
6. 6. Selection of new population, than 1 if the conditions of the stop are not fulfilled.

Now let’s discuss every clause. The search is carried out: at the random and using the roulette. In the first case the individual from the population is chosen by chance unlike the second case where for the most suitable individuals is defined the greater probability of becoming the parents. For forming the couples are applicable some other methods: panmixia – by chance, inbreeding – the most resembling species and outbreeding – the less resembling species.

Crossover is carried out for the pair of chromosomes and the result are 2 new chromosomes – the children. For 2 chromosomes, that have the binary code, the arbitrary part of the code of the other chromosome is taken and the missing part from the other. The operator mutation is executed as the inversion of each beat of the binary code of the chromosome with some probability as an independent test. After all stages of the formation of population the estimation of the suitability of the present individuals must be executed. It results in the formation of the new population. In case of 1 criterion are possible: tournament selection and elite selection. If there are more than 1 criteria it is possible to use the criterion function with the suspended criteria or the Pareto Set.

Search of the optimal parameters of the strategy as per each criteria is, obviously, the task that can be solved using the fingering; thus, even the multiobjective task can be solved in this way and their will be the parameters of GA under which the method will coincide. On the selection of the parameters greatly influences the form of the criterion function. And in special cases, for example for the tabular function, defined by the random numbers, the solution with the help of fingering can be found more quickly, but such an example is an exception to the rule. GA under another type and form of function may effectively use all possible features and regularities for the achieving of the optimum result. Thus the more natural the function looks like, the quicker coincides the algorithm.

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