CN103425838A - Path tracking method based on linux - Google Patents

Path tracking method based on linux Download PDF

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Publication number
CN103425838A
CN103425838A CN2013103485777A CN201310348577A CN103425838A CN 103425838 A CN103425838 A CN 103425838A CN 2013103485777 A CN2013103485777 A CN 2013103485777A CN 201310348577 A CN201310348577 A CN 201310348577A CN 103425838 A CN103425838 A CN 103425838A
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estimate
point
iteration
path
mode
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孟祥飞
吴楠
何志平
郭美思
宗栋瑞
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Inspur Electronic Information Industry Co Ltd
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Abstract

The present invention provides a kind of path following method based on linux, the a certain Nonlinear Programming Problems of solution can be attributed to for a kind of problem is encountered in the numerous areas such as economics, electronic circuitry design, automatic control, CAD and manufacture and computer graphics all, with extremely wide especially in neural network and graph and image processing. Path following method is by having constructed objective function
Figure DEST_PATH_IMAGE002
Track path function
Figure DEST_PATH_IMAGE004
It is modeled, and
Figure DEST_PATH_IMAGE006
Section, which is walked by initializationization, estimates step, correction walks, exchange step, verifies five steps of step is tracked, and objective function is finally obtained Solution.

Description

A kind of path following method based on linux
Technical field
The present invention relates to the Computer Applied Technology field, specifically a kind of path following method based on linux.
Background technology
In nearest decades, the method that solves some optimization problems (as equalization problem, nonlinear programming, multiple objective programming, variational inequality, Complementarity Problem etc.) is suggested in succession.Wherein, nonlinear programming problem is a very important problem, and it,, in economics, electronic circuitry design, control automatically, computer-aided design (CAD) and the numerous areas such as manufacture, neural network and graph and image processing, has a lot of application.
At first the work that solves the mathematical programming problem aspect propose in C. B. Garcia in 1979 and W. J. Zangwill.In order to solve convex programming problem, they have constructed the restricted problem with parameter:
Figure 2013103485777100002DEST_PATH_IMAGE001
Wherein,
Figure 75526DEST_PATH_IMAGE002
,
Figure 2013103485777100002DEST_PATH_IMAGE003
.Then, utilize non-smooth function
Figure 274426DEST_PATH_IMAGE004
With
Figure 2013103485777100002DEST_PATH_IMAGE005
Be equation by the Partial Conversion that contains inequality in the KKT system, so, just primal problem can be converted into to a Nonlinear System of Equations problem, then can construct one and homotopy it be solved.1984, a kind of new method was proposed to solve linear programming problem by N. Karmarkar, and the method is a kind of method of path trace in fact or is referred to as interior point method.This method solves linear programming problem and has polynomial complexity.The method has caused a lot of scholars' concern after proposing.
In order to solve nonconvex nonlinear programming problems, a kind of combined homotopy method (Combined Homotopy Interior Point Method, be called for short the CHIP method) by Feng Guochen, Lin Zhenghua, put forward in ripple, and the existence in homotopy path and global convergence also are proven.The positive China of woods, Li Yong reach the homotopy Method that will contain the nonlinear programming problem of inequality constrain in ripple in 1996 and have been generalized on the problem with equality constraint subsequently.Lin Zhenghua, in involving Feng Guochen, proposed to solve the homotopy Method of convex programming problem in 1997.And then, under harmonious condition, about global convergence and the polynomial complexity similar to the interior point method of current existence of the combined homotopy method of convex programming, also in ripple, Xu Qing, Feng Guochen, proved.In the last few years, some researchers had constantly improved this method, and combined homotopy interior point method constantly is improved and promotes.
