CN106447022A - Multi-objective particle swarm optimization based on globally optimal solution selected according to regions and individual optimal solution selected according to proximity - Google Patents
Multi-objective particle swarm optimization based on globally optimal solution selected according to regions and individual optimal solution selected according to proximity Download PDFInfo
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Abstract
In order to make the convergence and the diversity of an approximate Pareto front end of a multi-objective particle swarm optimization (PSO) optimize, the invention provides a multi-objective particle swarm optimization based on a globally optimal solution selected according to regions and an individual optimal solution selected according to proximity. The multi-objective particle swarm optimization comprises the steps of: averagely dividing a first quadrant region of a coordinate into certain fan-shaped regions according to an angle; enabling particles to select the globally optimal solution from non-inferior solutions of the same region with its own; meanwhile setting an individual external archive, and storing a Pareto optimal solution found by each individual particle; selecting a position, as an individual extreme value, corresponding to an individual external archive vector which has a minimum vetorial angle with the particles from the individual external archive; increasing an item of the particle for its own poor historical experience cognition in a basic speed updating formula; and proving the effectiveness by adopting a method for similarly analyzing the optimization mechanism of the standard PSO. Two multi-objective detection functions are optimized by use of the multi-objective particle swarm optimization; the multi-objective particle swarm optimization is compared with two other optimization algorithms so as to prove the better optimization effect of the improved particle swarm optimization.
Description
Technical field
The invention belongs to intelligent algorithm field is it is proposed that choose globally optimal solution, from outside individual shelves from institute subregion
Position outside individual archives vector corresponding to minimum with particle vector angle is chosen as individual extreme value in case, and in base
Increase particle in this speed more new formula to oneself cognitive item of bad historical experience, and mathematical proof is carried out to it, to obtain
Convergence and the more excellent approximate Pareto front end of diversity.
Background technology
Eberhart and Kennedy proposes one kind and is based on random population by studying the predation of flock of birds and the shoal of fish
Novel intelligent optimization method particle swarm optimization algorithm (Particle Swarm Optimization, PSO).This algorithm root
Individuality and colony intelligence produced by social combination according to particle in colony to instruct Optimizing Search.Particle group optimizing
Algorithm is because, the advantages of form is succinct, convergence is quick and parameter adjustment mechanism is flexible, can approach non-convex or discontinuous simultaneously
Pareto optimum front end, thus be considered as the effective ways solving multi-objective optimization question.With particle swarm optimization algorithm from
When single-objective problem expands to multi-objective problem, the choosing of the storage of Pareto optimal solution set and maintenance, the overall situation and individual optimal solution
The aspects such as the balance between taking and develop and exploiting also become the emphasis of research.
At present File Maintenance is concentrated mainly on to the research of multi-objective particle, globally optimal solution selects and plants
The aspects such as group's diversity increase, also have small part document that individual optimal solution selection aspect is studied.Document《Based on target
Weight guides the energy-saving and emission-reduction power system optimal dispatch of multi-objective particle swarm》The mesh with current particle is selected from Noninferior Solution Set
That the most close noninferior solution element of mark weighting factor values is as its global extremum, to keep the diversity of solution, but this choosing
The mode of selecting makes the selection of globally optimal solution excessively single, affects the diversity understanding to a certain extent.Document《Using probability
The self adaptation multi-objective particle swarm algorithm selecting》By the dominance hierarchy of the non-bad sequence of particle, set individual fitness numerical value,
For strengthening the dispersiveness of optimal solution set, using crowding distance, fitness is punished, and then acquisition is compared according to probability selection
Corresponding optimum individual, but the individual optimal solution so obtaining is not enough to describe individual Pareto optimum front-end information.Literary composition
Offer《Multi-objective particle based on Pareto entropy》Found using outside individual archives preservation individual particles
Pareto optimal solution, selects the member closest with globally optimal solution as individual optimal solution from outside individual archives, protects
Hold the diversity of individual optimal solution selection, but the selection of individual optimal solution has not taken into account the positional information considering particle.Literary composition
Offer《The some specificity analysises of particle group optimizing method》Propose the generality that standard particle group optimizing method is launched by iteration time
Description formula, analyzes the Optimization Mechanism of standard PSO, the feelings based on mass society information and itself historical experience on this basis
Under condition, the mathematical description in particle maximum search space of having derived, by being illustrated as history shape by the general description of Particles Moving
The form of state weighted sum, further demonstrates the accumulation with iteration cycle for the PSO parameter, the characteristic of oblivion on probability meaning.
