CN104598754A - Method for optimizing multi-objective coordination problem in progressive combination manner - Google Patents

Abstract

The invention discloses a method for optimizing a multi-objective coordination problem in a progressive combination manner. The method specifically comprises the following steps of (1) target decomposition, decomposing the multi-objective coordination problem layer by layer, forming a hierarchical structure for optimizing the problem, and converting the operation of solving the original problem into the operation of solving the lagrangian function relaxation problem by utilization of a lagrangian relaxation method, wherein penalty parameters in a lagrangian relaxation penalty function are updated in the second step; (2) updating the penalty parameters by utilization of an iterative method; (3) checking convergence, stopping and outputting an optimal solution when the number of iterations exceeds a set threshold value or the minimum value obtained in two successive functions is smaller than a given number, otherwise returning to the second step, updating the penalty parameters, and calculating and checking the convergence by utilization of the updated penalty parameters.

Description

A kind of gradual combined optimization method of multiple goal Research on Interactive Problem

Technical field

The present invention relates to Mechatronic Systems design field, the invention discloses a kind of gradual combined optimization method of multiple goal Research on Interactive Problem.

Background technology

At present, the multiobject cooperate optimization of many problems existed in engineering problem and the combinatorial optimization problem of hybrid variable and coupling thereof, Optimization Solution problem as complicated antenna-feedback system just belongs to this class problem, and the common method solving this kind of problem comprises intelligent method and law of planning etc.But intelligent method efficiency is low, be difficult to process continuous, Combinatorial Optimization optimization problem of different nature; And law of planning can produce local optimum, be only suitable for solving the few situation of variable, be difficult to process global optimization, Multivariable, therefore still there is not the appropriate method being applicable to solving this type of complex engineering problems.

The cooperative optimization method of standard adopts Distributed Design thought to reduce the complicacy of system, has certain effect, but is easily absorbed in local optimum, causes being difficult to obtain optimum solution; For this problem, scholars propose and introduce various innovative approach.Mainly contain numerical method and artificial intelligence method carries out improvement two kinds, the synergetic adopting numerical method to improve effectively can be searched for regional area around initial designs point; Have the problem of continuous, single-peaked feature for design space, this innovatory algorithm can carry out fast search along the fastest descent direction, due to very high to the requirement of initial point, equally easily enters local best points; And the advantage such as the adaptability adopting artificial intelligence approach to have and global search, but efficiency is extremely low.Another shortcoming of standard in combination optimization is only applicable to solving of system-level-two-layer problem of subject level, namely be the optimization method of a two-stage, very strict to the decomposition of problem, the analytical model after its decomposition must on same level, if decompose unreasonable, convergence is just difficult to ensure.This problem does not also have extraordinary improving one's methods so far.And complex engineering system is normally multi-level, so Cooperative Optimization Algorithm is difficult to ensure that it has the convergence determined.

Method of Lagrange multipliers is very convenient for solving convex function problem, but easily loses efficacy for non-convex problem, and is difficult to solve Large Scale Nonlinear optimization problem.Augmentation draws bright day multiplier method to improve it, but still is only applicable to the convex programming problem solving Linear Constraints, and the determination of penalty function is more difficult.

Summary of the invention

For the problems referred to above that the combined optimization method of prior art exists, the invention discloses a kind of gradual combined optimization method of multiple goal Research on Interactive Problem.

The invention discloses a kind of gradual combined optimization method of multiple goal Research on Interactive Problem, it specifically comprises the following steps: step one, goal decomposition: multiple goal Research on Interactive Problem successively decomposed, form the hierarchical structure of optimization problem, employing Lagrangian Relaxation asks the solution of Lagrangian function relaxation problem by asking the solution of former problem to be converted into, the punishment parameter wherein in Lagrange relaxation penalty function is upgraded by step 2; Step 2, employing process of iteration upgrade punishment parameter; Step 3, inspection convergence: be less than a given number when iterations exceedes the threshold value of setting or double functional minimum value, then stop and exporting optimum solution, otherwise get back to step 2 and upgrade punishment parameter, and calculate by the punishment parameter after upgrading and check convergence.