Calendar year 2001, a kind of method that is called cohesion homotopy (Aggregate Constraint Homotopy is called for short the ACH method) is proposed in ripple, Feng Guochen and Zhang Shaoliang, and the method can solve the nonconvex programming problem with multiple constraint.They have done some assumed conditions subsequently, i.e. feasible set bounded, border canonical and weak normal cone condition are proved existence and the convergence in homotopy path.In ACH, the number of its homotopy variable is
Figure 5622DEST_PATH_IMAGE006
, and the number of homotopy variable is in CHIP
Figure 2013103485777100002DEST_PATH_IMAGE007
.In cohesion is homotopy, by constraint condition, more nonlinear programming becomes single constraint planning and processes, and the scale of Solve problems is just greatly reduced like this.Therefore, if in the situation that in the middle of reality, run into the constraint number many, we preferentially select to condense homotopy.But, in some conditions, such as under " weak normal cone condition ", condensing the homotopy requirement to initial point higher, must be taken in the middle of certain subset in feasible set, this condition is difficult to reach or bad getting sometimes; Under other some condition, need some auxiliary functions could construct Homotopy equation, and these auxiliary functions often all are difficult to realize.Therefore, if want to accomplish also some difficulty of large-scale application.In 2006, a kind of combined homotopy of being more convenient for using was proposed in ripple, Shang Yufeng.This homotopy Method to the requirement of condition still less, but but can solve the constantly nonconvex programming problem of change of some borders.They have proved existence and the Global Convergence in the method path, and the method compares with some homotopy Method of previous proposition, more weak to the requirement of initial point, homotopy also easy structure.
More and more along with the problem in engineering, the scale of some problems of generation may be very large, also more and more higher to the requirement of the efficiency of the method that solves this class problem.For such demand, in 2011, straight cohesion homotopy Method brave, proposed leveling in ripple was condensed some calculating shortcomings of homotopy Method with improvement, and has provided the proof of the Global Convergence of this method.This method is mainly to utilize some function to be weighted division to original constraint function, and sets up a criterion to reduce the calculating that constraint function is asked first order derivative and second derivative.The nonlinear programming problem that the method is very many to the constraint number is very effective.
Although the theoretical research of nonlinear programming problem has a lot of breakthroughs, the problem that can only guarantee like this to adapt to different complexities is raised the efficiency when setting up the path trace model, can be for everybody but there is no a unified effective method in tracing process, this method, as an important tool that solves the nonlinear programming problem disaggregation, can well meet scientific research field and the central practical problems run into of engineering application.
In the middle of actual engineering project, nonlinear programming problem has following feature:
A) related data volume is very large, and requires performance very harsh (Millisecond often), and it is extremely important therefore how obtaining fast the target solution;
B) description of nonlinear programming problem converts mathematical model to often by vectorial form, and we take matrix structure to preserve and process these data usually.But, in the middle of these vectors, be all much asymmetrical, therefore, how we can guarantee that space complexity can guarantee again time complexity and track path is particularly important accurately in the middle of the design of data structure;
C) in the path trace process, method to constantly circulate estimate, proofread and correct, transposing, these four steps of verification, how making cycle index reduce but not lack precision is also the technical matters that we will capture.
Summary of the invention
The purpose of this invention is to provide a kind of path following method based on linux.
The objective of the invention is to propose a kind of path following method based under linux system.The method can draw the solution of the central primal problem (objective function) of nonlinear programming problem quickly and accurately.
The objective of the invention is to realize in the following manner, for solving Nonlinear System of Equations
Figure 668160DEST_PATH_IMAGE008
Result, at first to choose a simple equation , its solution
Figure 422489DEST_PATH_IMAGE010
Known or solve than being easier to, then construct one with parameter Smooth function
Figure 2013103485777100002DEST_PATH_IMAGE013
, it is met
Figure 886148DEST_PATH_IMAGE014
；
