CN103631758A - Method for solving non-linear programming and absolute value equation through improved harmony search algorithm - Google Patents

Abstract

The invention discloses a method for solving non-linear programming and an absolute value equation through an improved harmony search algorithm. The method for solving the non-linear programming includes the steps of defining a question and parameter values, initializing a harmony memory bank, generating a new harmony by learning the harmony memory bank, finely adjusting embedded tones which are normally distributed and selecting the tones at random, evaluating a new solution, and detecting whether an algorithm end condition is met or not. The method for solving the absolute value equation includes the steps of defining a question and parameter values, initializing a harmony memory bank, generating a new harmony by learning the harmony memory bank, finely adjusting tones and selecting the tones, setting a probability WSR, selecting a harmony to be updated by comparing a random number Rand with the WSR, a new solution is evaluated, and detecting whether an algorithm end condition is met or not. According to the method for solving the non-linear programming and the absolute value equation, the non-linear programming and the absolute value equation are solved through the improved algorithm, and numerical stability and arithmetic speed and accuracy are improved.

Description

A kind ofly improve the method that harmony searching algorithm solves nonlinear programming and Absolute Value Equation

Technical field

The invention belongs to and improve harmony searching algorithm technical field, relate in particular to a kind of method that harmony searching algorithm solves nonlinear programming and Absolute Value Equation of improving.

Background technology

Harmony search (Harmony search, HS) algorithm is the intelligent optimization algorithm of a kind of novelty of people's propositions such as calendar year 2001 Korea S scholar Geem Z W.Algorithm simulation in musical composition musicians rely on oneself memory, by repeatedly adjusting the tone of each musical instrument in band, finally reach the process of a beautiful harmony state.At present, the method is widely applied in the problems such as the optimization of multidimensional multi-extreme value function, pipeline optimal design, slope stability analysis.

Nonlinear programming is an important branch of planning strategies for, the extreme-value problem of its research objective function under certain constraint condition.Traditional mostly gradient class algorithm based on traditional of nonlinear programming approach that solves, and depend on choosing of initial point, and how to choose suitable initial point in some practical problemss, it itself is a more difficult problem, all in all, current these algorithms are the gradient class algorithm based on traditional mostly, the result of trying to achieve is also local optimum, solves at present globally optimal solution and remains a challenge difficult problem.

Along with the appearance of various intelligent algorithms, the research that solves nonlinear programming problem by intelligent algorithm also starts.Concrete grammar is exactly the fitness function of structure nonlinear programming problem, then adopts intelligent algorithm to solve.In recent years, genetic algorithm, particle cluster algorithm, differential evolution algorithm, ant group algorithm, Cultural Algorithm, the bacterium algorithm etc. of looking for food has been widely applied to solving of nonlinear programming problem, and makes great progress.

Absolute Value Equation refers to:

Ax-|x|=b

A ∈ R wherein ^{n * n}, x, b ∈ R ⁿ, | x| represents each component of x to take absolute value, and this problem is referred to as Absolute Value Equation (Absolute Value Equations), notes by abridging as AVE.It is the important subclass of the absolute value matrix equation of Jiri Rohn proposition.The research of Absolute Value Equation derives from two aspects, and one is Interval linear equation.Interval linear equation refers to that the coefficient of equation and constant term are not known definite, but is positioned at a certain interval, and equation number equates with unknown quantity number.Another source is linear complementary problem (linear complementary problem is the optimization problem that a class has broad practice background, and it unifies for linear programming, quadratic programming provide one framework of studying).Research for Absolute Value Equation at present mainly concentrates on theoretical and two aspects of algorithm, and the former mainly studies its existence of solution, uniqueness; And the latter mainly sets up its effective method for solving and corresponding convergence.Louis Caccetta has provided a Smoothing Newton Method that solves Absolute Value Equation in the article of delivering for 2010, and this algorithm substitutes the ABS function in former Absolute Value Equation with a smooth function, and has proved that this Smoothing Method has quadratic convergence.This is that at present relevant Absolute Value Equation is studied best notional result.The applied research of relevant Absolute Value Equation also starts.

Existing research be to have unique solution Absolute Value Equation, provide algorithm and carry out corresponding convergence.Algorithm is to solve the in the situation that of a given initial point, therefore can not obtain all solutions; For existing the Absolute Value Equation of a plurality of solutions how to obtain all solutions, this is a more difficult problem.Swarm Intelligence Algorithm is a kind of random search algorithm, this algorithm is the random initial point that produces in feasible zone, by a series of operation (selection, variation etc.), finally converge to the approximate solution of former problem, because Swarm Intelligence Algorithm is that multiple spot starts, therefore just likely look for solution as much as possible.

Summary of the invention

The object of the embodiment of the present invention is to provide a kind of method that harmony searching algorithm solves nonlinear programming and Absolute Value Equation of improving, be intended to solve traditional solve how nonlinear programming approach chooses that suitable initial point is more difficult, Absolute Value Equation method for solving is to existing the Absolute Value Equation of a plurality of solutions can not obtain the problem of all solutions.