Further, the detailed process adopting process of iteration to upgrade punishment parameter in above-mentioned steps two is: the renewal iteration expression formula of the transposed vector v of Lagrange multiplier is: v ^(k+1)=v ^(k)+ 2w ^(k)ow ^(k)oc ^(k), the linear update mechanism of w is: w ^(k+1)=β w ^(k), wherein k refers to outer loop iterations, v ^(k+1)calculating rely on the v of interior loop solution ^(k), w ^(k)with nonuniformity vector c ^(k), w is penalty weight.

Further, said method is specially the renewal and the inspection convergence that adopt number multiplication to realize punishing parameter.

Further, above-mentioned several multiplication specifically comprises the following steps: that a) problem definition is decomposed: initialization t ⁽⁰⁾, r ^(k), k=0 be set and be iteration v first ⁽⁰⁾and w ⁽⁰⁾definition punishment parameter; B) solve Inner eycle: put k=k+1, apply the v specified ^(k)and w ^(k)solve resolution problem, obtain new estimation solution t ^(k), x ^(k)and r ^(k); C) convergence is checked: if the i.e. each system acquisition of outer loop convergence is very little to dependent variable difference, put k=K stopping, otherwise, go to step d); D) outer loop, upgrades punishment parameter, forwards step b to).

Further, the objective function of above-mentioned multiple goal Research on Interactive Problem is meet: be the complete vector of all design variables, f is whole objective function, g and h is corresponding inequality and equality constraints function.

Further, above-mentioned objective function is divided into f=f by unit addition ₁₁+ ... + f _nM, be divided into the hierarchical structure of N layer M unit.

Further, said method also comprises to make each unit problem divide, thus can independently solve, and introduces response r _ijwith consistency constraint c _ij=t _ij-r _ij=0, initial target is mated in forced response, and consistency constraint penalty function relaxes and adds on objective function.

By adopting above technical scheme, beneficial effect of the present invention is: coordinate the coupling variable value in multiple-objection optimization, improve solution efficiency, the present invention is used in combinatorial optimization problem, converge on optimum solution, can solve the problem that there is not coupling variable, at this moment the penalty of bottom problem is degenerated, and upper strata coordination function is zero simultaneously.

Accompanying drawing explanation

Fig. 1 is destination layer solution schematic diagram.

Fig. 2 is problem solving, upgrades process flow diagram.

Fig. 3 is that the figure of problem describes.

Fig. 4 is the variation diagram of error along with iterations of problem.

Fig. 5 is gear reducer transmission principle schematic diagram.

Embodiment

Below in conjunction with Figure of description, describe the specific embodiment of the present invention in detail.

The invention discloses a kind of gradual combined optimization method of multiple goal Research on Interactive Problem, it specifically comprises the following steps:

Step one, disposable design problem is carried out goal decomposition, be specially: hierarchical structure objective function f being decomposed into N layer M unit, f=f ₁₁+ ... + f _nM, wherein i=1 ..., N, j=1 ..., M.As shown in Figure 1, the hierarchical structure of disposable Design problem decomposing 3 layers of 6 unit.Generally, can stop when not being coupled between the unit after decomposition decomposing, certainly also can continue to be decomposed into the less unit more easily calculated according to different target calls.Suppose that disposable design problem is meet: be the complete vector of all design variables, f is whole objective function, g and h is inequality and equality constraints function; x _ijrepresent the local variable of each unit, t _ijrepresent target variable, by target variable t between unit _ijcoupling, i=1 ..., N, j=1 ..., M; F=f ₁₁+ ... + f _nM; G=[g ₁₁, g ₁₂..., g _nM], h=[h ₁₁, h ₁₂..., h _nM], disposable Design problem decomposing is:

$\underset{{\stackrel{\‾}{x}}_{11}...{\stackrel{\‾}{x}}_{\mathrm{NM}}}{\min }\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\underset{j\∈E_i}{\mathrm{\Σ}}{f}_{\mathrm{ij}}\left({\stackrel{\‾}{x}}_{\mathrm{ij}}\right),$