And equation under certain conditions, The solution set comprise one from Start, level off to lineoid
Figure 2013103485777100002DEST_PATH_IMAGE017
Smooth curve
Figure 229722DEST_PATH_IMAGE018
,
Figure 504846DEST_PATH_IMAGE018
Other end limit point be
Figure 2013103485777100002DEST_PATH_IMAGE019
,
Figure 351579DEST_PATH_IMAGE020
Be
Figure 686745DEST_PATH_IMAGE008
Solution;
Described certain condition is at first to path function
Figure 415667DEST_PATH_IMAGE013
Carry out modeling, then exist
Figure 2013103485777100002DEST_PATH_IMAGE021
Interval by the initialization step, estimate step, proofread and correct step, transposing step, verification walk five steps and followed the tracks of, and finally obtains objective function
Figure 307531DEST_PATH_IMAGE022
Solution;
To path function
Figure 805508DEST_PATH_IMAGE013
Carry out in the process of modeling, the two-dimentional sparse matrix of taking reduces space complexity and time complexity as parameter; Mainly some global variables are carried out to initialization in the initialization step; In estimating step, take tangent line first to estimate the mode that all the other secants estimate and estimated, with this, reduced time complexity; Monitor tracing step and carry out the step-length renewal according to optimal strategy in the transposing step; Final step verification step is used for judging whether the error of trace point and actual point meets set requirement, if meet method in a word, otherwise continue to forward to, estimates step;
Due to five steps, following the tracks of is a process constantly circulated, therefore good communication and the parameter setting of each step, adopt the managed mixed mode of global variable and local static variable, the sparse matrix of taking when the design of key data structure and common matrix are mixed deposits the dynamic management structure, and concrete steps are as follows:
(a) design of pre-service and matrix of coefficients data structure
For the communication of well carrying out each step, we have taked the mode of global variable, such as precision, step-length, initial point, estimate point, check point, initially estimate direction, estimate direction etc. and all seal up for safekeeping and unifiedly in the middle of a global profile monitor renewal, the benefit of doing like this is to have lacked the assignment back and forth of a lot of intermediate variables in data exchange process, when data volume is large, the assignment time overhead is also very large, and be convenient to communication and debugging, therefore, before following the tracks of, at first these data to be carried out to pre-service;
In order to solve the asymmetry problem of vector in the middle of nonlinear programming problem, this method has been taked the data structure of sparse matrix and normal two-dimensional array and the data structure of depositing is carried out the processing of data, like this for different nonlinear programming problems, take two sets of data structures and homography computing method to be switched, the strategy of switching is: " number of non-zero entry is more than three times of null element number ", not only reduce space expense, and greatly reduced time complexity.Here we only introduce the data structure volume description of sparse matrix:
struct Trituple{
Int row_index; The rower of // non-zero entry
Int col_index; The row mark of // non-zero entry
Double value; // element value
}；
struct SparseMatrix{
Int Mrx_row; The row of // matrix
Int Mrx_col; // matrix column
Trituple* data; // tlv triple array
}；
Such as
Figure 2013103485777100002DEST_PATH_IMAGE023
Such matrix takes the structure of sparse matrix to describe, and effect is apparent, and the ratio that this structure occupies in the middle of actual nonlinear programming problem is more higher;
(b) path trace process
Follow the trail of by In the process in the homotopy path of determining, method is divided following five steps:
1. initialization walks: provide three amounts in the initialization step: an initial point
Figure 2013103485777100002DEST_PATH_IMAGE025
, an initialization step-length
Figure 199897DEST_PATH_IMAGE026
, carry out initialization in global variable step; Last allowable error
Figure 2013103485777100002DEST_PATH_IMAGE027
Take macrodefined mode to set in inf;
2. estimate step: after obtaining initial point, carry out estimating of next point, the first step takes tangent line to estimate, and what all the other were taked is that secant is estimated mode, at first obtains at initial point
Figure 285665DEST_PATH_IMAGE028
The unit tangent vector at place
Figure 2013103485777100002DEST_PATH_IMAGE029
,
Calculate
Figure 700466DEST_PATH_IMAGE030
Draw and estimate a little
Figure 2013103485777100002DEST_PATH_IMAGE031
, the follow-up process of estimating is that the difference by getting the point after the current check point obtained and a upper correction thereof is carried out secant and estimated; Here used with the Gauss method of elimination of pivot in a column and asked tangent vector