The embodiment of the present invention is achieved in that a kind of method that harmony searching algorithm solves nonlinear programming of improving, and the method that this improvement harmony searching algorithm solves nonlinear programming comprises the following steps:

The first step, problem definition and parameter value: suppose that problem is for minimizing, the following minf of form (x), s.t.x _i∈ X _i, i=1,2 ..., N, f (x) is objective function, x is by decision variable x _ithe solution vector forming (i=1,2 ..., N), the codomain of each variable is X _i: x _i ^l≤ X _i≤ x _i ^u, N is decision variable number, algorithm parameter has: the number of times of the size of harmony data base, study harmony data base probability, tone fine setting probability, tone fine setting bandwidth, creation;

Second step, initialization harmony data base:

HMS harmony x of random generation ¹, x ²..., x ^hMSput into harmony data base, harmony data base can be analogous to the population in genetic algorithm, and harmony data base is as follows:

$\mathrm{HM}=\left[\begin{array}{c}{x}^1\\ x^2\\ .\\ .\\ .\\ x^{\mathrm{HMS}}\end{array}\right.|\left.\begin{array}{c}f\left(x^1\right)\\ f\left(x^2\right)\\ .\\ .\\ .\\ f\left(x^{\mathrm{HMS}}\right)\end{array}\right]=\left[\begin{array}{cccc}{x}_1^1& x_2^1& ...& x_N^1\\ x_1^2& x_2^2& ...& x_{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="HSA0000098294050000071.png"\; state="\{\{state\}\}">N</figure-callout>}}^2\\ .& .& & .\\ .& .& ...& .\\ .& .& & .\\ x_1^{\mathrm{HMS}}& x_2^{\mathrm{HMS}}& ...& x_N^{\mathrm{HMS}}\end{array}\right.|\left.\begin{array}{c}f\left(x^1\right)\\ f\left(x^2\right)\\ .\\ .\\ .\\ f\left(x^{\mathrm{HMS}}\right)\end{array}\right];$

The 3rd step, generates a new harmony:

Generate new harmony x _i'=(x ' ₁, x ' ₂..., x ' _n), each tone x of new harmony _i' (i=1,2 ..., N) by following three kinds of mechanism, produce: study harmony data base, tone fine setting, selects tone at random;

First variable x ' of new explanation ₁there is the probability of HMCR to be selected from HM

any one value, have the probability of 1-HCMR to be selected from outside HM, and in variable range any one value, same, the generating mode of other variable is as follows:

$x_i^{\&prime;}=\left\{\begin{array}{cc}{x}_i^{\&prime;}\&Element;(x_i^1,x_i^2,...,x_i^{\mathrm{HMS}}),& \mathrm{ifrand}<\mathrm{HMCR}\\ x_i^{\&prime;}\&Element;X_i,& \mathrm{otherwise}\end{array}\right.;(i=\mathrm{1,2},...,N)$

Rand represents the equally distributed random number on [0,1];

Secondly, if new harmony x ' _ifrom harmony data base HM, during tone fine setting, adopt following operation

$x_i^{\&prime;}=\left\{\begin{array}{cc}{x}_i^{\&prime;}+\mathrm{NR}\&times;\mathrm{bw},& \mathrm{ifrand}<\mathrm{PAR}\\ x_i^{\&prime;},& \mathrm{otherwise}\end{array}\right.;(i=\mathrm{1,2},...,N)$

Wherein, bw is tone fine setting bandwidth, and PAR is tone fine setting probability; NR is a random number of obeying standardized normal distribution, here u ₁and u ₂all represent upper equally distributed random number;

The 4th step, upgrades harmony data base

New explanation in the 3rd step is assessed, if be better than the poorest one of functional value in harmony data base, new explanation is updated in harmony data base (HM), concrete operations are as follows:

$\mathrm{If\; f}\left(x^{\&prime;}\right)<f\left(x^{\mathrm{worst}}\right)=\underset{j=\mathrm{1,2},...,\mathrm{HMS}}{\max }f\left(x^j\right),\mathrm{then}{x}^{\mathrm{worst}}=x^{\&prime;};$

The 5th step, checks and whether reaches algorithm end condition

Repeat the 3rd step and the 4th step, until creation number of times reaches Tmax.

Further, in the first step, the number of times of the size of harmony data base, study harmony data base probability, tone fine setting probability, tone fine setting bandwidth, creation will be initialised.

Another object of the embodiment of the present invention is to provide a kind of method that harmony searching algorithm solves Absolute Value Equation of improving, and the method that this improvement harmony searching algorithm solves Absolute Value Equation comprises the following steps:

The first step, problem definition and parameter value: suppose that problem is for minimizing, the following minf of form (x), s.t.x _i∈ X _i, i=1,2 ..., N, f (x) is objective function here, x is by decision variable x _ithe solution vector forming (i=1,2 ..., N), the codomain of each variable is X _i: x _i ^l≤ X _i≤ x _i ^u, N is decision variable number, algorithm parameter has: and the number of times of the size of harmony data base, study harmony data base probability, tone fine setting probability, tone fine setting bandwidth, creation, each parameter all will be initialised in the first step;

Second step, initialization harmony data base:

HMS harmony x of random generation ¹, x ²..., x ^hMSput into harmony data base, harmony data base can be analogous to the population in genetic algorithm;

The 3rd step, generates a new harmony:

Generate new harmony x _i'=(x ' ₁, x ' ₂..., x ' _n), generating new harmony X _newtime, set a probability WSR, produce a random number R and, if Rand<WSR upgrades the poorest solution in current harmony storehouse HM; Otherwise, the preferably solution in current harmony storehouse HM is upgraded,, in population renewal process, with certain probability, in feasible zone, to choose meanwhile, the algorithm after note is improved is HSWB, each tone x of new harmony _i' (i=1,2 ..., N) by following three kinds of mechanism, produce: study harmony data base, tone fine setting, selects tone at random;

First variable x ' of new explanation ₁there is the probability of HMCR to be selected from HM

any one value, have the probability of 1-HCMR to be selected from any one value of HM outer (and in variable range), same, the generating mode of other variable is as follows:

$x_i^{\&prime;}=\left\{\begin{array}{cc}{x}_i^{\&prime;}\&Element;(x_i^1,x_i^2,...,x_i^{\mathrm{HMS}}),& \mathrm{ifrand}<\mathrm{HMCR}\\ x_i^{\&prime;}\&Element;X_i,& \mathrm{otherwise}\end{array}\right.;(i=\mathrm{1,2},...,N)$

Rand represents the equally distributed random number on [0,1];

Secondly, if new harmony x ' _ifrom harmony data base, finely tune by row tone, concrete operations are as follows:

The 4th step, upgrades harmony data base

New explanation in the 3rd step is assessed, if be better than the poorest one of functional value in HM, new explanation is updated in HM, concrete operations are as follows:

$\mathrm{If\; f}\left(x^{\&prime;}\right)<f\left(x^{\mathrm{worst}}\right)=\underset{j=\mathrm{1,2},...,\mathrm{HMS}}{\max }f\left(x^j\right),\mathrm{then}{x}^{\mathrm{worst}}=x^{\&prime;};$

The 5th step, checks and whether reaches algorithm end condition

Repeat the 3rd step and the 4th step, until creation (iteration) number of times reaches Tmax.

Further, in second step, harmony data base is expressed as:

$\mathrm{HM}=\left[\begin{array}{c}{x}^1\\ x^2\\ .\\ .\\ .\\ x^{\mathrm{HMS}}\end{array}\right.|\left.\begin{array}{c}f\left(x^1\right)\\ f\left(x^2\right)\\ .\\ .\\ .\\ f\left(x^{\mathrm{HMS}}\right)\end{array}\right]=\left[\begin{array}{cccc}{x}_1^1& x_2^1& ...& x_N^1\\ x_1^2& x_2^2& ...& x_{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="HSA0000098294050000071.png"\; state="\{\{state\}\}">N</figure-callout>}}^2\\ .& .& & .\\ .& .& ...& .\\ .& .& & .\\ x_1^{\mathrm{HMS}}& x_2^{\mathrm{HMS}}& ...& x_N^{\mathrm{HMS}}\end{array}\right.|\left.\begin{array}{c}f\left(x^1\right)\\ f\left(x^2\right)\\ .\\ .\\ .\\ f\left(x^{\mathrm{HMS}}\right)\end{array}\right].$

Improvement harmony searching algorithm provided by the invention solves the method for nonlinear programming and Absolute Value Equation, the present invention improves harmony searching algorithm, Algorithm for Solving after application enhancements nonlinear programming problem, and contrast with former algorithm, from numerical result, obviously see that algorithm improvement is that numerical stability or arithmetic speed and precision are all very successful, and there is speed of convergence faster.

The present invention is by having unique solution and separating solving of Absolute Value Equations more, result shows that the harmony searching algorithm after improvement has speed of convergence faster, and can find former problem solution as much as possible, algorithm simple, intuitive of the present invention, parameter is few, easily realizing, is a kind of efficient algorithm that solves Absolute Value Equation.

Accompanying drawing explanation

Fig. 1 is that the method flow diagram of nonlinear programming is provided in the improvement harmony searching algorithm providing the invention process;

Fig. 2 is that the invention process is in the schematic diagram of the distribution of two kinds of random numbers that provide;

Fig. 3 is that the invention process is in the f providing ₁(x) convergence curve schematic diagram;

Fig. 4 is that the invention process is in the f providing ₂(x) convergence curve schematic diagram;

Fig. 5 is that the invention process is in the f providing ₃(x) convergence curve schematic diagram;

Fig. 6 is that the invention process is in the f providing ₄(x) convergence curve schematic diagram;

Fig. 7 is that the invention process is in the f providing ₅(x) convergence curve schematic diagram;

Fig. 8 is that the invention process is in the f providing ₆(x) convergence curve schematic diagram;

Fig. 9 is that the method flow diagram of Absolute Value Equation is provided in the improvement harmony searching algorithm providing the invention process;

Figure 10 is that the invention process is in the AVE1 providing, AVE2, AVE3 adaptive value convergence curve schematic diagram;

Figure 11 is that the invention process is in the solving result schematic diagram of the AVE4 providing;

Figure 12 is that the invention process is in the solving result schematic diagram of the AVE5 providing.

Embodiment

In order to make object of the present invention, technical scheme and advantage clearer, below in conjunction with embodiment, the present invention is further elaborated.Should be appreciated that concrete enforcement described herein, only in order to explain the present invention, is not intended to limit the present invention.