Meet: $\begin{array}{c}{g}_{\mathrm{ij}}\left({\stackrel{\‾}{x}}_{\mathrm{ij}}\right)\≤0;h_{\mathrm{ij}}\left({\stackrel{\‾}{x}}_{\mathrm{ij}}\right)=0;c_{\mathrm{ij}}={\stackrel{\→}{t}}_{\mathrm{ij}}-{\stackrel{\→}{r}}_{\mathrm{ij}}=0;\\ {\stackrel{\‾}{x}}_{\mathrm{ij}}=[{\stackrel{\→}{x}}_{\mathrm{ij}},{\stackrel{\→}{r}}_{\mathrm{ij}},{\stackrel{\→}{t}}_{i+1k1},...,{\stackrel{\→}{t}}_{i+1{\mathrm{kn}}_{\mathrm{ij}}}];\∀j\∈E_i,i=1,...,N\end{array}.$

Step 2, employing Lagrangian Relaxation ask the solution of Lagrangian function relaxation problem by asking the solution of former problem to be converted into, the Lagrange relaxation penalty function of former problem is:

W _ijrepresent penalty function weight, represent Lagrange multiplier, E _irepresent the set belonging to all elements of i-th layer.C _ij=t _ij-r _ij=0, r _ijfor response, c=[c ₂₂... c _nM] be the nonuniformity vector that all nonuniformity elements are formed.

In order to make each unit problem divide, thus independently can solve, introducing response r _ijwith consistency constraint c _ij=t _ij-r _ij=0, initial target is mated in forced response, and consistency constraint penalty function relaxes and adds on objective function.C=[c ₁₁... c _nM] be the nonuniformity vector that all nonuniformity elements are formed.Due to the existence of penalty function (c), unit problem is generally coupling.By inseparable penalty function, maintain the consistance between unit problem, therefore must define the unit problem that coordination strategy solves coupling.After introducing penalty function, problem can be decomposed into following formula, the same g=[g of constraint condition ₁₁, g ₁₂..., g _nM], h=[h ₁₁, h ₁₂..., h _nM].

$\underset{{\stackrel{\‾}{x}}_{\mathrm{ij}}}{\min }{f}_{\mathrm{ij}}\left({\stackrel{\‾}{x}}_{\mathrm{ij}}\right)+\mathrm{\π}\left(c({\stackrel{\‾}{x}}_{11},...{\stackrel{\‾}{x}}_{\mathrm{ij}})\right)$

Exist in the subproblem of child each, the nonuniformity constraint that last child's item that penalty term depends on subproblem determines.So

M _krepresent the layer at minterm k (root) place, represent all minterms of i layer j variable.

Do not exist in the subproblem of child each, penalty term depends on the nonuniformity constraint that all paternal items determine.Adopt k _njrepresent the target number of the n-th layer father item of i floor j target.

If definition do not continue decompose layer be minimum target layer, outside minimum target layer is upper strata destination layer, and because upper strata objective function has been decomposed to minimum target layer, then the Optimization Solution of subproblem can be divided into two classes: a. for minimum target, f _ij≠ 0, η _ijrepresent the paternal target of all j targets on i layer.Problem representation is:

B. for upper strata target, f _ij=0, problem representation is:

Namely upper strata destination layer is coordinated, minimum target layer Optimization Solution.

Step 3, employing number multiplication iteration upgrade punishment parameter, until outer loop convergence.The iterative step of number multiplication is specific as follows:

A) problem definition is decomposed: initialization t ⁽⁰⁾, r ^(k), k=0 be set and be iteration v first ⁽⁰⁾and w ⁽⁰⁾definition punishment parameter.

B) solve Inner eycle: put k=k+1, apply the v specified ^(k)and w ^(k)solve resolution problem and obtain new estimation solution t ^(k), x ^(k)and r ^(k).

C) convergence is checked: if the i.e. each system acquisition of outer loop convergence is very little to dependent variable difference, put k=K stopping algorithm.Otherwise, go to step d).

D) outer loop, upgrades punishment parameter, and forwards step b to).