Figure 643014DEST_PATH_IMAGE029
, below simply introduce the Gauss method of elimination with pivot in a column;
Figure 80949DEST_PATH_IMAGE032
Choose , make
Figure 317545DEST_PATH_IMAGE034
Maximize, and carry out following row exchange
Figure 2013103485777100002DEST_PATH_IMAGE035
Figure 790114DEST_PATH_IMAGE036
It is that specifically the following formula of mistake obtains by with adjacent 2 of calculating, determining that is estimated a direction that secant is estimated main method:
Figure 2013103485777100002DEST_PATH_IMAGE037
。
Also having a kind of mode of estimating is that three Hermite estimate, and concrete formula is as follows:
Figure 965881DEST_PATH_IMAGE038
When estimating for the first time, only has point of initial value point, therefore, the mode that can only take tangent line to estimate, tangent line is estimated in the process of calculating tangent vector very consuming time, therefore, as long as obtain after first estimates a little, just taking simple and effective secant to estimate mode, this method can be fast a lot of on execution efficiency compared with other two kinds of modes, therefore taked first step tangent line to estimate, all the other steps are the mode that secant is estimated;
3. proofread and correct step: with
Figure 2013103485777100002DEST_PATH_IMAGE039
For initial value, by process of iteration, produce a sequence
Figure 563215DEST_PATH_IMAGE040
, make
Figure 2013103485777100002DEST_PATH_IMAGE041
For
Figure 787523DEST_PATH_IMAGE042
Approximate and the error of mid point is less than If iteration does not restrain, dwindle step-length and go back to and estimate step, adopt Newton iteration method here, step is as follows:
From a point
Figure 52283DEST_PATH_IMAGE045
Start, estimate
Figure DEST_PATH_IMAGE046
, it provides its error and is
Figure 336633DEST_PATH_IMAGE047
Coarse estimation
Figure DEST_PATH_IMAGE048
, can prevent this error propagation and be controlled through some step Newton iteration corrections, the initial value of this iterative process is discreet value
Figure 562209DEST_PATH_IMAGE049
, wish iteration convergence in
Figure DEST_PATH_IMAGE050
Solution
Figure 590208DEST_PATH_IMAGE051
Vector
Figure DEST_PATH_IMAGE052
With path
Figure 834108DEST_PATH_IMAGE053
The point
Figure DEST_PATH_IMAGE054
Tangent, through point
Figure 23781DEST_PATH_IMAGE049
And with tangent vector
Figure 736653DEST_PATH_IMAGE052
Vertical
Figure 302763DEST_PATH_IMAGE055
The dimension lineoid
Figure DEST_PATH_IMAGE056
The point
Figure 666749DEST_PATH_IMAGE057
(wherein
Figure DEST_PATH_IMAGE058
) locate cutting path, trimming process is exactly in order to obtain this point, order
Figure 27323DEST_PATH_IMAGE059
',
Figure DEST_PATH_IMAGE060
Mean
Figure 224562DEST_PATH_IMAGE049
Component, lineoid
Figure 594363DEST_PATH_IMAGE056
Can be write as:
Figure 484959DEST_PATH_IMAGE061
So trimming process is exactly solving equations
The Iteration of Newton iteration is:
Figure 344330DEST_PATH_IMAGE063
Newton iteration is local convergence, exists
Figure DEST_PATH_IMAGE064
In exist a little
Figure 953166DEST_PATH_IMAGE051
A certain field
Figure 939708DEST_PATH_IMAGE065
, work as initial value
Figure DEST_PATH_IMAGE066
The time, the Newton iteration convergence, if from
Figure 950389DEST_PATH_IMAGE049
The Newton iteration set out is not restrained, and by dwindling, estimates step-length
Figure 918345DEST_PATH_IMAGE026
, make
Figure 76794DEST_PATH_IMAGE049
Drop in certain neighborhood of Newton iteration convergence;
The distinguishing rule of iteration stopping has two kinds: the first, do not converge on a close point if estimate iteration, and should stop immediately iteration; The second, if iteration convergence, when approximate solution all in error up to specification
Figure 788398DEST_PATH_IMAGE027
In, also stop iteration, in the design of method, adopt well to comprise both of these case with the if structure, mistake estimate a little and perpendicular to the lineoid of estimating direction on proofreaied and correct, the correction equation group
Figure 653586DEST_PATH_IMAGE067
，
Wherein, vector dMean to estimate direction;
4. adjust step-length: this step mainly is considered for precision and cycle index, in trimming process, if Meet the demands, adjustment step-length that will be suitable , otherwise control flow turns back to and estimates step, in the process of method design, step-length carried out to mark successively, and in the Predictor Corrector process each time, this variable is becoming always, so realizes with the local variable of a static state;
5. verification walks: this step is the termination signal of whole path trace, once
Figure 188921DEST_PATH_IMAGE068
Middle component
Figure 438636DEST_PATH_IMAGE012
Equal 0 or close to 0, just stop following the tracks of, in the Newton iteration method process, at every turn all right tDo inspection, if the condition meet stopped, method just allow iterative loop stop and final result to out.
