Optimization_by_a_new_improved_Contraction-Expansion_algorithm_and_application.pdf

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1. Optimization by a new improved contraction-expansion algorithm and its application Meng Luo, Shiliang Gu Yangzhou University March 13, 2016 Annual Report 2015
2. PART ONE  Curve and surface fitting using Contraction-expansion optimization algorithm PART TWO  A New Regression Method: Screening Stepwise Regression PART THREE  The Traveling Salesman Problem (TSP)
3. Introduction  General methods for curve and surface fitting Contraction-expansion algorithm Instance verification (Experimental part) Discussion and conclusion Future work OUTLINE
4. INTRODUCTION:FITTING • Fitting with a curve and surface, or fitting a statistical/mathematical model to data finds its application in almost all empirical sciences, viz., physics, chemistry, zoology, botany, environmental sciences and economics, etc. That the processing of data in experimental methods, i.e., such as UV spectroscopy, X-ray analysis, IR spectrometry or chromatographic techniques, etc., is a key part of any research work, in which curve and surface fitting has become a standard in the last few years. • It has four objectives: the first, to describe the observed (or experimentally obtained) dataset by a statistical/mathematical formula that select the appropriate model; the second, to estimate the parameters of the formula so obtained and interpret them, which the interpretation is consistent with the generally accepted principles of the discipline concerned; the third, to predict, interpolate or extrapolate the expected values of the dependent variable with the estimated formula; and the last, to use the formula for designing, controlling or planning. • There are many principles of curve and surface fitting, such as the Least Squares (of errors), the Least Absolute Errors, the Generalized Method of Moments and so on.
5.  Lest sum of squares to deviation from regression(RSS, or Q) is usually used as objective function of nonlinear regression. It may be noted that a nonlinear least squares problem is fundamentally a problem in optimization of nonlinear functions. And the elementary optimization main included Univariate Optimization, Multivariate Optimization and Constrained Optimization. Initially optimization of nonlinear functions was methodologically based on the Lagrange-Leibniz-Newton principles and therefore could not easily escape local optima.  According to all above, the methods of parameter estimation can be divided into two categories. One is analysis method characterized by figuring out Q’s partial derivative simultaneous equations. This method main includes Gradient method, Newton method, Gauss-Newton method and so on. Another kind is a direct method which directly using iterative estimation from nonlinear equations. It main includes Genetic Algorithm(GA), Simulated Annealing(SA), Evolutionary Algorithm(EA), Particle Swarm Optimization(PSO) and Geometric Fitting Algorithms, etc. OPTIMIZATION
6. Objectives of the Present Work • The objective of the present work is to evaluate the performance of the Contraction-expansion algorithm at nonlinear curve and surface fitting. • For this purpose, we have collected problems-models and datasets-mostly from two main sources; the first from the website of NIST [National Institute of Standards and Technology (NIST), US Department of Commerce, USA at http://www.itl.nist.gov/div898/strd/nls/nls_m and the second, the website of the CPC-X Software (makers of the AUTO2FIT Software at http://www.geocities.com/neuralpower now new website at http://www.7d- soft.com/en/auto2fit.htm, And also now the new soft name is 1stOpt). Also some models (and datasets) have been obtained from other sources also (www.MathWorkS.com). • It also may be noted that the difficulty level of a Least Squares curve and surface fitting problem depends on: (Ⅰ) the (statistical) model, (Ⅱ) the dataset, (Ⅲ) the algorithm used for optimization, and (Ⅳ) the guessed range (or the starting points of search that any arbitrary value) of parameters.
