Solving non linear system

Question

Ahmed Elsayed 2021-2-14

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/744767-solving-non-linear-system

评论： Alex Sha 2021-2-17

采纳的回答： Walter Roberson

在 MATLAB Online 中打开

I'm trying to solve a non-linear min-max systems using the below code. and i'm geeting this error

Error using symengine

Input arguments must be convertible to floating-point numbers.

Error in sym/privBinaryOp (line 1030)

Csym = mupadmex(op,args{1}.s, args{2}.s, varargin{:});

Error in sym/min (line 78)

C = privBinaryOp(X, Y, 'symobj::zipWithImplicitExpansion', 'symobj::minComplex');

Any suggestions to fix it?

syms x111 x114 x121 x124 x211 x214 x221 x224
eq1 = min((min(3*x111,8*x111)+min(2*x211,11*x211)),(6*min(3*x111,8*x111)+6*min(2*x211,11*x211)))+min((7*min(3*x121,8*x121)+7*min(2*x221,11*x221)),(-3*max(3*x124,8*x124)-3*max(2*x224,11*x224)))+min((2*min(x111,7*x111)+2*min(3*x211,7*x211)),(-5*max(x114,7*x114)-5*max(3*x214,7*x214)))+min((min(x121,7*x121)+min(3*x221,7*x221)),(5*min(x121,7*x121)+5*min(2*x221,7*x221)))==-964;
eq2 = min ((6* min (3* x111, 8 *x111)+6* min (2* x211, 11 *x211)),(-max (3*x114, 8* x114) -max (2* x214, 11* x214)))+min ((min (3 *x121, 8* x121)+min (2* x221, 11* x221)),(7* min (3* x121, 8* x121)+7 *min (2 *x221, 11 *x221)))+ min ((3 *min (x111, 7* x111)+3 *min (3* x211, 7* x211)), (6 *min (x111, 7* x111)+6* min (3 *x211, 7* x211)))+min ((4 *min (x121, 7 *x121) +4 *min (3* x221, 7* x221)),(-3 *max (x124, 7 *x124) -3* max (3* x224, 7 *x224)))== -793;
eq3 = min ((min (-8 *x114, -x114) +min (-4 *x214, 6* x211)), (6 *min (-8* x114, -x114) +6 *min (-4* x214, 6* x211)))+min ((7* min (-8* x124, -x124) +7* min (-4* x224, 6* x221)), (-3* max (-x121, -8* x121) -3* max (6* x224, -4* x221)))+min ((2* min (-5* x114, -x114) +2* min (-6 *x214, 2* x211)), (-5* max (-x111, -5* x111) -5* max (2* x214, -6* x211)))+min ((min (-5* x124, -x124) +min (-6* x224, 2* x221)), (5* min (-5* x124, -x124) +5 *min (-6* x224, 2* x221))) == -1331;
eq4 = min ((6* min (-8* x114, -x114) +6* min (-4 *x214, 6* x211)), (-max (-x111, -8* x111) -max (6* x214, -4* x211)))+min ((min (-8* x124, -x124) +min (-4 *x224, 6* x221)), (7* min (-8* x124, -x124) +7 *min (-4* x224, 6* x221)))+min ((3* min (-5* x114, -x114) +3* min (-6* x214, 2* x211)), (6* min (-5* x114, -x114) +6* min (-6* x214, 2* x211)))+min ((4* min (-5 *x124, -x124) +4* min (-6* x224, 2* x221)), (-3* max (-x121, -5* x121) -3* max (2* x224, -6* x221))) == -1476;
eq5 = max ((max (3* x114, 8* x114) +max (2* x214, 11* x214)), (6* max (3* x114, 8* x114) +6 *max (2 *x214, 11 *x214)))+max ((-3 *min (3 *x121, 8* x121) -3* min (2 *x221, 11* x221)), (7 *max (3* x124, 8* x124) +7* max (2* x224, 11* x224)))+max ((-5* min (x111, 7* x111) -5 *min (3* x211, 7* x211)), (2 *max (x114, 7 *x114) + 2* max (3* x214, 7* x214)))+max ((max (x124, 7 *x124) +max (3 *x224, 7* x224)), (5 *max (x124, 7* x124) +5 *max (3 *x224, 7* x224))) == 1816;
eq6 = max ((-min (3* x111, 8* x111) -min (2* x211, 11* x211)), (6* max (3* x114, 8* x114) +6 *max (2* x214, 11* x214)))+max ((max (3* x124, 8* x124) +max (2* x224, 11* x224)), (7 *max (3 *x124, 8* x124) +7* max (2* x224, 11* x224)))+max ((3* max (x114, 7* x114) +3 *max (3 *x214, 7* x214)), (6* max (x114, 7* x114) +6* max (3 *x214, 7* x214)))+max ((-3* min (x121, 7* x121) -3* min (3 *x221, 7* x221)), (4* max (x124, 7* x124) +4* max (3 *x224, 7 *x224))) == 2019;
eq7 = max ((max (-x111, -8* x111) +max (6 *x214, -4 *x211)), (6* max (-x111, -8* x111) +6* max (6 *x214, -4 *x211)))+max ((-3* min (-8* x124, -x124) -3 *min (-4* x224, 6 *x221)), (7 *max (-x121, -8* x121) +7 *max (6 *x224, -4* x221)))+max ((-5* min (-5* x114, -x114) -5 *min (-6 *x214, 2* x211)), (2* max (-x111, -5* x111) +2 *max (2* x214, -6* x211)))+max ((max (-x121, -5 *x121) +max (2* x224, -6 *x221)), (5* max (-x121, -5 *x121) +5 *max (2 *x224, -6* x221))) == 968;
eq8 = max ((-min (-8 *x114, -x114) -min (-4 *x214, 6 *x211)), (6* max (-x111, -8* x111) +6* max (6* x214, -4 *x211)))+max ((max (-x121, -8 *x121) +max (6* x224, -4* x221)), (7 *max (-x121, -8 *x121) +7 *max (6* x224, -4* x221)))+max ((3 *max (-x111, -5 *x111) +3 *max (2* x214, -6 *x211)), (6 *max (-x111, -5* x111) +6* max (2 *x214, -6 *x211)))+max ((-3* min (-5* x124, -x124) -3 *min (-6* x224, 2* x221)), (4* max (-x121, -5 *x121) +4 *max (2 *x224, -6 *x221))) == 843;
        