Below in conjunction with drawings and the specific embodiments, application principle of the present invention is further described.

As shown in Figure 1, the method that the improvement harmony searching algorithm of the embodiment of the present invention solves nonlinear programming comprises the following steps:

S101: the parameter value of the number of times of the size of problem definition and harmony data base, study harmony data base probability, tone fine setting probability, tone fine setting bandwidth, creation;

S102: initialization harmony data base;

S103: finely tune, select tone to generate new harmony at random by study harmony data base, tone;

S104: new explanation is assessed, if be better than the poorest one of functional value in harmony data base, new explanation is updated in harmony data base;

S105: check and whether reach algorithm end condition, until creation (iteration) number of times reaches Tmax.

Harmony searching algorithm in the present invention has been simulated musicians in musical composition and, by means of the memory of oneself, by repeatedly adjusting the tone of each musical instrument in band, has finally been reached the process of a beautiful harmony state.Harmony searching algorithm is analogous to the harmony of musical instrument tone the solution vector of optimization problem, and evaluate is each corresponding target function value.Algorithm is introduced two major parameters, i.e. data base probability (HMCR) and fine setting probability (PAR).First algorithm produces HMS initial solution (harmony) and puts into harmony data base HM (Harmony memory); Then, random search new explanation in harmony data base, the probability that produces new explanation is determined by HMCR, produces at random 0～1 random number rand.If rand<HMCR, new explanation random search in HM obtains; Otherwise outside harmony data base, and be the interior search of the feasible zone value of variable.With fine setting probability P AR, the new explanation of taking from HM is carried out to local dip again.Finally, judge whether new explanation target function value is better than the poorest solution in HM, if so, upgrade harmony storehouse, and continuous iteration, until reach predetermined iterations Tmax.

The method concrete steps that improvement harmony searching algorithm of the present invention solves nonlinear programming are:

The first step, problem definition and parameter value:

Suppose that problem is for minimizing, the following minf of its form (x), s.t.x _i∈ X _i, i=1,2 ..., N, f (x) is objective function here, x is by decision variable x _ithe solution vector forming (i=1,2 ..., N), the codomain of each variable is X _i: x _i ^l≤ X _i≤ x _i ^u, N is decision variable number.Algorithm parameter has: 1. the size of harmony data base (HMS), 2. learn harmony data base probability (HMCR), 3. tone fine setting probability (PAR), 4. tone fine setting bandwidth (bw), the 5. number of times (Tmax) of creation, each parameter all will be initialised in the first step.

Second step, initialization harmony data base:

HMS harmony x of random generation ¹, x ²..., x ^hMSput into harmony data base, harmony data base can be analogous to the population in genetic algorithm.Harmony data base is as follows:

$\mathrm{HM}=\left[\begin{array}{c}{x}^1\\ x^2\\ .\\ .\\ .\\ x^{\mathrm{HMS}}\end{array}\right.|\left.\begin{array}{c}f\left(x^1\right)\\ f\left(x^2\right)\\ .\\ .\\ .\\ f\left(x^{\mathrm{HMS}}\right)\end{array}\right]=\left[\begin{array}{cccc}{x}_1^1& x_2^1& ...& x_N^1\\ x_1^2& x_2^2& ...& x_{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="HSA0000098294050000071.png"\; state="\{\{state\}\}">N</figure-callout>}}^2\\ .& .& & .\\ .& .& ...& .\\ .& .& & .\\ x_1^{\mathrm{HMS}}& x_2^{\mathrm{HMS}}& ...& x_N^{\mathrm{HMS}}\end{array}\right.|\left.\begin{array}{c}f\left(x^1\right)\\ f\left(x^2\right)\\ .\\ .\\ .\\ f\left(x^{\mathrm{HMS}}\right)\end{array}\right]$

The 3rd step, generates a new harmony:

Generate new harmony x _i'=(x ' ₁, x ' ₂..., x ' _n), each tone x of new harmony _i' (i=1,2 ..., N) by following three kinds of mechanism, produce: 1. learn harmony data base, 2. tone fine setting, 3. selects tone at random.

First variable x ' of new explanation ₁there is the probability of HMCR to be selected from HM

any one value, have the probability of 1-HCMR to be selected from any one value of HM outer (and in variable range).Same, the generating mode of other variable is as follows:

$x_i^{\&prime;}=\left\{\begin{array}{cc}{x}_i^{\&prime;}\&Element;({\mathrm{<figure-callout\; id="1"\; label="x\; i"\; filenames="HSA0000098294050000071.png"\; state="\{\{state\}\}">x</figure-callout>}}_i^{},{\mathrm{<figure-callout\; id="2"\; label="x\; i"\; filenames="HSA0000098294050000071.png"\; state="\{\{state\}\}">x</figure-callout>}}_i^{},...,x_i^{\mathrm{HMS}}),& \mathrm{ifrand}<\mathrm{HMCR}\\ x_i^{\&prime;}\&Element;X_i,& \mathrm{otherwise}\end{array}\right.;(i=\mathrm{1,2},...,N)$

Rand represents the equally distributed random number on [0,1].