The invention has the beneficial effects as follows: all can be summed up as and solve a certain Nonlinear Programming Problems for the class problem that runs in economics, electronic circuitry design, control automatically, computer-aided design (CAD) and the numerous areas such as manufacture and computer graphics, especially use very extensive in neural network and graph and image processing.Path following method is by establishing target function The track path function
Figure DEST_PATH_IMAGE070
Carry out modeling, and
Figure 795985DEST_PATH_IMAGE071
Interval by the initialization step, estimate step, proofread and correct step, transposing step, verification walk five steps and followed the tracks of, and finally can access objective function
Figure 866710DEST_PATH_IMAGE069
Solution.
The accompanying drawing explanation
Fig. 1 is that tangent line is estimated with secant and estimated figure;
Fig. 2 is correction chart;
Fig. 3 path trace algorithm flow chart;
Fig. 4 Dynamic Data Processing process flow diagram.
Embodiment
Below in conjunction with echoing embodiment, content of the present invention is described in further detail.
Concrete steps are as follows:
For solving Nonlinear System of Equations
Figure DEST_PATH_IMAGE072
Result, at first to choose a simple equation , its solution
Figure 572946DEST_PATH_IMAGE010
Known or solve than being easier to, then construct one with parameter
Figure 115922DEST_PATH_IMAGE012
Smooth function
Figure 408363DEST_PATH_IMAGE070
, it is met
Figure DEST_PATH_IMAGE074
；
And equation under certain conditions,
Figure 265461DEST_PATH_IMAGE050
The solution set comprise one from
Figure 787445DEST_PATH_IMAGE075
Start, level off to lineoid
Figure DEST_PATH_IMAGE076
Smooth curve
Figure 439006DEST_PATH_IMAGE018
, Other end limit point be
Figure 941849DEST_PATH_IMAGE077
,
Figure 490642DEST_PATH_IMAGE020
Be
Figure 578684DEST_PATH_IMAGE072
Solution;
Affiliated certain condition, at first to path function
Figure 658766DEST_PATH_IMAGE070
Carry out modeling, then exist
Figure 857666DEST_PATH_IMAGE071
Interval by the initialization step, estimate step, proofread and correct step, transposing step, verification walk five steps and followed the tracks of, and finally obtains objective function
Figure 260966DEST_PATH_IMAGE069
Solution;
To path function Carry out in the process of modeling, the two-dimentional sparse matrix of taking reduces space complexity and time complexity as parameter; Mainly some global variables are carried out to initialization in the initialization step; In estimating step, take tangent line first to estimate the mode that all the other secants estimate and estimated, with this, reduced time complexity; Monitor tracing step and carry out the step-length renewal according to optimal strategy in the transposing step; Final step verification step is used for judging whether the error of trace point and actual point meets set requirement, if meet method in a word, otherwise continue to forward to, estimates step;
Due to five steps, following the tracks of is a process constantly circulated, therefore good communication and the parameter setting of each step, adopt the managed mixed mode of global variable and local static variable, the mixed dynamic management structure of depositing of the sparse matrix of taking and common matrix when the design of key data structure.
Flow process according to Fig. 3 starts, and when entering pretreatment stage, we are mainly to path function Be described, and particular content is placed in the middle of the Fun.c file, its function model is as follows:
Path_track(const Matrix& x,const Matrix& y,double t,const Matrix& g)
Then overall communication variable is arranged, such as accuracy requirement, step-length, initial point, estimate point, check point, initially estimate direction, estimate direction etc.Then enter the initialization step, the initialization step is carried out the initialization initial point according to the path function of following the tracks of, three primary variabless of step-length and error, here lay special stress on a bit, must be chosen data save mode (sparse and non-sparse) according to the feature of nonlinear programming problem when being exactly the initialization initial point.Then enter and estimate step; as shown in Figure 1; our first step takes tangent line to estimate; be the mode that secant estimates later and estimated processing; wherein; tangent line is estimated and is taked Gaussian elimination method to obtain estimating direction, and algorithm is static Matrix GaussLin (const Matrix &amp in the public function storehouse; GsA, const Matrix & GsB) in, realized.