Because the solution of Lagrangian function relaxation problem is also not equal to the solution of former problem.Introduce error by laxization, can consistance be obtained to precision penalty two kinds of solutions.But many this precision penalty embody the difficulty of calculating, as non-differentiability with without minimum value.

Under Augmented Lagrangian Functions, lax error can be reduced by two kinds of mechanism:

A) select v close

B) larger w is selected.

be that Lagrangian number when revising disposable design problem problem and meeting optimum solution is taken advantage of because of subvector, meet consistency constraint method.Adopt automatic weight selection algorithm to solve, avoid arranging very large weight.In the interior loop of this algorithm, decompose disposable design problem problem and apply the weight of specifying and solve, and outer loop upgrades punishment weight according to the information of interior loop.

The successful Application part of augmentation Lagrangian Relaxation is that the update mechanism of outer loop makes v close

The renewal iteration expression formula of the transposed vector v of Lagrange multiplier is:

v ^(k+1)＝v ^(k)+2w ^(k)ow ^(k)oc ^(k)

K refers to outer loop iterations, v ^(k+1)calculating rely on the v of interior loop solution ^(k), w ^(k)with nonuniformity vector c ^(k).Update mechanism and combining of Lagrangian penalty are called several multiplication.

Under the hypothesis of convex function, number multiplication converges on optimum solution, and sequence w ⁽⁰⁾..., w ^(k)for non-increasing.The linear update mechanism of w is:

w ^(k+1)＝βw ^(k)

For convex objective function, β strictly must be more than or equal to 1, can accelerating convergence as 2< β <3.For non convex objective function and larger weight w, penalty quadratic term has convexity.Number multiplication converges on the optimum solution of former problem.

Main flow of the present invention as shown in Figure 2.First carry out case study, then carry out the decomposition of problem, for each subproblem, k=0 is set, successively carry out solving of problem.

Embodiment 1:

A problem comprises two subproblem A and B, and each subproblem comprises 3 targets, intercouples between target, and two subproblems have coupling variable, as shown in Figure 3.This is a three-layer problem, and because cooperate optimization problem is applicable to solving of double-deck problem, for the convergence that multilayer problem is not determined, the method for therefore carrying out according to Cooperative Optimization Algorithm improving or combining all is not suitable for solving of this problem.And according to the method that this patent adopts, very easily problem can be carried out decomposition and solve.

Former target is F={f ₁, f ₂, f ₃, f ₄, f ₅, f ₆, ground floor target is for coordinating coupling, the second layer is coordinate in subproblem with coupling, and to coordinate in subproblem with coupling, third layer target is f ₁, f ₂, f ₃, f ₄, f ₅, f ₆.

According to gradual combined optimization method, each subproblem that above-mentioned example divides is:

Ground floor

Target is:

$\min \{\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_d}-\stackrel{\&RightArrow;}{x_{\mathrm{dlow}1}})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_d}-\stackrel{\&RightArrow;}{x_{\mathrm{dlow}2}})\}$ Variable is

The second layer

Target is:

${\min }_A\{\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_a}-\stackrel{\&RightArrow;}{x_{\mathrm{alow}1}})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_a}-\stackrel{\&RightArrow;}{x_{\mathrm{alow}2}})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_b}-\stackrel{\&RightArrow;}{x_{\mathrm{blow}1}})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_b}-\stackrel{\&RightArrow;}{x_{\mathrm{blow}3}})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_c}-\stackrel{\&RightArrow;}{x_{\mathrm{clow}2}})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_c}-\stackrel{\&RightArrow;}{x_{\mathrm{clow}3}})\}$

${\min }_B\{\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_e}-\stackrel{\&RightArrow;}{x_{\mathrm{elow}4}})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_e}-\stackrel{\&RightArrow;}{x_{\mathrm{elow}5}})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_f}-\stackrel{\&RightArrow;}{x_{\mathrm{flow}4}})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_f}-\stackrel{\&RightArrow;}{x_{\mathrm{flow}6}})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_g}-\stackrel{\&RightArrow;}{x_{\mathrm{glow}5}})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_g}-\stackrel{\&RightArrow;}{x_{\mathrm{glow}6}})\}$

Variable is with

Third layer is:

${\min }_{A1}\{f_1+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{aup}}}-\stackrel{\&RightArrow;}{x_a})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{bup}}}-\stackrel{\&RightArrow;}{x_b})\}$

${\min }_{A2}\{f_2+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{aup}}}-\stackrel{\&RightArrow;}{x_a})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{cup}}}-\stackrel{\&RightArrow;}{x_c})\}$

${\min }_{A3}\{f_3+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{bup}}}-\stackrel{\&RightArrow;}{x_a})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{cup}}}-\stackrel{\&RightArrow;}{x_c})\}$

${\min }_{B4}\{f_4+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{eup}}}-\stackrel{\&RightArrow;}{x_e})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{fup}}}-\stackrel{\&RightArrow;}{x_f})\}$

${\min }_{B5}\{f_5+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{eup}}}-\stackrel{\&RightArrow;}{x_e})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{gup}}}-\stackrel{\&RightArrow;}{x_g})\}$

${\min }_{B6}\{f_6+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{fup}}}-\stackrel{\&RightArrow;}{x_f})+\mathrm{\&pi;}(\stackrel{\&RightArrow;}{x_{\mathrm{gup}}}-\stackrel{\&RightArrow;}{x_g})\}$

Embodiment 2:

Problem: $f=f_1+f_2=z_1^2+z_2^2$

Meet: $g_1=(z_3^{-2}+z_4^2)z_5^{-2}-1\&le;0$

$g_2=(z_6^{-2}+z_5^2)z_7^{-2}-1\&le;0$

$h_1=(z_4^{-2}+z_3^2+z_5^2)z_1^{-2}-1=0$

$h_2=(z_6^2+z_5^2+z_7^2)z_2^{-2}-1\&le;0$

z ₁,z ₂,…，z ₇≥0

Solution procedure:

1) be, sublayer and destination layer by PROBLEM DECOMPOSITION

Sublayer:

f ₁: $f_1=z_2^2+{\mathrm{\&lambda;}}_{z_{51}}\&times;(z_{5\mathrm{up}}-z_5)+w_{z_{51}}\&times;{(z_{5\mathrm{up}}-z_5)}^2$

Wherein: $z_2={(z_3^2+z_4^{-2}+z_5^2)}^{1/2}$

f ₂: $f_2=z_1^2+{\mathrm{\&lambda;}}_{z_{52}}\&times;(z_{5\mathrm{up}}-z_5)+w_{z_{52}}\&times;{(z_{5\mathrm{up}}-z_5)}^2$

Wherein: $z_1={(z_5^2+z_6^2+z_7^2)}^{1/2}$

Destination layer:

f _up: $f_{\mathrm{up}}={\mathrm{\&lambda;}}_{z_{51}}\&times;(z_5-z_{5\mathrm{low}1})+w_{z_{51}}\&times;{(z_{5\mathrm{up}}-z_{5\mathrm{low}1})}^2+{\mathrm{\&lambda;}}_{z_{52}}\&times;(z_5-z_{5\mathrm{low}2})+w_{z_{52}}\&times;{(z_{5\mathrm{up}}-z_{5\mathrm{low}2})}^2$

2), initial point gets z ⁽⁰⁾=[3,3,3,3,3,3,3] are punishment parameter v initially ⁽¹⁾=0, w ⁽¹⁾=0.1

3), solve Inner eycle: put k=k+1, application, substitute into 1) in solve, obtain functional minimum value and z now.

4), checking convergence: if double functional minimum value is less than a given number, then algorithm stops.Otherwise, forward step 5 to).

5), outer loop, upgrade punishment parameter obtain.Forward step 3 to).

Error in solution procedure obtains according to the square-error with exact solution, as shown in Figure 4.

Through circulation, the optimum solution that can obtain function is: z ^(k)=[2.140,2.100,1.27,0.75,1.09,1.03,1.47], f=8.947, and the exact solution of function is: z ^*=[2.152,2.073,1.321,0.759,1.072,1.003,1.464], f=8.9285.Relative error is about 0.207%, can see that exact solution and optimum solution are very close.