Secondly, if new harmony x ' _ifrom harmony data base HM, during tone fine setting, adopt following operation

$x_i^{\&prime;}=\left\{\begin{array}{cc}{x}_i^{\&prime;}+\mathrm{NR}\&times;\mathrm{bw},& \mathrm{ifrand}<\mathrm{PAR}\\ x_i^{\&prime;},& \mathrm{otherwise}\end{array}\right.;(i=\mathrm{1,2},...,N)$

Wherein, bw is tone fine setting bandwidth, and PAR is tone fine setting probability; NR is a random number of obeying standardized normal distribution, and NR adopts Box-Muller method to generate,

here u ₁and u ₂all represent upper equally distributed random number.

The 4th step, upgrades harmony data base

New explanation in the 3rd step is assessed, if be better than the poorest one of functional value in harmony data base (HM), new explanation is updated in harmony data base (HM), concrete operations are as follows:

$\mathrm{If\; f}\left(x^{\&prime;}\right)<f\left(x^{\mathrm{worst}}\right)=\underset{j=\mathrm{1,2},...,\mathrm{HMS}}{\max }f\left(x^j\right),\mathrm{then}{x}^{\mathrm{worst}}=x^{\&prime;}.$

The 5th step, checks and whether reaches algorithm end condition

Repeat the 3rd step and the 4th step, until creation number of times reaches Tmax.

The method that solves nonlinear programming by following checking improvement of the present invention harmony searching algorithm is described further:

After improvement of the present invention, harmony searching algorithm adopts standardized normal distribution random number, Fig. 2 has provided the distribution of two kinds of random numbers, obviously, the population diversity that adopts standardized normal distribution random number can improve population (has expanded hunting zone, effectively flee from local optimum), thus the algorithm after improving has good convergence.

Numerical experiment

Performance for harmony search (HS) algorithm after relatively improving, the present invention uses following 6 Benchmark trial functions (choosing n=30), as shown in table 1, harmony search (HS) algorithm routine is write with Matlab7.1, parameter is chosen: HMS=10, HMCR=0.85, PAR=0.45, bw=0.01, the greatest iteration algebraically Tmax=5000 that algorithm stops.During tone fine setting, adopt respectively [0,1] upper uniform random number and standardized normal distribution random number, dotted line adopts [0 while representing tone fine setting, result while 1] going up uniform random number, result when solid line adopts standardized normal distribution random number while representing tone fine setting, the convergence curve of example 1～example 6 is as shown in Fig. 3～Fig. 8.

A table 1.6 Benchmark trial function

As shown in Figure 9, the method that the improvement harmony searching algorithm that the embodiment of the present invention provides solves Absolute Value Equation comprises the following steps:

S901: the parameter value of the number of times of the size of problem definition and harmony data base, study harmony data base probability, tone fine setting probability, tone fine setting bandwidth, creation;

S902: initialization harmony data base;

S903: finely tune, select tone to generate new harmony at random by study harmony data base, tone, set a probability WSR, produce a random number R and, determine whether to upgrade harmony data base;

S904: new explanation is assessed, if be better than the poorest one of functional value in harmony data base, new explanation is updated in harmony data base;

S905: check and whether reach algorithm end condition, until creation (iteration) number of times reaches Tmax.

The method concrete steps that improvement harmony searching algorithm of the present invention solves Absolute Value Equation are:

The first step, problem definition and parameter value:

Suppose that problem is for minimizing, the following minf of its form (x), s.t.x _i∈ X _i, i=1,2 ..., N, f (x) is objective function here, x is by decision variable x _ithe solution vector forming (i=1,2 ..., N), the codomain of each variable is X _i: x _i ^l≤ X _i≤ x _i ^u, N is decision variable number.Algorithm parameter has: 1. the size of harmony data base (HMS), 2. learn harmony data base probability (HMCR), 3. tone fine setting probability (PAR), 4. tone fine setting bandwidth (bw), the 5. number of times (Tmax) of creation, each parameter all will be initialised in the first step.

Second step, initialization harmony data base:

HMS harmony x of random generation ¹, x ²..., x ^hMSput into harmony data base, harmony data base can be analogous to the population in genetic algorithm.Harmony data base is as follows:

$\mathrm{HM}=\left[\begin{array}{c}{x}^1\\ x^2\\ .\\ .\\ .\\ x^{\mathrm{HMS}}\end{array}\right.|\left.\begin{array}{c}f\left(x^1\right)\\ f\left(x^2\right)\\ .\\ .\\ .\\ f\left(x^{\mathrm{HMS}}\right)\end{array}\right]=\left[\begin{array}{cccc}{x}_1^1& x_2^1& ...& x_N^1\\ x_1^2& x_2^2& ...& x_{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="HSA0000098294050000071.png"\; state="\{\{state\}\}">N</figure-callout>}}^2\\ .& .& & .\\ .& .& ...& .\\ .& .& & .\\ x_1^{\mathrm{HMS}}& x_2^{\mathrm{HMS}}& ...& x_N^{\mathrm{HMS}}\end{array}\right.|\left.\begin{array}{c}f\left(x^1\right)\\ f\left(x^2\right)\\ .\\ .\\ .\\ f\left(x^{\mathrm{HMS}}\right)\end{array}\right]$

The 3rd step, generates a new harmony:

Generate new harmony x _i'=(x ' ₁, x ' ₂..., x ' _n), when generating new harmony, set a probability WSR, produce a random number R and, if Rand<WSR upgrades " the poorest solution " in current harmony storehouse HM; Otherwise, " preferably separating " in current harmony storehouse HM upgraded., in population renewal process, with certain probability (RSR), in feasible zone, choose, the algorithm after note is improved is HSWB (Harmony Search with Worst and Best: the worst best harmony is searched for), each tone x of new harmony meanwhile _i' (i=1,2 ..., N) by following three kinds of mechanism, produce: 1. learn harmony data base, 2. tone fine setting, 3. selects tone at random.