Figure DEST_PATH_IMAGE078
Choose
Figure 73567DEST_PATH_IMAGE079
, make
Figure DEST_PATH_IMAGE080
Maximize, and carry out following row exchange
Figure 410002DEST_PATH_IMAGE081
Figure DEST_PATH_IMAGE082
Secant is estimated main algorithm thinking and is exactly in brief by with adjacent 2 of calculating, determining that one is estimated direction, specifically obtains by following formula:
Then according to estimating direction and dynamic step length obtains estimating a little
Figure DEST_PATH_IMAGE084
, in estimating this step of step, update local static variable at every turn and estimate a little, and a little be judged in order to make while estimating the data processing method of taking next time estimating, then proceed to proofread and correct and walk, as shown in Figure 2, after we have had and have estimated a little
Figure 878209DEST_PATH_IMAGE084
, with For initial value, by process of iteration, produce a sequence
Figure DEST_PATH_IMAGE086
, make
Figure 875432DEST_PATH_IMAGE087
For
Figure DEST_PATH_IMAGE088
Approximate and the error of mid point is less than
Figure 210599DEST_PATH_IMAGE089
.If iteration does not restrain, dwindle step-length and go back to and estimate step.Here we have adopted Newton iteration method.
Vector
Figure 1837DEST_PATH_IMAGE052
With path The point Tangent.Through point
Figure 84697DEST_PATH_IMAGE049
And with tangent vector Vertical
Figure 993539DEST_PATH_IMAGE055
The dimension lineoid
Figure 346023DEST_PATH_IMAGE056
The point
Figure 288571DEST_PATH_IMAGE057
(wherein
Figure 788823DEST_PATH_IMAGE058
) locate cutting path.Trimming process is exactly in order to obtain this (see figure 2).Order ', Mean
Figure 342929DEST_PATH_IMAGE049
Component.Lineoid
Figure 268160DEST_PATH_IMAGE056
Can be write as:
。
So trimming process is exactly solving equations
The Iteration of Newton iteration is:
Figure 166212DEST_PATH_IMAGE063
Equally, in this step, we still will upgrade local static variable check point and check point is judged so that the data processing method (as shown in Figure 3) that on making, once timing is taked, the transposing step is mainly to monitor and upgrade an accuracy variable according to accuracy requirement at every turn, in this step, we need to design many marker bits and judge how to carry out the step-length renewal, in our algorithm, we have designed four kinds of marks, mainly to design according to the feature of path trace function, this relates to the research of the central nonlinear programming problem of mathematics, here just do not do and introduced.Final step verification step is exactly the judgement of an end condition, as long as meet our set end condition, so just can draw the solution of target problem otherwise go back to and estimate step.Whole embodiment just can complete according to top mode.
Except the described technical characterictic of instructions, be the known technology of those skilled in the art.

Claims (1)

1. the path following method based on linux, is characterized in that for solving Nonlinear System of Equations
Figure 247448DEST_PATH_IMAGE001
Result, at first to choose a simple equation , its solution
Figure 835741DEST_PATH_IMAGE003
Known or solve than being easier to, then construct one with parameter
Figure 6DEST_PATH_IMAGE004
Smooth function
Figure 241632DEST_PATH_IMAGE005
, it is met
；
And equation under certain conditions, The solution set comprise one from Start, level off to lineoid Smooth curve
Figure 444074DEST_PATH_IMAGE010
,
Figure 942051DEST_PATH_IMAGE010
Other end limit point be
Figure 448119DEST_PATH_IMAGE011
,
Figure 664337DEST_PATH_IMAGE012
Be
Figure 359891DEST_PATH_IMAGE001
Solution;
Described certain condition is at first to path function
Figure 712375DEST_PATH_IMAGE005
Carry out modeling, then exist
Figure 389344DEST_PATH_IMAGE013
Interval by the initialization step, estimate step, proofread and correct step, transposing step, verification walk five steps and followed the tracks of, and finally obtains objective function
Figure 889596DEST_PATH_IMAGE014
Solution;
To path function
Figure 575792DEST_PATH_IMAGE005
Carry out in the process of modeling, the two-dimentional sparse matrix of taking reduces space complexity and time complexity as parameter; Mainly some global variables are carried out to initialization in the initialization step; In estimating step, take tangent line first to estimate the mode that all the other secants estimate and estimated, with this, reduced time complexity; Monitor tracing step and carry out the step-length renewal according to optimal strategy in the transposing step; Final step verification step is used for judging whether the error of trace point and actual point meets set requirement, if meet method in a word, otherwise continue to forward to, estimates step;