Trial function:

(1) Sphere function

$f_1\left(x\right)=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i^2,$ n＝30,|x _i|≤100

Exact solution: x _i=0, f ₁(x)=0

Optimum solution: x _i=0, f ₁(x)=0

(2) De Jong function

$f_2(x_1,x_2)=100\&times;{(x_1^2-x_2)}^2+{(1-x_1)}^2$ x ₁∈(-2,2),x ₂∈(-2,2)

Initial point: x ₁=0, x ₂=0

Exact solution: x ₁=1, x ₂=1, f ₂(x ₁, x ₂)=0

Optimum solution: x ₁=1, x ₂=1, f ₂(x ₁, x ₂)=0 represents can not fall into local optimum

(3) Sinusoidal nonlinear function

f ₃(x)＝sin(x ₁+x ₂)+(x ₁-x ₂) ²-1.5x ₁+2.5x ₂+1,x ₁,x ₂∈[-20,20]

Initial point: x ₁=0, x ₂=0

Exact solution: x ₁=-19.2, x ₂=-20

Optimum solution: x ₁=-19.2, x ₂=-20

(4) engineering practice project 1

Certain complicated antenna-feedback system existing optimizes project, in the bus plane of this system heterosphere interconnected in, require that realize the optimization of bus plane, two performances are respectively with shear resistance y using mechanical property and thermal behavior as two indexs ₁with thermal resistance y ₂for judgment criteria.Performance weights is w ₁, w ₂, each performance arget value is respectively, y _obj1and y _obj2.The final function investigated is using the relative value of performance value and desired value as objective function.Because shear resistance is larger, the mechanical property of structure is better, so get as the judgment criteria of mechanical property; The thermal behavior of thermal resistance more minor structure is better, so heat-obtaining resistance relative value as the judgment criteria of thermal behavior, namely objective function is in order to avoid falling into oblivion the generation of situation, arranging matching degree minimum is constraint condition $\frac{y_1}{y_{\mathrm{obj}1}}\&GreaterEqual;{\mathrm{mat}}_1,$ $\frac{y_{\mathrm{obj}2}}{y_2}\&GreaterEqual;{\mathrm{mat}}_2.$

Power P, pcb board size s are fixed variable, with copper base size x for optimized variable.To the test specification of copper base experimentally, copper base range of size is set to 0 ~ 60.Performance index are bond strength y _obj1=15, thermal resistance y _obj2=20.The constraint of the matching degree of performance is set to 0.7.Performance weights is all 0.5.

FR4/R04350 and the interconnected index thermal resistance y of copper base ₂with the relation of variable:

Intermediate quantity $f\left(s\right)=\left\{\begin{array}{c}4.28,s=60\\ 2.08,s=80\\ 0.98,s=100\\ -0.2,s=120\\ -0.92,s=150\end{array}\right.;$

$y_2=\left\{\begin{array}{c}\frac{(1-0.0277P)\&times;(291.5\&times;x^{-0.856}+f\left(s\right))}{-0.016x+1.5},x<10\\ 0.87\&times;(1-0.0277P)\&times;(291.5\&times;x^{-0.856}+f\left(s\right)),x>10\end{array}\right.$

The relation F=0.501+0.0741*x of FR4 and the interconnected adhesion F (KN) of copper base and copper base size, thickness of slab is T, then adhesion F and bond strength y ₁between pass be

y ₁＝F/(4×x×T)

T is about 2mm, therefore the relation of FR4 and the interconnected index bond strength of copper base (unit is N) and variable:

y ₁＝(0.501+0.0741×x)×1000/(4×x×T)＝1000/4/2×(0.501+0.0741×x)

y ₁＝125*(0.501+0.0741*x)/x

In like manner, according to the relation F=0.215+0.0218*x of R04350 and the interconnected adhesion F (KN) of copper base with copper base size, the thickness of slab of test board is that T is about 2mm, obtains the relation of index bond strength and variable:

y ₁＝125*(0.215+0.0218*x)/x

Problem objective function:

$\mathrm{Fun}=0.5\&times;\frac{y_1}{14}+0.5\&times;\frac{10}{y_2}$

The constraint condition of problem

0<x≤60

y ₁≥14×0.7

y ₂≤10/0.7

Power is 2w, when pcb board size gets 150mm.