In order that the globally optimal solution selected from external archive can preferably guide population balance exploitation and exploitation target
Convergence and multifarious approximate Pareto front end are taken into account to obtain in space, and coordinate first quartile zone leveling is divided by the present invention
Become certain sector region, make particle with the noninferior solution of oneself the same area in choose globally optimal solution, so can be preferable
Balanced the probability that noninferior solution in external archive is chosen as globally optimal solution, effectively keep the distribution of Pareto optimal solution equal
Even property.The Pareto optimal solution that setting outside individual archives preservation individual particles find simultaneously, selects from outside individual archives
With the position corresponding to the outside individual archives vector of particle vector angle minimum as individual extreme value, to ensure individual history letter
The integrality of breath and the diversity of individual optimal solution selection.In order to improve the search efficiency of particle, strengthen Pareto optimal solution
Convergence, the present invention increased particle to oneself cognitive item of bad historical experience in base speed more new formula, and adopts
Method similar to the Optimization Mechanism of analytical standard PSO proves its validity.
Content of the invention
In order that the convergence of multi-objective particle swarm algorithm approximate Pareto front end and diversity are more excellent, the present invention proposes
Coordinate first quartile region is angularly divided equally into certain sector region, allows particle non-bad with oneself the same area
Globally optimal solution is chosen in solution, the Pareto optimal solution that setting each individual particles of outside individual archives preservation find simultaneously, from
Position outside individual archives vector corresponding to minimum with particle vector angle is chosen as individual pole in outside individual archives
Value, and increase particle in base speed more new formula to the cognitive item of oneself bad historical experience, and using similar to point
The method of the Optimization Mechanism of analysis standard PSO proves its validity.With 2 multi-target detection functions of this algorithm optimization, with other 2 kinds
Optimized algorithm compares, to verify this innovatory algorithm more preferably effect of optimization.
Specific embodiment
The present invention comprises the following steps：
1 multi-objective optimization question.
The general type of multi-objective optimization question is
Wherein fi(X) it is i-th object function, m is object function number；X ties up decision vector, X=[x for n1,...,
xn]；gj(X) it is equality constraints functions；hk(X) it is inequality constraints function.
2nd, the speed of particle and location updating formula.
In order to realize fully sharing of information, strengthen the validity of guidance information, the frame to speed more new formula for the present invention
Frame improves, and increases particle in base speed more new formula to oneself cognitive item of bad historical experience. and so permissible
Strengthen the utilization to self information in learning process, strengthen the efficiency of guidance information, to history poor solution is reduced or avoided empty
Between repeat search, improve the convergence of approximate Pareto front end that multi-objective particle swarm algorithm obtains.
In formula (2-1), c1C2And c3For acceleration factor .c1Embody the impact to particle running orbit for the particle individual experience
Power, c2Embody the influence power to particle running orbit for the particle colony experience, c3Embody particle bad learning experience particle is run
The influence power of track；r1R2And r3For equally distributed random number in (0,1)；ω successively decreases with the increase of iterations；
Represent particle i self compare in t iterative process fitness value worst when position.
The mathematical proof of the speed more new formula of the 2.1 bad historical experience items of consideration
2.1.1 consider PSO mono- information maximization search space description of bad empirical term
Consider bad empirical term PSO particle single information maximization search space expression formula be (2-3)
Wherein,Variable band bracket
Subscript or subscript represent iteration cycle, not parenthesized subscript or subscript represent power.
Single information maximization search space expression formula of standard PSO particle is formula (2-4)
In formula
Comparison expression (2-3) and formula (2-4) can draw and consider that the PSO particle of bad empirical term can be prevented effectively from search once
Through searching for the position that lives the worst life, reduce invalid search space, improve search efficiency.
2.1.2 the descriptive analysis of single information maximization search space
General description to information maximization search space is derived and is arranged, and obtains the form of formula (2-5).