Due to five steps, following the tracks of is a process constantly circulated, therefore good communication and the parameter setting of each step, adopt the managed mixed mode of global variable and local static variable, the sparse matrix of taking when the design of key data structure and common matrix are mixed deposits the dynamic management structure, and concrete steps are as follows:
The design of pre-service and matrix of coefficients data structure
For the communication of well carrying out each step, we have taked the mode of global variable, such as precision, step-length, initial point, estimate point, check point, initially estimate direction, estimate direction etc. and all seal up for safekeeping and unifiedly in the middle of a global profile monitor renewal, the benefit of doing like this is to have lacked the assignment back and forth of a lot of intermediate variables in data exchange process, when data volume is large, the assignment time overhead is also very large, and be convenient to communication and debugging, therefore, before following the tracks of, at first these data to be carried out to pre-service;
In order to solve the asymmetry problem of vector in the middle of nonlinear programming problem, this method has been taked the data structure of sparse matrix and normal two-dimensional array and the data structure of depositing is carried out the processing of data, like this for different nonlinear programming problems, take two sets of data structures and homography computing method to be switched, the strategy of switching is: " number of non-zero entry is more than three times of null element number ", not only reduced space expense, and greatly reduce time complexity, here we only introduce the data structure volume description of sparse matrix:
struct Trituple{
Int row_index; The rower of // non-zero entry
Int col_index; The row mark of // non-zero entry
Double value; // element value
};
struct SparseMatrix{
Int Mrx_row; The row of // matrix
Int Mrx_col; // matrix column
Trituple* data; // tlv triple array
}；
Such as
Figure 782782DEST_PATH_IMAGE015
Such matrix takes the structure of sparse matrix to describe, and effect is apparent, and the ratio that this structure occupies in the middle of actual nonlinear programming problem is more higher;
The path trace process
Follow the trail of by
Figure 896232DEST_PATH_IMAGE016
In the process in the homotopy path of determining, method is divided following five steps:
1. initialization walks: provide three amounts in the initialization step: an initial point
Figure 634512DEST_PATH_IMAGE017
, an initialization step-length , carry out initialization in global variable step; Last allowable error
Figure 185896DEST_PATH_IMAGE019
Take macrodefined mode to set in inf;
2. estimate step: after obtaining initial point, carry out estimating of next point, the first step takes tangent line to estimate, and what all the other were taked is that secant is estimated mode, at first obtains at initial point
Figure 470247DEST_PATH_IMAGE020
The unit tangent vector at place
Figure 679511DEST_PATH_IMAGE021
,
Calculate Draw and estimate a little
Figure 889093DEST_PATH_IMAGE023
, the follow-up process of estimating is that the difference by getting the point after the current check point obtained and a upper correction thereof is carried out secant and estimated; Here used with the Gauss method of elimination of pivot in a column and asked tangent vector
Figure 154465DEST_PATH_IMAGE021
, below simply introduce the Gauss method of elimination with pivot in a column;
Figure 54287DEST_PATH_IMAGE024
Choose
Figure 620398DEST_PATH_IMAGE025
, make
Figure 656487DEST_PATH_IMAGE026
Maximize, and carry out following row exchange
Figure 344957DEST_PATH_IMAGE027
Figure 466497DEST_PATH_IMAGE028
It is that specifically the following formula of mistake obtains by with adjacent 2 of calculating, determining that is estimated a direction that secant is estimated main method:
Also having a kind of mode of estimating is that three Hermite estimate, and concrete formula is as follows:
Figure 805523DEST_PATH_IMAGE030