7. • According to the level of difficulty, the problems can be classified into four categories: Lower, Average, Higher, and Extra Hard. The list of problems (dealt with in the present study) so categorized is given below table1: Table1: Classification of Problems according to Difficulty Level Difficulty level Problem Names Source of Problem Classified by Lower Misrela, MisrelB, Chwirut1, Chwirut2, Lanczos3, Gauss1, Gauss2, DanWood, NIST NIST Judge Goffe Author Average ENSO, Gauss3, Hahn, Kirby, Lanczos1,Lanczos2, MGH17, Misra1c, Misra1d, Nelson, Roszman1 NIST NIST Higher Bennett, BoxBOD, Eckerle4, MGH09,MGH10, Ratkowsky42, Ratkowsky43,Thurber NIST NIST Hougen Mathworks.com Author Extra Hard 1stOpt problems (all 9 functions), Mount , SinCos 1stOpt(Auto2Fit) 1stOpt(Auto2Fit)
8. GENERAL METHODS FOR CURVE AND SURFACE FITTING Objective function of fitting multi-variable material： ) , , , ( , , 2 1 i im ik i i Y X X X X     nonlinear function： ) , 2 , 1 ; , 2 , 1 ( ), ; ( ) ; , , , , ( ˆ 2 1 m k n i b X f b X X X X f Y i im ik i i i          Where is the regression value of the nonlinear function. ˆ i Y ) , , , , , ( ' 2 1 p j b b b b b        is p nonzero nonlinear regression parameters. So the sum of squares to deviation from regression(Q):          n i n i i i i i b F b X f Y Y Y Q 1 1 2 2 ) ( )) ; ( ( ) ˆ ( So called optimal fitting that is the estimation of must be content: min ) ( )) ; ( ( 1 2       opt n i opt i i b F b X f Y Q ) , , , , ( , , , 2 , 1 opt p opt j opt opt opt b b b b b         to content Q to be minimum in the p-dimensional space. (1) (2)
9. CONTRACTION-EXPANSION ALGORITHM Contraction stage (C stage) Contraction stage constitutes an initial searching domain begin with initial values and shrinks the searching range constantly to get the optimal parameters. (1) (0) (1) (1) (0) , 2, 1,0,1,2 kj j j j j b b kl k l b             To the j-th parameter (0) j b (1) j l For the center of initial value For the initial step length 5 equal distance searching points small positive number 1,2, j p  L Calculate the searching points in turn to find the step points with minimum Q-value as the next searching center points . (1) j b
10.  If there are p parameters, the number of the searching points in this cycle is 5p. Calculate the searching points in turn to find the points with minimum Q-value and obtain the optimal point as the center of next circle. Use these optimal points with reduced step length to search the similar points. got the optimal points ( ) ( 1) ( ) ( ) ( 1) , 2, 1,0,1,2 / 2.05 t t t kj j j t t j j b b kl k l l              the (t-1)-th cycle ( ) t b the jth parameter as searching points 1,2, j p  L ( ) min C Q ( ) C opt b Note : Objective function , the optimal parameter . Because the searching step length reduces the half after each circle, the searching domain will decrease quickly with the increase of t, and the value of objective function will decrease increasingly. ) , , , , ( ) 1 ( ) 1 ( ) 1 ( 2 ) 1 ( 1 ) 1 ( p j b b b b b      Because Q usually is not the monotone function of b, it usually cannot get the optimal parameters after contraction stage, and it always turns up . ( ) min min C Q Q 
11. Fig.1 the procedure of contraction step
12. Expansion stage (E stage)  The Expansion stage still adopt the five-point-method, which starts with the optimal points after the Contraction stage and expands the searching areas in order to search the parameters better than in the areas nearby V=4-6 the (t-1)-th cycle got the optimal points ( ' 1) t b  The t’-th circle of searching points ( ') ( ' 1) ( ') ( ) ( ' 1) , 2, 1,0,1,2 t t t kj j j t t j j b b kl k l vl              1,2, j p  L Note : the searching area will expand quickly as a result of the step length expand v times after each circle. If the former optimal parameter points are local optimal points, E stage can skip the local pitfalls and reach the global optimum.        circle next the of points center the be t can' points current the circle next the of points center the be can points current the D D  Set a critical value D and let ) ( ) ( ' ' t old t new Q Q   
13. Fig.2 the procedure of expansion step
14. IMPROVED CONTRACTION-EXPANSION ALGORITHM Step points method’s improvement There are p parameters Five points method Three points method The total calculated points of each cycle 5p 3p The more calculation The less calculation Note :when the parameters are few and it would be still better to use the five points method.  Confirmed the centre point of the next cycle.  During the searching process, the parameter point can be the centre point of the current cycle instantly as long as the new optimal point (spring point) appears.