Optimum Solution is x=50.6, adopt FR4 and copper base interconnected.

(5) engineering practice project 2

For verifying that the gradual combined optimization method of the strategy based on Combinatorial Optimization proposed has engineering practicability, now illustrate for a gear reducer MDO Problem.Speed reduction unit MDO Problem is one of ten standard examples of assessment multidisciplinary design optimization performance, especially in the process studying cooperative optimization method validity and efficiency.

The speed reduction unit model adopted as shown in Figure 5, design variable geared surface engaging width x ₁, module x ₂, pinion wheel number of teeth x ₃, distance between bearings x ₄and x ₅, pinion shaft diameter x ₆with Large Gear Shaft During diameter x ₇, constraint comprises Gear Root bending strength and contact strength of tooth surface g ₁and g ₂, axle transversely deforming constraint g ₃and g ₄, axle stress constraint g ₅and g ₆, the rule of thumb geometric condition g that determines ₇~ g ₁₁and the boundary constraint of design variable.

$\begin{array}{c}\min :f_1\left(X\right)=0.7854x_1{x}_2^2(3.3333x_3^2+14.9334x_3-43.0934)\\ -1.508x_1(x_6^2+x_7^2)+7.477(x_6^3+x_7^3)+0.7854(x_4{x}_6^2+x_5{x}_7^2)\end{array}$

$g_1=\frac{27}{x_1{x}_2^2{x}_3}-1.0\&le;0$

$g_2=\frac{397.5}{x_1{x}_2^2{x}_3^2}-1.0\&le;0$

$g_3=\frac{1.93x_4^3}{x_2{x}_3{x}_6^4}-1.0\&le;0$

$g_4=\frac{1.93x_5^3}{x_2{x}_3{x}_7^4}-1.0\&le;0$

$g_5=\frac{\sqrt{\frac{{745}^2{x}_4^2}{x_2^2{x}_3^2}+1.69\&times;{10}^7}}{{110x}_6^3}-1.0\&le;0$

$g_6=\frac{\sqrt{\frac{{745}^2{x}_5^2}{x_2^2{x}_3^2}+1.575\&times;{10}^8}}{{85x}_7^3}-1.0\&le;0$

$g_7=\frac{x_2{x}_3}{40}-1.0\&le;0$

$g_8=\frac{x_1}{x_2}-12\&le;0$

$g_9=5-\frac{x_1}{x_2}\&le;0$

$g_{10}=\frac{{1.5x}_6+1.9}{x4}-1.0\&le;0$

$g_{11}=\frac{{1.1x}_7+1.9}{x_5}-1.0\&le;0$

The design variable span of Optimized model is 2.6≤x ₁≤ 3.6; 0.7≤x ₂≤ 0.8; 17≤x ₃≤ 28; 7.3≤x ₄≤ 8.3; 7.3≤x ₅≤ 8.3; 2.9≤x ₆≤ 3.9; 5.0≤x ₇≤ 5.5.

Gradual combined optimization method framework is system-level by three sons, a system-level composition.Subsystem unification is for asking with x ₁, x ₂, x ₃for the optimum of design variable, subsystem two is for asking with x ₁, x ₂, x ₃, x ₄, x ₆for the optimum of design variable, subsystem three is for asking with x ₁, x ₂, x ₃, x ₅, x ₇for the optimum of design variable, system-level primary responsibility coupling variable x ₁, x ₂, x ₃coordination problem.

The theoretical optimum solution of this example is { 3.50,0.70,17.00,7.30,7.715,3.35,5.2866}.The theoretially optimum value of objective function is 2994.35.Choosing arbitrarily one group of feasible solution is initial point, and the optimum results applying gradual combined optimization method is as shown in table 4.The iterations adopting gradual combined optimization method is 23 times, and choose another initial point and find that the optimum solution of iteration does not change, just the number of times of iteration changes, and iterations remains within 30 times on the whole.And adopt other algorithms, as Modified feasible direction method carries out Optimizing Search iteration (PSO+MMFD-CO) on the basis of particle group optimizing, iterations is 130 times.