In formulaForWithFactorial decay factor；ForWithSpacing
From random weight；Weighting parameters for initial time velocity vector；
ForWithBetween distance random weight.
Expression formula to the general description in standard PSO particle search space is formula (2-6),
φ ' in formula(n)δ '(n)WithY′(n)It is that γ (m) in respective items in (2-5) formula changes γ ' (m) gained afterwards, its thing into
Reason meaning is identical.
(1) the analysis of initial position factorial decay factor
Factorial decay factorY (n)Reflection particle is in initial time individuality positional information respectivelyWith colony's optimal locationWith
The characteristic of time change.WithIt is respectively (0, ri) (0, rg) and (0, rw) on be uniformly distributed.Obtain Y(n)Expectation
And variance, expression formula respectively formula (2-7) and (2-8)：
Coefficient Y ' before respective items in standard PSO(n)Expectation and variance be respectively formula (2-9) and (2-10),
Analysis draws：E (Y (n)) ＞ E (Y' (n)), show that the PSO algorithm considering bad empirical term is more individual than in PSO algorithm
Longer to forgeing the required time with colony optimal location information attenuation, increase the base of the good experience of history instructing particle search
Number, effectively strengthens the accuracy of guidance information；D (Y (n)) ＞ D (Y ' (n)), illustrates that the PSO algorithm considering bad empirical term compares standard
Individual more active in attenuation process, during enhancing particle search with colony's optimal location information in PSO algorithm
Activity, maintains the diversity of particle, it is to avoid particle is absorbed in Local Search too early.
(2) the weighting parameters analysis of initial velocity
With factorial decay factor Y(n)Derivation in the same manner, analogizes the weighting parameters δ obtaining initial velocity vector(n)Expectation
For formula (2-11).
The weighting parameters δ ' of PSO algorithm initial velocity(n)Be desired for formula (2-12).
E (δ can be drawn(n)) ＞ E (δ '(n)), show that the PSO algorithm considering bad empirical term is more first than in PSO algorithm
Beginning speed weighting parameters take the probability of higher value bigger, make initial velocity higher to the position influence of search so that particle
Single information maximization search space bigger, improve the probability searching globally optimal solution.
3 external archive maintenance strategies
The reasonable setting of external archive and effective renewal can preserve high-quality Pareto optimal solution set.Screen current grain
Non-dominant collection in subgroup, and reject inferior solution therein, it is subsequently adding filing.If number of particles has exceeded filing and has been permitted in filing
The maximum-norm permitted, then the particle randomly selecting the maximum-norm number allowing equal to filing from the particle of filing constitutes outside shelves
Case.
4 globally optimal solution selection strategies.
In particle swarm optimization algorithm, globally optimal solution guides the Evolutionary direction of whole colony.But in multiple target grain
In the optimized algorithm of subgroup, globally optimal solution is one group of Pareto optimal solution set, if can not effective selection strategy reasonable in design,
Then it is likely to result in selection pressure too small and evolutionary process can not be effectively pushed and effectively improve the diversity of approximate Pareto front end.
In order to produce the selection pressure of appropriateness and effectively improve the diversity of approximate Pareto front end, subregion is pressed in special setting
Select the strategy of globally optimal solution.The present invention the vector of the location point of particle itself and origin of coordinates composition be referred to as particle to
Amount, is referred to as external archive vector the vector of the point in external archive and initial point composition simultaneously.Each particle carries out speed renewal
When, the method choosing globally optimal solution is：[0, pi/2] interval in coordinate system is divided equally into maximum for overall external archive
1/2 of capacity is fan-shaped interval, is then referred to corresponding sector to external archive vector according to the corner dimension of abscissa
Interval, which interval particle vector falls in, and just randomly selecting an external archive point in this interval is when this particle updates
Globe optimum, if this interval does not have external archive point, chooses and this particle vector angle minimum in external archive
Point corresponding to external archive vector is as globe optimum.
5th, individual optimal solution selection strategy.
In the most of multi-objective particles having existed, each particle only may be selected individual optimal solution,
Its update method is to take the principle of " not arranging, do not update ", that is, only when the particle of new domination current individual extreme value
More new individual extreme value, is otherwise always maintained at constant.The update method of this single individual optimal solution is simply efficient, but an individual
Optimal solution is not enough to express the information of the individual approximate Pareto front end that individual particles found and experienced, and loses individual optimum
Solve the diversity of selection.