15. COMBINING NUMERICAL DERIVATIVE WITH CONTRACTION-EXPANSION ALGORITHM Numerical differentiation  It is hard for us to get the partial derivative function, so it isn’t suitable to put it become the first condition of curve and surface fitting. If we use numerical differentiation, we can get the estimation value of the partial derivative function in b . Expressed by the formula: ' 0 ( ) ( ) ( ) ( ) lim ( ) df b f b f b f b db b b           Two nearby points If is a small number, the derivative of the point can be approximate to:  ' ( ) ( ) ( ) ( ) ( ) ( ) f b f b f b f b f b b b            
16.  When the surface and curve fitting involve multi-parameter, the approximate partial derivative of the j-th parameter of the formula in the optimal parameter point b (0) ： (0) (0) (0) (0) 0 ( ) ( ) ( ) ( ) ( ) i i j i i j i j j j j j f f b f b f b f b b b b               ,is a minor difference of the j-th parameter, which can direct be the step length after the C-stage. Improved Gauss-Newton: 1 A K     (0) b b     When there are some difference between b and b(0) , it must let b instead of b(0) figure out the new .  ) 0 , , , , 0 , 0 ( '   j j   
17. Instance verification (Experimental part)  In what follows, we present our findings on the performance of the Contraction-expansion algorithm at optimization of the Least Squares problems. The datasets and the models are available at the source (NIST, Auto2Fit). Table 2 8 datasets with higher level of difficulty Dataset name Nonlinear function Number of Parameters(p) Number of Observations (n) Level of difficulty Thurber 7 37 High MGH09 4 11 High BoxBOD 2 6 High Rat42 3 9 High MGH10 3 16 High Eckerle4 3 35 High Rat43 4 15 High Bennett5 3 154 High 2 3 1 2 3 4 2 3 5 6 7 ˆ 1 b b b X b X Y b X b X b X        2 1 2 2 3 4 ( ) ˆ b X b X Y X b X b     1 2 ˆ (1 exp( )) Y b b X    4 1 1/ 2 3 ˆ (1 exp( )) b b Y b b X    2 1 3 ˆ exp( ) b Y b X b   2 1 3 2 2 2 ( ) ˆ exp( ) 2 b X b Y b b    1 2 3 ˆ 1 exp( ) b Y b b X    3 1/ 1 2 ˆ ( ) b Y b b X   
18. Datasets name b1 B2 b3 b4 RSS MHG09 NIST 1.928069346e-1 1.912823287e-1 1.230565069e-1 1.360623307e-1 3.075056039e-4 C-E 1.928069348e-1 1.912823214e-1 1.230565037e-1 1.360623277e-1 3.075056039e-4 BoxBOD NIST 2.138094089e-2 5.472374854e-1 —— —— 1.168008877e-3 C-E 2.138094092e-2 5.472374813e-1 —— —— 1.168008877e-3 Rat42 NIST 7.246223758e+1 2.618076840e+0 6.735920007e-2 —— 8.056522934e+0 C-E 7.246223748e+1 2.618076843e+0 6.735920014e-2 —— 8.056522934e+0 MGH10 NIST 5.609636471e-3 6.181346346e+3 3.452236346e+2 —— 8.484585517e+1 C-E 5.609636614e-3 6.181346326e+3 3.452236340e+2 —— 8.484585517e+1 Eckerle4 NIST 1.556382718e+0 4.088832175e+0 4.515412184e+2 —— 1.463588749e-3 C-E 1.554382716e+0 4.088832156e+0 4.515412184e+2 —— 1.463588749e-3 Rat43 NIST 6.996415127e+2 5.277125303e+0 7.596293833e-1 1.279248386e+0 8.786404908e+3 C-E 6.996415123e+2 5.277125330e+0 7.596293876e-1 1.279248391e+0 8.786404908e+3 Bennett5 NIST -2.52350580e+3 4.673656464e+1 9.321848319e-1 —— 5.240474407e-4 C-E -2.52196804e+3 4.673018418e+1 9.322900378e-1 —— 5.240432639e-4 Table 3 The certified results of NIST and estimated by improved C-E algorithm for 7 datasets • Except the last dataset of the bennett5 et al( NIST,1994), the improved C-E algorithm fitting the results that objective function is equal to all the questions the public NIST results ( Table 2 ), in which the present algorithm that come true various issues of optimal fitting. It also verified the NIST that the majority of the results are correct .