First variable x ' of new explanation ₁there is the probability of HMCR to be selected from HM

any one value, have the probability of 1-HCMR to be selected from any one value of HM outer (and in variable range).Same, the generating mode of other variable is as follows:

$x_i^{\&prime;}=\left\{\begin{array}{cc}{x}_i^{\&prime;}\&Element;({\mathrm{<figure-callout\; id="1"\; label="x\; i"\; filenames="HSA0000098294050000071.png"\; state="\{\{state\}\}">x</figure-callout>}}_i^{},{\mathrm{<figure-callout\; id="2"\; label="x\; i"\; filenames="HSA0000098294050000071.png"\; state="\{\{state\}\}">x</figure-callout>}}_i^{},...,x_i^{\mathrm{HMS}}),& \mathrm{ifrand}<\mathrm{HMCR}\\ x_i^{\&prime;}\&Element;X_i,& \mathrm{otherwise}\end{array}\right.;(i=\mathrm{1,2},...,N)$

Rand represents the equally distributed random number on [0,1].

Secondly, if new harmony x ' _ifrom harmony data base HM, carry out tone fine setting to it, concrete operations are as follows:

The 4th step, upgrades harmony data base

New explanation in the 3rd step is assessed, if be better than the poorest one of functional value in HM, new explanation is updated in HM, concrete operations are as follows:

$\mathrm{If\; f}\left(x^{\&prime;}\right)<f\left(x^{\mathrm{worst}}\right)=\underset{j=\mathrm{1,2},...,\mathrm{HMS}}{\max }f\left(x^j\right),\mathrm{then}{x}^{\mathrm{worst}}=x^{\&prime;}.$

The 5th step, checks and whether reaches algorithm end condition

Repeat the 3rd step and the 4th step, until creation (iteration) number of times reaches Tmax.

The method that solves Absolute Value Equation by following checking improvement of the present invention harmony searching algorithm is described further:

Selecting All Parameters WSR=0.7 of the present invention * (1-k/Tmax)+0.3, parameter WSR dynamically reduces along with the increase of iterations k, and this mainly upgrades " the poorest solution " in HM with regard to guaranteeing iteration originally, and the later stage mainly upgrades in HM and " preferably separates "; This algorithm can upgrade " the poorest solution " (pusher) like this, also can improve " preferably separating " (front drawing) simultaneously, can effectively prevent precocity, and the accelerating convergence (train of climbing the mountain as same, respectively there is a headstock front and back, headstock below promotes train from back to front, and headstock above constantly draws, and train just can reach the top of the hill quickly like this).Parameters R SR chooses the diversity that can improve population, generally between [0.1,0.3], chooses better.

Numerical experiment

In order to test the performance that solves of this paper algorithm, extensive AVE problem is tested, and algorithm routine is write with MatlabR2009a, algorithm parameter is chosen: HMS=15, HMCR=0.8, PAR=0.5, bw=0.01, the maximum evolutionary generation Tmax=10000 that algorithm stops, RSR=0.15; For eliminating the impact of randomness on algorithm, HS algorithm and HSWB algorithm respectively move 10 times, AVE1, and AVE2, AVE3 has unique solution; AVE4, AVE5 has a plurality of solutions.

AVE1. matrix A meets A ^t=A, its element is a _ii=500, a _ij=1+rand, i ≠ j, rand represents the random number in [0,1] here, b=(A-I) e; This problem meets the condition (singular value of A is greater than 1) of theorem 1, so this problem exists unique solution, and unique solution be x=(1,1 ..., 1) ^t.

AVE2. matrix A meets

A _ii=4n, a _{i, i+1}=a _{i+1, i}=n; Remaining a _ij=0.5, i, j=1,2 ..., n,

B=(A-I) e; This problem meets the condition (singular value of A is greater than 1) of theorem 1, so this problem exists unique solution, and unique solution be x=(1,1 ..., 1) ^t.

AVE3. singular value is greater than 1 matrix A by the random generation of following Matlab order

rand(′state′,0)；

A=rand(n,n)′*rand(n,n)+n*eye(n)；

b＝rand(n，1)；

Illustrate: provide after matrix dimension n, reader can produce and identical herein data with above-mentioned code, and this problem has unique solution.

Table 2, table 3 have provided respectively 100 dimensions and have moved the optimal-adaptive value of 10 times, average adaptive value, the poorest adaptive value and standard deviation with the AVE problem of 200 dimensions.

The statistics comparison (n=100) that two kinds of algorithm operations of table 2 are 10 times

The statistics comparison (n=200) that two kinds of algorithm operations of table 3 are 10 times

Statistics from table 2 and table 3 can find out, HSWB Algorithm for Solving result is significantly better than HS algorithm, and this shows that HSWB algorithm has stronger search capability.

