CN105701332A - Eigenvalue non-precision optimization method based on multi-factor complex control problem - Google Patents

Abstract

The present invention relates to an eigenvalue non-precision optimization method based on a multi-factor complex control problem. According to the method, a control field problem is solved by an optimization method, a similarity cutting plane model of functions f and c is defined, and after introducing a cutting plane model of an I-th modified function, eigenvalue non-precision optimization method design and calculation are performed. The method provided by the present invention seeks a balance between establishing a mathematical model and numerical accuracy, and is widely applied to signal processing and competition decisions in robot design, anti Chebyshev approximation, optimal time control, centering design and filtration design.

Description

Eigenvalue non-precision optimization method based on multifactor complicated problem

Technical field

The invention belongs to technical field of automatic control, have and relate to a kind of eigenvalue non-precision optimization method based on multifactor complicated problem。

Background technology

Automation control area, Based Intelligent Control problem relates to the various aspects of industry, agricultural and national life, in control process, because of the change of environmental factors, the Intelligent adjustment problem to control parameter is the difficult problem that controllers faces, especially in gene amplification control process, DNA circle border factor affects the parameter of control process and regulates, in control process, need to find a general non-precision optimization method, then need to solve a class non-precision optimization problem:

minf(x)

S.tc (x)≤0,

Wherein f (x), c (x) are non-smooth functions。Even if it is known that Non-smooth surface unconstrained problem is difficult to direct solution。The method solving non-precision optimization problem can simply be divided into: subgradient method, cutting plane method, analytical center cutting plane method, Shu Fangfa。Shu Fangfa and analytical center cutting plane method be regarded as cutting plane method modified version, and more stable, reliable。Wherein, analytical center cutting plane method is based on a kind of given separate information program, so the method is not based on Oralce to a certain extent。Lemarechal and Wolfe does not propose the Shu Fangfa solving convex optimization problem 1975 fraction of the year。Thereafter, scholars have devised a series of outstanding Shu Fangfa, but these methods are all based on accurate functional value and subgradient information, namely accurate Shu Fangfa。

Until calendar year 2001, Hintermuller proposes one and compels Shu Fangfa based on the near of non-precision subgradient information。Kiwiel gives a Shu Fangfa improved, and the method is based only on non-precision subgradient information equally。Algorithm above, all just for the convex problem of Non-smooth surface, there is presently no and solves non-precision optimization problem optimization method。

Summary of the invention

In order to overcome the deficiencies in the prior art, propose a kind of eigenvalue non-precision optimization method based on multifactor complicated problem, described method optimization method solves practical problems, it is most important that need to find balance between founding mathematical models and Numerical accuracy。Needing more many information, often this model is just more difficult to solve。Therefore, non-precision Shu Fangfa has a wider application, and the mathematical model of the present invention can be widely used in Robot Design, anti-Chebyshev approaches, signal processing in time optimal control, Center, design of filter, in competitive decision。

The technical scheme is that the eigenvalue non-precision optimization method based on multifactor complicated problem, described method optimization method solves control field problem, the approximate cutting plane model of defined function f and c:

${\hat{f}}_l\left(y\right)=f_{x^k}+\underset{i\&Element;L_l^f}{\max }\{-{\hat{e}}_{f_i}^k+<g_{f_{y^i}}+{\mathrm{\&rho;}}_l{\mathrm{\&Delta;}}_i^k,y-x^k>\},$

${\hat{c}}_l\left(y\right)=c_{x^k}+\underset{i\&Element;L_l^c}{\max }\{-{\hat{e}}_{c_i}^k+<g_{c_{y^i}}+{\mathrm{\&rho;}}_l{\mathrm{\&Delta;}}_i^k,y-x^k>\}.$

Wherein

$e_{f_i}^k=f_{x^k}-f_{y^i}-<g_{f_{y^i}},x^k-y^i>,$

$e_{c_i}^k=c_{x^k}-c_{y^i}-<g_{c_{y^i}},x^k-y^i>.$

Have with superior function, introduced following the l and improve the cutting plane model of function:

Initial step:

Choose initial point y⁰, make x⁰=y⁰, calculate non-precision functional value with OracleAnd non-precision subgradientOrder declines and walks index k=k (l)=0, iteration index l=0,Index set

Step1 (calculating iteration point):