When estimating for the first time, only has point of initial value point, therefore, the mode that can only take tangent line to estimate, tangent line is estimated in the process of calculating tangent vector very consuming time, therefore, as long as obtain after first estimates a little, just taking simple and effective secant to estimate mode, this method can be fast a lot of on execution efficiency compared with other two kinds of modes, therefore taked first step tangent line to estimate, all the other steps are the mode that secant is estimated;
3. proofread and correct step: with
Figure 602578DEST_PATH_IMAGE031
For initial value, by process of iteration, produce a sequence
Figure 211413DEST_PATH_IMAGE032
, make
Figure 119327DEST_PATH_IMAGE033
For
Figure 192325DEST_PATH_IMAGE034
Approximate and the error of mid point is less than
Figure 160281DEST_PATH_IMAGE035
If iteration does not restrain, dwindle step-length and go back to and estimate step, adopt Newton iteration method here, step is as follows:
From a point
Figure 256413DEST_PATH_IMAGE036
Start, estimate
Figure 781066DEST_PATH_IMAGE037
, it provides its error and is
Figure 911833DEST_PATH_IMAGE038
Coarse estimation
Figure 50691DEST_PATH_IMAGE039
, can prevent this error propagation and be controlled through some step Newton iteration corrections, the initial value of this iterative process is discreet value
Figure 368539DEST_PATH_IMAGE040
, wish iteration convergence in
Figure 946151DEST_PATH_IMAGE041
Solution
Figure 931425DEST_PATH_IMAGE042
Vector
Figure 241183DEST_PATH_IMAGE043
With path
Figure 859378DEST_PATH_IMAGE044
The point
Figure 912784DEST_PATH_IMAGE045
Tangent, through point
Figure 752564DEST_PATH_IMAGE040
And with tangent vector
Figure 233224DEST_PATH_IMAGE043
Vertical
Figure 853561DEST_PATH_IMAGE046
The dimension lineoid
Figure 445080DEST_PATH_IMAGE047
The point
Figure 404946DEST_PATH_IMAGE048
(wherein
Figure 56507DEST_PATH_IMAGE049
) locate cutting path, trimming process is exactly in order to obtain this point, order
Figure 663942DEST_PATH_IMAGE050
',
Figure 59151DEST_PATH_IMAGE051
Mean
Figure 873523DEST_PATH_IMAGE040
Component, lineoid
Figure 758303DEST_PATH_IMAGE047
Can be write as:
Figure 25336DEST_PATH_IMAGE052
So trimming process is exactly solving equations
Figure 224236DEST_PATH_IMAGE053
The Iteration of Newton iteration is:
Newton iteration is local convergence, exists
Figure 699528DEST_PATH_IMAGE055
In exist a little
Figure 453857DEST_PATH_IMAGE042
A certain field
Figure 456448DEST_PATH_IMAGE056
, work as initial value
Figure 42150DEST_PATH_IMAGE057
The time, the Newton iteration convergence, if from
Figure 206416DEST_PATH_IMAGE040
The Newton iteration set out is not restrained, and by dwindling, estimates step-length
Figure 448041DEST_PATH_IMAGE018
, make
Figure 988744DEST_PATH_IMAGE040
Drop in certain neighborhood of Newton iteration convergence;
The distinguishing rule of iteration stopping has two kinds: the first, do not converge on a close point if estimate iteration, and should stop immediately iteration; The second, if iteration convergence, when approximate solution all in error up to specification
Figure 179685DEST_PATH_IMAGE019
In, also stop iteration, in the design of method, adopt well to comprise both of these case with the if structure, mistake estimate a little and perpendicular to the lineoid of estimating direction on proofreaied and correct, the correction equation group
Figure 780430DEST_PATH_IMAGE058
，
Wherein, vector dMean to estimate direction;
4. adjust step-length: this step mainly is considered for precision and cycle index, in trimming process, if
Figure 243773DEST_PATH_IMAGE059
Meet the demands, adjustment step-length that will be suitable
Figure 588166DEST_PATH_IMAGE018
, otherwise control flow turns back to and estimates step, in the process of method design, step-length carried out to mark successively, and in the Predictor Corrector process each time, this variable is becoming always, so realizes with the local variable of a static state;
5. verification walks: this step is the termination signal of whole path trace, once
Figure 882882DEST_PATH_IMAGE059
Middle component
Figure 654528DEST_PATH_IMAGE004
Equal 0 or close to 0, just stop following the tracks of, in the Newton iteration method process, at every turn all right tDo inspection, if the condition meet stopped, method just allow iterative loop stop and final result to out.
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Publication number Priority date Publication date Assignee Title
CN104362664A (en) * 2014-07-28 2015-02-18 浙江工业大学 Grid connection method of medium-voltage microgrid system
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