In order more clearly to provide the search capability of two kinds of algorithms, when Figure 10 has provided n=200, two kinds of algorithms move the average adaptive value convergence curve of 10 times.

AVE4. consider following problem

$A=\left[\begin{array}{cc}0.1& 0.02\\ 0.2& 0.01\end{array}\right],b=\left[\begin{array}{c}-1\\ -2\end{array}\right],$

Due to γ=min _i| b _i|/max _i| b _i|=0.5, || A|| _∞=0.21< γ/2, so this problem has 2 ²=4 solutions. these four solutions are x ⁱ=D _i(AD _i-I) ^-1b, i=1,2,3,4.

Here ${\mathrm{<figure-callout\; id="1"\; label="D"\; filenames="HSA0000098294050000071.png"\; state="\{\{state\}\}">D</figure-callout>}}_1=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="HSA0000098294050000071.png"\; state="\{\{state\}\}">D</figure-callout>}}_2=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],D_3=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right],D_4=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right];$

Therefore

$x^1=\left[\begin{array}{c}1.1612\\ 2.2548\end{array}\right],x^2=\left[\begin{array}{c}1.0624\\ -2.1906\end{array}\right],x^3=\left[\begin{array}{c}-0.9424\\ 1.8298\end{array}\right],x^4=\left[\begin{array}{c}-0.8762\\ -1.8067\end{array}\right];$

HS algorithm and HSWB algorithm respectively move 10 times, and result as shown in figure 11.

AVE5. consider following problem

$A=\left[\begin{array}{ccc}0.01& 0.02& 0.03\\ 0.02& 0.03& 0.01\\ 0.03& 0.02& 0.01\end{array}\right],b=\left[\begin{array}{c}-1\\ -2\\ -3\end{array}\right],$

Due to γ=min _i| b _i|/max _i| b _i|=1/3, || A|| _∞=0.06< γ/2, so this problem has 2 ³=8 solutions. these eight solutions are

x ¹=D _i(AD _i-I) ^-1b，i=1，2，…，8，D _i=diag(d ₁，d ₂，d ₃)，d _i=±1，i=1，2，3.

HS algorithm and HSWB algorithm respectively move 10 times, and result as shown in figure 12.

More example shows, HSWB convergence of algorithm performance is good, and by less cost (number of run), HSWB algorithm can find former problem solution as much as possible, and it is basically identical to obtain the probability of each solution.

The present invention improves harmony searching algorithm, Algorithm for Solving after application enhancements nonlinear programming problem, and contrast with former algorithm, from numerical result, obviously see that algorithm improvement is that numerical stability or arithmetic speed and precision are all very successful, and there is speed of convergence faster.

The present invention is by having unique solution and separating solving of Absolute Value Equations more, result shows that the harmony searching algorithm after improvement has speed of convergence faster, and can find former problem solution as much as possible, algorithm simple, intuitive of the present invention, parameter is few, easily realizing, is a kind of efficient algorithm that solves Absolute Value Equation.

The foregoing is only preferred embodiment of the present invention, not in order to limit the present invention, all any modifications of doing within the spirit and principles in the present invention, be equal to and replace and improvement etc., within all should being included in protection scope of the present invention.

Claims (3)

1. improve the method that harmony searching algorithm solves nonlinear programming, it is characterized in that, the method that this improvement harmony searching algorithm solves nonlinear programming comprises the following steps:

The first step, problem definition and parameter value: suppose that problem is for minimizing, the following minf of form (x), s.t.x _i∈ X _i, i=1,2 ..., N, f (x) is objective function, x is by decision variable x _ithe solution vector forming (i=1,2 ..., N), the codomain of each variable is X _i: x _i ^l≤ X _i≤ x _i ^u, N is decision variable number, algorithm parameter has: the number of times of the size of harmony data base, study harmony data base probability, tone fine setting probability, tone fine setting bandwidth, creation;

Second step, initialization harmony data base:

HMS harmony x of random generation ¹, x ²..., x ^hMSput into harmony data base, harmony data base can be analogous to the population in genetic algorithm, and harmony data base is as follows:

$\mathrm{HM}=\left[\begin{array}{c}{x}^1\\ x^2\\ .\\ .\\ .\\ x^{\mathrm{HMS}}\end{array}\right.|\left.\begin{array}{c}f\left(x^1\right)\\ f\left(x^2\right)\\ .\\ .\\ .\\ f\left(x^{\mathrm{HMS}}\right)\end{array}\right]=\left[\begin{array}{cccc}{x}_1^1& x_2^1& ...& x_N^1\\ x_1^2& x_2^2& ...& x_N^2\\ .& .& & .\\ .& .& ...& .\\ .& .& & .\\ x_1^{\mathrm{HMS}}& x_2^{\mathrm{HMS}}& ...& x_N^{\mathrm{HMS}}\end{array}\right.|\left.\begin{array}{c}f\left(x^1\right)\\ f\left(x^2\right)\\ .\\ .\\ .\\ f\left(x^{\mathrm{HMS}}\right)\end{array}\right];$

The 3rd step, generates a new harmony:

Generate new harmony x _i'=(x ' ₁, x ' ₂..., x ' _n), each tone x of new harmony _i' (i=1,2 ..., N) by following three kinds of mechanism, produce, study harmony data base, tone fine setting, selects tone at random;