In order to obtain iteration point y^l+1, calculate following quadratic programming:

Thus, prediction slippage is calculated

And calculated by OracleAnd

Step2 (Shu Gengxin):

Calculate new bundle information:

${\mathrm{\&Delta;}}_{l+1}^k=y^{l+1}-x^k,d_{l+1}^k=\frac{{\left|\right|y^{l+1}-x^k\left|\right|}^2}{2},$

$e_{f_{l+1}}^k=f_{x^k}-f_{y^{l+1}}-<g_{f_{y^{l+1}}},x^k-y^{l+1}>,$

$e_{c_{l+1}}^k=c_{x^k}-c_{y^{l+1}}-<g_{c_{y^{l+1}}},x^k-y^{l+1}>.$

Correspondingly update,

${\hat{e}}_{f_{l+1}}^k=e_{f_{l+1}}^k+{\mathrm{\&mu;}}_l{d}_{l+1}^k,{\hat{e}}_{c_{l+1}}^k=e_{c_{l+1}}^k+{\mathrm{\&mu;}}_l{d}_{l+1}^k.$

Finally, new iteration index collection is selected

Step3 (decline pacing examination):

If δ_l≤tol_stopThen iteration stopping, otherwise sounds out following decline pacing examination:

$\left\{\begin{array}{cc}{f}_{y^{l+1}}\&le;f_{x^k}-m{\mathrm{\&delta;}}_l,c_{y^{l+1}}\&le;0,& \mathrm{if}{c}_{x^k}\&le;0,\\ c_{y^{l+1}}\&le;c_{x^k}-m{\mathrm{\&delta;}}_l,& \mathrm{if}{c}_{x^k}>0.\end{array}\right.$

If y^l+1Meet formula above, then y^l+1It is the step that declines, then makes x^k+1=y^l+1, k (l+1)=k+1, k=k+1, andOtherwise make x^k+1=x^k, k (l+1)=k,

Step4 (updates convexification parameter):

${\mathrm{\&rho;}}_{l+1}=\left\{\begin{array}{cc}{\mathrm{\&rho;}}_l,& \mathrm{if}{\mathrm{\&rho;}}_{l+1}^{\min }\&le;{\mathrm{\&rho;}}_l,\\ N_0{\mathrm{\&rho;}}_{l+1}^{\min }& \mathrm{if}{\mathrm{\&rho;}}_{l+1}^{\min }>{\mathrm{\&rho;}}_l,\end{array}\right.$

Wherein

${\mathrm{\&rho;}}_{l+1}^{\min }=\underset{d_i^k\&NotEqual;0}{\max }\{\underset{i\&Element;L_{l+1}^f}{\max }-\frac{e_{f_i}^k}{d_i^k},\underset{i\&Element;L_{l+1}^c}{\max }-\frac{e_{c_i}^k}{d_i^k}\}.$

Update iteration index, make l=l+1,

Step5 (updates approaching parameter):

If meet such as lower inequality to meet simultaneously,

$\left\{\begin{array}{c}{f}_{y^{l+1}}>f_{x^k}+M_0\\ c_{y^{l+1}}>c_{x^k}+M_0,\end{array}\right.$

Then this iteration point is considered as unacceptable, it is necessary to quickly increase approaching parameter μ_l+1

μ_l+1=N₁μ_l.

Wherein,

Parameter m ∈ (0,1), M₀> 0, N₀> 1, N₁> 1, initial approaching parameter μ₀, initial convexification parameter ρ₀, iteration ends parameter tol_stop=10^-6；

Reparametrization and bundle information:

ρ₀=ρ_l+1, μ₀=μ_l+1, x⁰=x^k, k=l=0,

$e_{f_0}^0=0,e_{c_0}^0=0,d_0^0=0,{\mathrm{\&Delta;}}_0^0=0.$

Otherwise, μ is made_l+1=μ_l, proceed to Step1。

Beneficial effect of the present invention

1) the method for the invention solves to control problem encountered with optimization method, finds balance between founding mathematical models and Numerical accuracy。

2) present invention has wider application at control field, the mathematical model of the present invention can be widely used in Robot Design, anti-Chebyshev approaches, signal processing in time optimal control, Center, design of filter, in competitive decision。

3) present invention uses Inaccurate information, and functional value and subgradient information all right and wrong accurate。