In the present invention, in order to give full expression to the information of individual approximate Pareto front end, using outside individual archives preservation
The Pareto optimal solution that the individual particles of each particle find, its renewal process is, when the new explanation of particle is dominant old solution, then
New explanation is stored in outside individual archives, if both are not mutually dominant, randomly choose one and is stored to outside individual archives, if old
Solution is dominant new explanation, then do not store.In order to reduce computation complexity, the maximum capacity of outside individual archives is set to the outside shelves of the overall situation
The 1/2 of case maximum capacity.During particle group optimizing, the present invention will select and particle vector angle from outside individual archives
The minimum position corresponding to outside individual archives vector, as individual extreme value, presss from both sides if there is two and minimum equal above
Angle, then from wherein randomly selecting a position as individual extreme value.
6th, improve the calculation process of multi-objective particle swarm algorithm.
Step1：The position of random initializtion particle it is ensured that each particle position is feasible solution, that is, meets the institute of operation
Prescribed Properties.Initialization speed, and limit two extreme values V of speedmax、Vmin.
Step2：Judge the dominance relation between each particle, initialize Noninferior Solution Set.
Step3：The individual extreme value of initialization is particle itself, and the corresponding global extremum of each particle is true according to 3 above and 4
Fixed.
Step4：According to speed V of formula (2-1) more new particle, if V is ＞ VmaxOr V ＜ Vmin, then it is adjusted to V=V respectivelymax
Or V=Vmin.Position according to formula (2-2) more new particle.
Step5：If the solution difference meeting maximum iteration time or adjacent generations is sufficiently small, exports optimal solution, optimized
Journey terminates；Otherwise turn Step6.
Step6：Obtain new Noninferior Solution Set.
Step7：Global extremum and individual extreme value are chosen according to the method in 4 and 5, and turns Step4.
7th, test and its analyze.
7.1 noninferior solution performance evaluations
For convergence and the distributing homogeneity of the approximate Pareto front end of effective assessment algorithm acquisition, the present invention is using anti-
Reincarnation generation distance (inverted generationaldistance, abbreviation IGD) is as Performance Evaluation index.IGD is that tolerance is true
The distance between the approximate Pareto front end that real Pareto front end obtains to optimized algorithm index.This desired value is lower, shows excellent
The convergence of approximate Pareto front end and the diversity of changing algorithm acquisition are better, closer to true Pareto front end.P is made to be true
Pareto optimal solution set, A is the approximate Pareto optimal solution set that optimized algorithm obtains, then IGD can count according to formula (7-1)
Calculate:
Wherein,For minimum normalization Euclidean distance；WithIt is respectively q mesh in P
The maximum put on and minimum of a value, q=1,2 ..., Q, Q are target number；pi∈ P, i=1,2 ..., | P |；aj∈ A, j=1,
2,...,|A|.In this experiment, sample number | P | of the true Pareto optimal solution set of test function ZDT1 and ZDT2 is
1000.
7.2 detection functions and the parameter setting of algorithm
In order to test the superior function of this paper algorithm, select 2 multi-target detection functions, i.e. function ZDT1 and ZDT2, often
Individual detection function contains two object functions and 30 dimension variables.The algorithm being compared with algorithm proposed by the present invention is:Classical
Multi-objective optimization algorithm MOPSO and NSGA-II, the iterations of all algorithms is 100, and the scale that achieves is 100, particle scale
100.With 3 kinds of algorithms, above-mentioned 2 kinds of multi-target detection functions are carried out respectively 20 times test, take 20 suboptimization IGD value average
It is worth variance and optimal value, and average four indexs of used time are compared.
Claims (4)
1., based on the multi-objective particle swarm algorithm selected globally optimal solution by area and select individual optimal solution nearby, walk including following
Suddenly：
(1) multi-objective optimization question；
(2) the speed of particle and location updating formula；
(3) external archive maintenance strategy；
(4) globally optimal solution selection strategy；
(5) individual optimal solution selection strategy；
(6) improve the calculation process of multi-objective particle swarm algorithm；
(7) test and its analyze.