sol = fsolve([eq1 eq2 eq3 eq4 eq5 eq6 eq7 eq8], [x111 x114 x121 x124 x211 x214 x221 x224]);
x1Sol = sol.x111;
x2Sol = sol.x114;
x3Sol = sol.x121;
x4Sol = sol.x124;
x5Sol = sol.x211;
x6Sol = sol.x214;
x7Sol = sol.x221;
x8Sol = sol.x224;
        

6 个评论
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Alex Sha 2021-2-14

在 MATLAB Online 中打开

Hi, Ahmed, as Walter explained, the results provided above have been obtained by a software package named 1stOpt, the code is very simple like below:

Function
min((min(3*x111,8*x111)+min(2*x211,11*x211)),(6*min(3*x111,8*x111)+6*min(2*x211,11*x211)))+min((7*min(3*x121,8*x121)+7*min(2*x221,11*x221)),(-3*max(3*x124,8*x124)-3*max(2*x224,11*x224)))+min((2*min(x111,7*x111)+2*min(3*x211,7*x211)),(-5*max(x114,7*x114)-5*max(3*x214,7*x214)))+min((min(x121,7*x121)+min(3*x221,7*x221)),(5*min(x121,7*x121)+5*min(2*x221,7*x221)))==-964;
min ((6* min (3* x111, 8 *x111)+6* min (2* x211, 11 *x211)),(-max (3*x114, 8* x114) -max (2* x214, 11* x214)))+min ((min (3 *x121, 8* x121)+min (2* x221, 11* x221)),(7* min (3* x121, 8* x121)+7 *min (2 *x221, 11 *x221)))+ min ((3 *min (x111, 7* x111)+3 *min (3* x211, 7* x211)), (6 *min (x111, 7* x111)+6* min (3 *x211, 7* x211)))+min ((4 *min (x121, 7 *x121) +4 *min (3* x221, 7* x221)),(-3 *max (x124, 7 *x124) -3* max (3* x224, 7 *x224)))== -793;
min ((min (-8 *x114, -x114) +min (-4 *x214, 6* x211)), (6 *min (-8* x114, -x114) +6 *min (-4* x214, 6* x211)))+min ((7* min (-8* x124, -x124) +7* min (-4* x224, 6* x221)), (-3* max (-x121, -8* x121) -3* max (6* x224, -4* x221)))+min ((2* min (-5* x114, -x114) +2* min (-6 *x214, 2* x211)), (-5* max (-x111, -5* x111) -5* max (2* x214, -6* x211)))+min ((min (-5* x124, -x124) +min (-6* x224, 2* x221)), (5* min (-5* x124, -x124) +5 *min (-6* x224, 2* x221))) == -1331;
min ((6* min (-8* x114, -x114) +6* min (-4 *x214, 6* x211)), (-max (-x111, -8* x111) -max (6* x214, -4* x211)))+min ((min (-8* x124, -x124) +min (-4 *x224, 6* x221)), (7* min (-8* x124, -x124) +7 *min (-4* x224, 6* x221)))+min ((3* min (-5* x114, -x114) +3* min (-6* x214, 2* x211)), (6* min (-5* x114, -x114) +6* min (-6* x214, 2* x211)))+min ((4* min (-5 *x124, -x124) +4* min (-6* x224, 2* x221)), (-3* max (-x121, -5* x121) -3* max (2* x224, -6* x221))) == -1476;
max ((max (3* x114, 8* x114) +max (2* x214, 11* x214)), (6* max (3* x114, 8* x114) +6 *max (2 *x214, 11 *x214)))+max ((-3 *min (3 *x121, 8* x121) -3* min (2 *x221, 11* x221)), (7 *max (3* x124, 8* x124) +7* max (2* x224, 11* x224)))+max ((-5* min (x111, 7* x111) -5 *min (3* x211, 7* x211)), (2 *max (x114, 7 *x114) + 2* max (3* x214, 7* x214)))+max ((max (x124, 7 *x124) +max (3 *x224, 7* x224)), (5 *max (x124, 7* x124) +5 *max (3 *x224, 7* x224))) == 1816;