Detailed description of the invention

The present invention is applied to the non-precision optimization problem that a class is important, and namely object function f (x) and constraint function c (x) are all non-convex functions。Method introduces improvement function

$h_{x^k}\left(x\right)=\max \{f\left(x\right)-f_{x^k};c\left(x\right)\},$

Wherein x^kIt is the step that declines, also referred to as iteration center。By improving function, non-precision is optimized restricted problem and changes into the Non-smooth surface unconstrained problem being easier to solve

$\begin{array}{cc}\min & h_{x^k}\left(x\right).\end{array}$

The present invention uses Inaccurate information, and functional value and subgradient information all right and wrong accurate。Our Shu Fangfa uses under information:

$\begin{array}{cc}\left\{\begin{array}{c}{f}_x=f\left(x\right)-{\mathrm{\&eta;}}_x,\\ g_{f_x}\&Element;B(g_{f\left(x\right)},{\mathrm{\&gamma;}}_x);\end{array}\right.& \left\{\begin{array}{c}{c}_x=f\left(x\right)-{\mathrm{\&eta;}}_x,\\ g_{c_x}\&Element;B(g_{c\left(x\right)},{\mathrm{\&gamma;}}_x),\end{array}\right.\end{array}$

WhereinAnd η_x, γ_xIt it is non-accurate parameters。

Before describing the approaching Shu Fangfa of non-precision, first there is following information:

$d_i^k=\frac{{|y^i-x^k|}^2}{2},{\mathrm{\&Delta;}}_i^k=y^i-x^k,$

Wherein yⁱIt is iteration point, x^kIt it is iteration center。Then, following linearized stability:

$e_{f_i}^k=f_{x^k}-f_{y^i}-<g_{f_{y^i}},x^k-y^i>,$

$e_{c_i}^k=c_{x^k}-c_{y^i}-<g_{c_{y^i}},x^k-y^i>.$

In order to process non-convex function, linearized stability adds quadratic term,

${\hat{e}}_{f_i}^k=e_{f_i}^k+{\mathrm{\&rho;}}_l{d}_i^k\text{,}{\hat{e}}_{c_i}^k=e_{c_i}^k+{\mathrm{\&rho;}}_l{d}_i^k,$

Wherein ρ_lIt is convexification parameter, and follows following principle

${\mathrm{\&rho;}}_l\&GreaterEqual;{\mathrm{\&rho;}}_{\min }=\underset{d_i^k\&NotEqual;0}{\max }\{\underset{i\&Element;L_l^f}{\max }-\frac{e_{f_i}^k}{d_i^k},\underset{i\&Element;L_l^c}{\max }-\frac{e_{c_i}^k}{d_i^k}\}.$

WhereinWithIt is corresponding index set。By this strategy, it is ensured thatWithIt it is all non-negative。

The present invention is after the cutting plane model introducing the l following improvement function, it is possible to being analyzed calculating, the step of calculating is。

0th, i.e. initially step:

In the calculating of this step, first choose initial data point, in order to calculate iteration point, e.g., choose initial point y⁰, make x⁰=y⁰, calculate non-precision functional value with OracleAnd non-precision subgradientOrder declines and walks index k=k (l)=0, iteration index l=0,Index set

Step1 (calculating iteration point):

In order to obtain iteration point y^l+1, calculate following quadratic programming:

Thus, prediction slippage is calculated

And calculated by OracleAnd

Step2 (Shu Gengxin):

Calculate new bundle information:

${\mathrm{\&Delta;}}_{l+1}^k=y^{l+1}-x^k,d_{l+1}^k=\frac{{\left|\right|y^{l+1}-x^k\left|\right|}^2}{2},$

$e_{f_{l+1}}^k=f_{x^k}-f_{y^{l+1}}-<g_{f_{y^{l+1}}},x^k-y^{l+1}>,$

$e_{c_{l+1}}^k=c_{x^k}-c_{y^{l+1}}-<g_{c_{y^{l+1}}},x^k-y^{l+1}>.$

Correspondingly update,

${\hat{e}}_{f_{l+1}}^k=e_{f_{l+1}}^k+{\mathrm{\&mu;}}_l{d}_{l+1}^k,{\hat{e}}_{c_{l+1}}^k=e_{c_{l+1}}^k+{\mathrm{\&mu;}}_l{d}_{l+1}^k.$