19. 50 60 70 80 90 100 110 120 0.5 1 1.5 2 2.5 3 x 104 X Y data fit MGH10 400 410 420 430 440 450 460 470 480 490 500 0.05 0.1 0.15 0.2 0.25 0.3 0.35 X Y data fit Eckerle4 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 200 400 600 800 1000 1200 1400 X Y data fit Thurber Fig.3 The curve fitting figure of result MGH10, Thurber and Eckerle4 E.g fitting figure
20. From the 1stOpt datasets analysis Test Dataset Regression Equations R2 Notes Dimension(m) Parameters(p) 1 0.996780 1 5 2 0.934642 4 9 3 0.969930 1 3 4 0.805143 4 9 5 0.994633 2 8 6 0.999644 1 7 7 0.971547 2 7 8 0.995372 3 6 9 0.970475 1 4 5 3 4 1 2 1 ˆ ( ) b b Y b X b b X    1 2 1 3 2 4 3 5 4 6 1 7 2 8 3 9 4 ˆ 1 b b X b X b X b X Y b X b X b X b X          1 2 3 ˆ (1 ) b Y b X X b    1 2 1 3 2 4 3 5 4 6 1 7 2 8 3 9 4 ˆ 1 b b X b X b X b X Y b X b X b X b X          3 5 7 8 1 2 1 4 2 6 1 2 ˆ b b b b Y b b X b X b X X     3 5 7 1 2 4 6 ˆ b b b Y b b X b X b X     1 2 1 3 2 4 1 2 5 1 6 2 7 1 2 ˆ 1 b b X b X b X X Y b X b X b X X        6 1 5 3 2 2 1 3 2 3 4 ˆ ( )(1 )( ) b b Y b X b X b X X b      4 1 2 3 ˆ exp( ) b Y b b X b   Table 4 The 1stOpt testing problems and result (R2)  How performance with C-E algorithm curve and surface fitting software systems and other nonlinear fitting ability to achieve the global optimum were required testing by 1stOpt test questions. These tests are really challenging problems, but using the C-E algorithm can be achieved fitting optimal that through give any initial value, and fundamentally achieved the global optimal solution. For the nine test questions, the results suggestion that C-E through the test problems, and found that for most results are consistent with the results of 1stOpt.
21. N. b1/b6 b2/b7 b3/b8 b4/b9 b5 RSS SSY R2 1 1.77400378E-04 7.11350731E-33 16.7253406 1.21591936E-03 3.04365188 1939218.994 586570000 0.996694 2 4.583533 0.954423E-04 0.262201E-03 -0.30464E-05 -0.795377E-05 -0.66982E-02 -0.270534E-01 0.006681996 0.331803E-01 - 3.479215815 25.2536 0.862229 3 -101.078842 -1258.50245 -170.113551 - - 7.68651757 255.5656 0.969924 4 674.679627 0.572582863 227.745913 5.55642598 2120.32839 0.033438616 1.64255958 -0.560015938 -176.051256 53118.24153 272600 0.805143 5 1.02849313 2.67993125E-3 -2.75083053E-14 0.253301683 4.04558865 1.26806961 -1.36709592E- 03 - 1.62322193 - 1.753875407 326.7796 0.994633 6 -2.43500445 1.74691786 8.88851139 0.818860686 -1.16451411 - -5.5959852E-04 - 4.02126762 - 0.004610731 12.9610 0.999644 7 92.0801585 -3.03915462e-2 -2.67381750e-2 1.07051864e-5 -2.72093319 - 7.44537711e-4 - -3.84596257e-4 - 21.26377223 747.3331 0.971547 8 178962.386 -2.59652299 3672.73182 - 0.530213052 - 27.8050105 - 195.346878 - 0.010560486 2.2822 0.995373 9 19.1581777 -0.362592753 -29.8159227 2.29795107 - 14.66642182 488.5455 0.969979 Table 5 The fitting results of 1stOpt testing problems with improved C-E algorithm Compared 1stOpt results
22. N. b1 b2 b3 b4 b5 b6 b7 RSS R2 7-1 92.0801585 -2.67382e-2 -2.7209332 7.445377e-4 -3.84596e-4 -3.03916e-2 1.070517e-5 21.263772 0.971547 7-2 66.313876 -0.013405 -2.0433683 3.60357e-4 -2.1008e-4 -3.2448e-2 5.78165e-6 34.325346 0.954070 7-3 -7190171.8 4480.2531 459000.36 -235.14145 59.44377 1914.3001 -3.20806 49.945287 0.933169 Table 6 Global and local optimal for the problem (7) Fig.4 The surface fitting figure of result 7-1(center), 7-2(left) and 7-3(right) Test 7
23. Fig. 5 The surface fitting figure of result 8 N. b1 b2 b3 b4 b5 b6 RSS R2 8 178962.39 3672.7318 0.5302131 27.805011 195.34688 -2.596523 0.0105605 0.995373 6 1 5 3 2 2 1 3 2 3 4 ˆ ( )(1 )( ) b b Y b X b X b X X b      Test 8