There is the poor astringency and local optimal problem that consistency constraint causes in standard in combination optimization method, and solving of double-deck problem can only be used for, although the cooperate optimization problem improved improves the problem of poor astringency and local optimum, or can only be used for solving of two-stage problem.Although adopt augmentation Lagrange penalty function method can carry out the release of problem consistance, be only applicable to convex programming problem, and penalty factor is difficult to determine.The present invention proposes a kind of new method of the gradual Combinatorial Optimization based on augmented vector approach, further goal decomposition and element paritng can be carried out to problem, thus simplify the complexity of problem, derivation algorithm is made to have higher solution efficiency, and multilayer problem can be applicable to, avoid the situation that can only be used for solving two-layer problem of cooperate optimization, also solving augmentation draws bright day multiplier method can only be used for the situation of convex programming problem, and avoid the impact of initial point on algorithm net result, can restrain simultaneously and obtain the higher optimum solution of precision.

Coefficient given in the above embodiments and parameter; be available to those skilled in the art to realize or use invention; invention does not limit only gets aforementioned disclosed numerical value; when not departing from the thought of invention; those skilled in the art can make various modifications or adjustment to above-described embodiment; thus the protection domain invented not limit by above-described embodiment, and should be the maximum magnitude meeting the inventive features that claims are mentioned.

Claims (7)

1. the gradual combined optimization method of a multiple goal Research on Interactive Problem, it specifically comprises the following steps: step one, goal decomposition: multiple goal Research on Interactive Problem successively decomposed, form the hierarchical structure of optimization problem, employing Lagrangian Relaxation asks the solution of Lagrangian function relaxation problem by asking the solution of former problem to be converted into, the punishment parameter wherein in Lagrange relaxation penalty function is upgraded by step 2; Step 2, employing process of iteration upgrade punishment parameter; Step 3, inspection convergence: be less than a given number when iterations exceedes the threshold value of setting or double functional minimum value, then stop and exporting optimum solution, otherwise get back to step 2 and upgrade punishment parameter, and calculate by the punishment parameter after upgrading and check convergence.

2. the gradual combined optimization method of multiple goal Research on Interactive Problem as claimed in claim 1, is characterized in that the detailed process adopting process of iteration to upgrade punishment parameter in described step 2 is: the transposed vector of Lagrange multiplier renewal iteration expression formula be: , linear update mechanism be: , wherein k refers to outer loop iterations, calculating to rely on interior loop solution , with nonuniformity vector , for penalty weight, be more than or equal to 1.

3. the gradual combined optimization method of multiple goal Research on Interactive Problem as claimed in claim 1, is characterized in that described method is specially the renewal and the inspection convergence that adopt number multiplication to realize punishing parameter.

4. the gradual combined optimization method of multiple goal Research on Interactive Problem as claimed in claim 3, is characterized in that described several multiplication specifically comprises the following steps: that a) problem definition is decomposed: initialization , , k=0 be set and be iteration first with definition punishment parameter; B) solve Inner eycle: put k=k+1, application is specified with solve resolution problem, obtain new estimation solution , with ; C) convergence is checked: if the i.e. each system acquisition of outer loop convergence is very little to dependent variable difference, put k=K stopping, otherwise, go to step d); D) outer loop, upgrades punishment parameter, forwards step b to).

5. the gradual combined optimization method of multiple goal Research on Interactive Problem as claimed in claim 1, is characterized in that the objective function of described multiple goal Research on Interactive Problem is , meet: , the complete vector of all design variables, whole objective function, with it is corresponding inequality and equality constraints function.

6. the gradual combined optimization method of multiple goal Research on Interactive Problem as claimed in claim 1, is characterized in that described objective function is divided into by unit addition , be divided into the hierarchical structure of N layer M unit.

7. the gradual combined optimization method of multiple goal Research on Interactive Problem as claimed in claim 1, is characterized in that described method also comprises to make each unit problem divide, thus can independently solve, and introduces response and consistency constraint , initial target is mated in forced response, and consistency constraint penalty function relaxes and adds on objective function.