Finally, new iteration index collection is selected

Step3 (decline pacing examination):

If δ_l≤tol_stopThen iteration stopping, otherwise sounds out following decline pacing examination:

$\left\{\begin{array}{cc}{f}_{y^{l+1}}\&le;f_{x^k}-m{\mathrm{\&delta;}}_l,c_{y^{l+1}}\&le;0,& \mathrm{if}{c}_{x^k}\&le;0,\\ c_{y^{l+1}}\&le;c_{x^k}-m{\mathrm{\&delta;}}_l,& \mathrm{if}{c}_{x^k}>0.\end{array}\right.$

If y^l+1Meet formula above, then y^l+1It is the step that declines, then makes x^k+1=y^l+1, k (l+1)=k+1, k=k+1, andOtherwise make x^k+1=x^k, k (l+1)=k,

Step4 (updates convexification parameter):

${\mathrm{\&rho;}}_{l+1}=\left\{\begin{array}{cc}{\mathrm{\&rho;}}_l,& \mathrm{if}{\mathrm{\&rho;}}_{l+1}^{\min }\&le;{\mathrm{\&rho;}}_l,\\ N_0{\mathrm{\&rho;}}_{l+1}^{\min }& \mathrm{if}{\mathrm{\&rho;}}_{l+1}^{\min }>{\mathrm{\&rho;}}_l,\end{array}\right.$

Wherein

${\mathrm{\&rho;}}_{l+1}^{\min }=\underset{d_i^k\&NotEqual;0}{\max }\{\underset{i\&Element;L_{l+1}^f}{\max }-\frac{e_{f_i}^k}{d_i^k},\underset{i\&Element;L_{l+1}^c}{\max }-\frac{e_{c_i}^k}{d_i^k}\}.$

Update iteration index, make l=l+1,

Step5 (updates approaching parameter):

If meet such as lower inequality to meet simultaneously,

$\left\{\begin{array}{c}{f}_{y^{l+1}}>f_{x^k}+M_0\\ c_{y^{l+1}}>c_{x^k}+M_0,\end{array}\right.$

Then this iteration point is considered as unacceptable, it is necessary to quickly increase approaching parameter μ_l+1

μ_l+1=N₁μ_l.

Wherein,

Parameter m ∈ (0,1), M₀> 0, N₀> 1, N₁> 1, initial approaching parameter μ₀, initial convexification parameter ρ₀, iteration ends parameter tol_stop=10^-6；

Reparametrization and bundle information:

ρ₀=ρ_l+1, μ₀=μ_l+1, x⁰=x^k, k=l=0,

$e_{f_0}^0=0,e_{c_0}^0=0,d_0^0=0,{\mathrm{\&Delta;}}_0^0=0.$

Otherwise, μ is made_l+1=μ_l, proceed to Step1。

In the algorithm of the present invention, draw and entered to improve function, restricted problem is converted into unconstrained problem, simplify a problem, for the convergence of non-precision Shu Fangfa, cutting plane model adds quadratic term, to guarantee the Fast Convergent of Shu Fangfa, use non-precision functional value and non-precision subgradient information, can better solve the model of complexity, such as semi-infinite problem, two stage stochastic programming problem。

Claims (1)

1. based on the eigenvalue non-precision optimization method of multifactor complicated problem, it is characterized in that: described method optimization method solves control field problem, the approximate cutting plane model of defined function f and c:

${\hat{f}}_l\left(y\right)=f_{x^k}+\underset{i\&Element;L_l^f}{\max }\{-{\hat{e}}_{f_i}^k+<g_{f_{y^i}}+{\mathrm{\&rho;}}_l{\mathrm{\&Delta;}}_i^k,y-x^k>\},$

${\hat{c}}_l\left(y\right)=c_{x^k}+\underset{i\&Element;L_l^c}{\max }\{-{\hat{e}}_{c_i}^k+<g_{c_{y^i}}+{\mathrm{\&rho;}}_l{\mathrm{\&Delta;}}_i^k,y-x^k>\}.$

Wherein

$e_{f_i}^k=f_{x^k}-f_{y^i}-<g_{f_{y^i}},x^k-y^i>,$

$e_{c_i}^k=c_{x^k}-c_{y^i}-<g_{c_{y^i}},x^k-y^i>.$

Have with superior function, introduced following the l and improve the cutting plane model of function:

Initial step:

Choose initial point y⁰, make x⁰=y⁰, calculate non-precision functional value with OracleAnd non-precision subgradientOrder declines and walks index k=k (l)=0, iteration index l=0,Index set

Step1 (calculating iteration point):

In order to obtain iteration point y^l+1, calculate following quadratic programming:

Thus, prediction slippage is calculated

And calculated by OracleAnd

Step2 (Shu Gengxin):

Calculate new bundle information:

${\mathrm{\&Delta;}}_{l+1}^k=y^{l+1}-x^k,d_{l+1}^k=\frac{{\left|\right|y^{l+1}-x^k\left|\right|}^2}{2},$

$e_{f_{l+1}}^k=f_{x^k}-f_{y^{l+1}}-<g_{f_{y^{l+1}}},x^k-y^{l+1}>,$

$e_{c_{l+1}}^k=c_{x^k}-c_{y^{l+1}}-<g_{c_{y^{l+1}}},x^k-y^{l+1}>.$

Correspondingly update,

${\hat{e}}_{f_{l+1}}^k=e_{f_{l+1}}^k+{\mathrm{\&mu;}}_l{d}_{l+1}^k,{\hat{e}}_{c_{l+1}}^k=e_{c_{l+1}}^k+{\mathrm{\&mu;}}_l{d}_{l+1}^k.$

Finally, new iteration index collection is selected

Step3 (decline pacing examination):

If δ_l≤tol_stopThen iteration stopping, otherwise sounds out following decline pacing examination:

$\left\{\begin{array}{cc}{f}_{y^{l+1}}\&le;f_{x^k}-m{\mathrm{\&delta;}}_l,c_{y^{l+1}}\&le;0,& \mathrm{if}{c}_{x^k}\&le;0,\\ c_{y^{l+1}}\&le;c_{x^k}-m{\mathrm{\&delta;}}_l,& \mathrm{if}{c}_{x^k}>0.\end{array}\right.$

If y^l+1Meet formula above, then y^l+1It is the step that declines, then makes x^k+1=y^l+1, k (l+1)=k+1, k=k+1, andOtherwise make x^k+1=x^k, k (l+1)=k,

Step4 (updates convexification parameter):

${\mathrm{\&rho;}}_{l+1}=\left\{\begin{array}{cc}{\mathrm{\&rho;}}_l,& \mathrm{if}{\mathrm{\&rho;}}_{l+1}^{\min }\&le;{\mathrm{\&rho;}}_l,\\ N_0{\mathrm{\&rho;}}_{l+1}^{\min },& \mathrm{if}{\mathrm{\&rho;}}_{l+1}^{\min }>{\mathrm{\&rho;}}_l,\end{array}\right.$

Wherein

${\mathrm{\&rho;}}_{l+1}^{\min }=\underset{d_i^k\&NotEqual;0}{\max }\{\underset{i\&Element;L_{l+1}^f}{\max }-\frac{e_{f_i}^k}{d_i^k},\underset{i\&Element;L_{l+1}^c}{\max }-\frac{e_{c_i}^k}{d_i^k}\}.$

Update iteration index, make l=l+1,

Step5 (updates approaching parameter):

If meet such as lower inequality to meet simultaneously,

$\left\{\begin{array}{c}{f}_{y^{l+1}}>f_{x^k}+M_0\\ c_{y^{l+1}}>c_{x^k}+M_0,\end{array}\right.$

Then this iteration point is considered as unacceptable, it is necessary to quickly increase approaching parameter μ_l+1

μ_l+1=N₁μ_l.

Wherein,

Parameter m ∈ (0,1), M₀> 0, N₀> 1, N₁> 1, initial approaching parameter μ₀, initial convexification parameter ρ₀, iteration ends parameter tol_stop=10^-6；

Reparametrization and bundle information:

ρ₀=ρ_l+1, μ₀=μ_l+1, x⁰=x^k, k=l=0,

$e_{f_0}^0=0,e_{c_0}^0=0,d_0^0=0,{\mathrm{\&Delta;}}_0^0=0.$

Otherwise, μ is made_l+1=μ_l, proceed to Step1。