First variable x ' of new explanation ₁there is the probability of HMCR to be selected from HM

any one value, have the probability of 1-HCMR to be selected from outside HM, and in variable range any one value, same, the generating mode of other variable is as follows:

$x_i^{\&prime;}=\left\{\begin{array}{cc}{x}_i^{\&prime;}\&Element;(x_i^1,x_i^2,...,x_i^{\mathrm{HMS}}),& \mathrm{ifrand}<\mathrm{HMCR}\\ x_i^{\&prime;}\&Element;X_i,& \mathrm{otherwise}\end{array}\right.;(i=\mathrm{1,2},...,N)$

Rand represents the equally distributed random number on [0,1];

Secondly, if new harmony x ' _ifrom harmony data base HM, during tone fine setting, adopt following operation

$x_i^{\&prime;}=\left\{\begin{array}{cc}{x}_i^{\&prime;}+\mathrm{NR}\&times;\mathrm{bw},& \mathrm{ifrand}<\mathrm{PAR}\\ x_i^{\&prime;},& \mathrm{otherwise}\end{array}\right.;(i=\mathrm{1,2},...,N)$

Wherein, bw is tone fine setting bandwidth, and PAR is tone fine setting probability; NR is a random number of obeying standardized normal distribution,

here u ₁and u ₂all represent upper equally distributed random number;

The 4th step, upgrades harmony data base

New explanation in the 3rd step is assessed, if be better than the poorest one of functional value in harmony data base, new explanation is updated in harmony data base (HM), concrete operations are as follows:

$\mathrm{If\; f}\left(x^{\&prime;}\right)<f\left(x^{\mathrm{worst}}\right)=\underset{j=\mathrm{1,2},...,\mathrm{HMS}}{\max }f\left(x^j\right),\mathrm{then}{x}^{\mathrm{worst}}=x^{\&prime;};$

The 5th step, checks and whether reaches algorithm end condition

Repeat the 3rd step and the 4th step, until creation number of times reaches Tmax.

2. improvement harmony searching algorithm as claimed in claim 1 solves the method for nonlinear programming, it is characterized in that, in the first step, the number of times of the size of harmony data base, study harmony data base probability, tone fine setting probability, tone fine setting bandwidth, creation will be initialised.

3. improve the method that harmony searching algorithm solves Absolute Value Equation, it is characterized in that, the method that this improvement harmony searching algorithm solves Absolute Value Equation comprises the following steps:

The first step, problem definition and parameter value: suppose that problem is for minimizing, the following minf of form (x), s.t.x _i∈ X _i, i=1,2 ..., N, f (x) is objective function here, x is by decision variable x _ithe solution vector forming (i=1,2 ..., N), the codomain of each variable is X _i: x _i ^l≤ X _i≤ x _i ^u, N is decision variable number, algorithm parameter has: and the number of times of the size of harmony data base, study harmony data base probability, tone fine setting probability, tone fine setting bandwidth, creation, each parameter all will be initialised in the first step;

Second step, initialization harmony data base:

HMS harmony x of random generation ¹, x ²..., x ^hMSput into harmony data base, harmony data base can be analogous to the population in genetic algorithm;

The 3rd step, generates a new harmony:

Generate new harmony x _i'=(x ' ₁, x ' ₂..., x ' _n), when generating new harmony, set a probability WSR, produce a random number R and, if Rand<WSR upgrades the poorest solution in current harmony storehouse HM; Otherwise, the preferably solution in current harmony storehouse HM is upgraded, meanwhile, in population renewal process, probability is chosen in feasible zone, and the algorithm after note is improved is HSWB, each tone x of new harmony _i' (i=1,2 ..., N) by following three kinds of mechanism, produce: study harmony data base, tone fine setting, selects tone at random;

First variable x ' of new explanation ₁there is the probability of HMCR to be selected from HM

any one value, have the probability of 1-HCMR to be selected from any one value of HM outer (and in variable range), same, the generating mode of other variable is as follows:

$x_i^{\&prime;}=\left\{\begin{array}{cc}{x}_i^{\&prime;}\&Element;(x_i^1,x_i^2,...,x_i^{\mathrm{HMS}}),& \mathrm{ifrand}<\mathrm{HMCR}\\ x_i^{\&prime;}\&Element;X_i,& \mathrm{otherwise}\end{array}\right.;(i=\mathrm{1,2},...,N)$

Rand represents the equally distributed random number on [0,1];

Secondly, if new harmony x ' _ifrom harmony data base, finely tune by row tone, concrete operations are as follows:

The 4th step, upgrades harmony data base

New explanation in the 3rd step is assessed, if be better than the poorest one of functional value in HM, new explanation is updated in HM, concrete operations are as follows:

$\mathrm{If\; f}\left(x^{\&prime;}\right)<f\left(x^{\mathrm{worst}}\right)=\underset{j=\mathrm{1,2},...,\mathrm{HMS}}{\max }f\left(x^j\right),\mathrm{then}{x}^{\mathrm{worst}}=x^{\&prime;};$

The 5th step, checks and whether reaches algorithm end condition;

Repeat the 3rd step and the 4th step, until creation iterations reaches Tmax.

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