2. according to claim 1 based on the multiple target grain selected globally optimal solution by area and select individual optimal solution nearby
Swarm optimization it is characterised in that：The present invention improves to the framework of speed more new formula, that is, in base speed more new formula
Increase particle to oneself cognitive item of bad historical experience, so can strengthen the utilization to self information in learning process,
Strengthen the efficiency of guidance information, repeat search to history poor solution space is reduced or avoided, improve multi-objective particle swarm algorithm
The convergence of the approximate Pareto front end obtaining,
In formula, c1C2And c3For acceleration factor, c1Embody the influence power to particle running orbit for the particle individual experience, c2Embody
The influence power to particle running orbit for the particle colony experience, c3Embody the shadow to particle running orbit for the bad learning experience of particle
The power of sound；r1R2And r3For equally distributed random number in (0,1)；ω successively decreases with the increase of iterations；Represent particle
I self compare in t iterative process fitness value worst when position.
The mathematical proof of the speed more new formula of the 2.1 bad historical experience items of consideration
2.1.1 consider PSO mono- information maximization search space description of bad empirical term
The single information maximization search space expression formula considering the PSO particle of bad empirical term is
Wherein,Variable is parenthesized
Subscript or subscript represent iteration cycle, and not parenthesized subscript or subscript represent power；
2.1.2 the descriptive analysis of single information maximization search space
General description to information maximization search space is derived and is arranged, and the form obtaining is
In formulaForWithFactorial decay factor；ForWithBetween distance
Random weight；Weighting parameters for initial time velocity vector；For
WithBetween distance random weight；
(1) the analysis of initial position factorial decay factor
Factorial decay factor Y(n)Reflection particle is in initial time individuality positional information respectivelyWith colony's optimal locationIn time
The characteristic of change,WithIt is respectively (0, ri) (0, rg) and (0, rw) on be uniformly distributed, obtain Y(n)Expectation and
Variance, expression formula is as follows respectively：
(2) the weighting parameters analysis of initial velocity
With factorial decay factor Y(n)Derivation in the same manner, analogizes the weighting parameters δ obtaining initial velocity vector(n)Be desired for
Formula：
3. according to claim 1 based on the multiple target grain selected globally optimal solution by area and select individual optimal solution nearby
Swarm optimization it is characterised in that：In order to produce the selection pressure of appropriateness and effectively improve the diversity of approximate Pareto front end, special
Setting selects the strategy of globally optimal solution by subregion, the vector that the present invention is formed the location point of particle itself with the origin of coordinates
Referred to as particle vector, is referred to as external archive vector the vector of the point in external archive and initial point composition simultaneously, and each particle enters
When scanning frequency degree updates, the method choosing globally optimal solution is：[0, the pi/2] interval in coordinate system be angularly divided equally into for
1/2 of overall external archive maximum capacity is fan-shaped interval, then to external archive vector according to the corner dimension with abscissa
It is referred to fan-shaped accordingly interval, which interval particle vector falls in, just randomly selects an external archive point in this interval
Globe optimum when updating for this particle, if this interval does not have external archive point, chooses and this grain in external archive
The minimum point corresponding to external archive vector of subvector angle is as globe optimum.
4. according to claim 1 based on the multiple target grain selected globally optimal solution by area and select individual optimal solution nearby
Swarm optimization it is characterised in that：In the present invention, for the information of the individual approximate Pareto front end of effective expression, using individuality
External archive preserves the Pareto optimal solution that the individual particles of each particle find, its renewal process is, when the new explanation of particle accounts for
During excellent old solution, then new explanation is stored in outside individual archives, if both are not mutually dominant, random selection one is stored to individual external
Portion's archives, the new explanation if old solution is dominant, do not store, in order to reduce computational complexity, the maximum capacity of outside individual archives
It is set to the 1/2 of overall external archive maximum capacity, during particle group optimizing, the present invention will select from outside individual archives
Position outside individual archives vector corresponding to minimum with particle vector angle as individual extreme value, if there is two and with
Upper equal minimum angle, then from wherein randomly selecting a position as individual extreme value.
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Application publication date: 20170222 |