24. Fig. 7 The curve fitting plot of Mount function Mount function 2 1 2 3 2 ˆ ( / )exp( 0.5(( ) / ) ) Y b b X b b    Function Algorithms b1 b2 b3 RSS R2 Mount Maximum Inherit Optimization 1.5413091 4.0173585 450.89195 0.005159 0.997148464 Mount Contraction Expansion Optimization 1.5412806 4.0172844 450.89201 0.005159 0.99715 400 410 420 430 440 450 460 470 480 490 500 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 data fit Table 7 compared the different Optimization Algorithm for the Mount Function
25. Fig. 7 The surface fitting plot of SinCos function SinCos function 1 1 2 3 2 3 4 1 2 3 / ( / ) ˆ ( ) b X b COS b X X Y b SIN X X X      Function Algorithms b1 b2 b3 b4 RSS R2 SinCos Maximum Inherit Optimization 0.493369 2.937520810.999959 5.837468 0.03861956 0.9974046 SinCos Contraction Expansion Optimization 0.4934612 2.939078410.9999625.8368422 0.038619065 0.9974 Table 8 compared the different Optimization Algorithm for the SinCos Function
26. Simulated data analysis b’=(5.004019,3.0379797,10.007446,0.19857432,1.0006819,-0.35678459,0.70202144, 1.0002495,0.99617730,0.99930820,0.96377287), Q=0.02780356233, R2=0.999923. 2 2 2 2 2 2 1 8 2 5 1 10 2 1 11 7 2 [ ( 1) ] ( ) [ ( ) ] 2 3 5 1 2 1 3 4 1 1 9 2 6 ˆ (1 ) ( ) X b X b X b X X b b X Y b b X e b b X X b X e b e                Fig. 7 The surface fitting plot of MathWorks Peaks function simulation data
27. CONCLUSION AND FUTURE WORK • The contraction-expansion (CE) algorithm (method) applied to fit functions to datasets given by NIST and others has exhibited a mixed performance. It has been successful at the job for all problems, of various degrees of difficulty, given by NIST, and 1stOpt(was Auto2Fit) test questions. • Being with only the contraction-expansion algorithm part, while most of the problems can be achieved global optimization, but more cycles and more seizes a long time to calculate. Also having only the numerical differentiation and improved Gauss-Newton method section, while faster achieved global optimization; but if take any arbitrary improper initial value, it is easy to fall into the pitfalls of the local optimum, which the estimated parameters are basically not optimal. Therefore, this paper contraction-expansion algorithm will combination numerical differentiation and improved Gauss- Newton methods. Not only can overcome their shortcomings, but also made curve and surface fitting achieved optimal.
28. • we may conclude that its success rate is appreciably high and it may be used for solving nonlinear curve and surface fitting problem with some good degree of reliability and dependability. It may be noted that there cannot be any ‘sure success method’ to solve all the problem of nonlinear least squares curve and surface fitting. • “Some of those test data are very hard, and may never get right answers without using 1stOpt(Auto2Fit). Even for 1stOpt(Auto2Fit), it does not ensure every run will be successful. … In some cases, you may try to change the control parameter of ‘Population Size’” They have suggested that to solve these problems one should use Global Levenberg-Marquard or Global BFGS method. • For the new contraction expansion algorithm that has been compiled practical program based on Matlab (License No. 271282) software platform. One-dimensional (curve) and two-dimensional (surface) of the fitting process that can intuitive and eye-catching dynamic display, and it can easily offer curve and surface fitting user to application. • Apart from the improvements needed to make our algorithm more robust, the next step could be to find new applications for our method. I.e., Traveling Salesman Problems(TSP) .
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