max ((-min (3* x111, 8* x111) -min (2* x211, 11* x211)), (6* max (3* x114, 8* x114) +6 *max (2* x214, 11* x214)))+max ((max (3* x124, 8* x124) +max (2* x224, 11* x224)), (7 *max (3 *x124, 8* x124) +7* max (2* x224, 11* x224)))+max ((3* max (x114, 7* x114) +3 *max (3 *x214, 7* x214)), (6* max (x114, 7* x114) +6* max (3 *x214, 7* x214)))+max ((-3* min (x121, 7* x121) -3* min (3 *x221, 7* x221)), (4* max (x124, 7* x124) +4* max (3 *x224, 7 *x224))) == 2019;
max ((max (-x111, -8* x111) +max (6 *x214, -4 *x211)), (6* max (-x111, -8* x111) +6* max (6 *x214, -4 *x211)))+max ((-3* min (-8* x124, -x124) -3 *min (-4* x224, 6 *x221)), (7 *max (-x121, -8* x121) +7 *max (6 *x224, -4* x221)))+max ((-5* min (-5* x114, -x114) -5 *min (-6 *x214, 2* x211)), (2* max (-x111, -5* x111) +2 *max (2* x214, -6* x211)))+max ((max (-x121, -5 *x121) +max (2* x224, -6 *x221)), (5* max (-x121, -5 *x121) +5 *max (2 *x224, -6* x221))) == 968;
max ((-min (-8 *x114, -x114) -min (-4 *x214, 6 *x211)), (6* max (-x111, -8* x111) +6* max (6* x214, -4 *x211)))+max ((max (-x121, -8 *x121) +max (6* x224, -4* x221)), (7 *max (-x121, -8 *x121) +7 *max (6* x224, -4* x221)))+max ((3 *max (-x111, -5 *x111) +3 *max (2* x214, -6 *x211)), (6 *max (-x111, -5* x111) +6* max (2 *x214, -6 *x211)))+max ((-3* min (-5* x124, -x124) -3 *min (-6* x224, 2* x221)), (4* max (-x121, -5 *x121) +4 *max (2 *x224, -6 *x221))) == 843;

running above code in 1stOpt, a number of qualified outcomes could be obtained, some more results:

4:

x111: -3.58695295911305
x211: -1.07445850909051
x121: 2.28498000867595
x221: -2.28614733165542
x124: 7.8754041983795
x224: 2.89049819889564
x114: 4.51941882428882
x214: 5.9927411320934

5:

x111: -17.7235672005484
x211: 11.2510753617849
x121: 108.850694404822
x221: 118.545735063581
x124: 7.37522534014823
x224: -6.20277399969955
x114: 20.7395783564568
x214: -10.1936285553793

the exactised one should be:

x111: -3
x211: 1
x121: 3
x221: -5
x124: 6
x224: 5
x114: 5
x214: 5

Alex Sha 2021-2-16

在 MATLAB Online 中打开

Hi, some error or wrong input：

1: "6x112" should be "6*x112";

2: "==-206&&" should be "==-206;";

3: "-3X223,5X222" should be "-3*X223,5*X222";

4: "X*213" should be "X213";

5: "2Min(" should be "2*Min(".

After doing above modifications, the result will be:

x112: -6.7405060869832
x212: 2.36971338564915
x122: 9.82757565025085
x222: 2.73979655042066
x123: 2.66935775949842
x223: 3.48702836315802
x113: 8.53397905414663
x213: -0.859553149820282

Alex Sha 2021-2-17

在 MATLAB Online 中打开

One more solution:

x112:	-9.12751728247536
x212:	-4.85242981963661
x122:	40.4685284423888
x222:	18.2309312718438
x123:	-30.3270190436273
x223:	17.4965264223753
x113:	13.8173196565403
x213:	-4.695181312283

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Answer 1

Walter Roberson 2021-2-14

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/744767-solving-non-linear-system#answer_623002

在 MATLAB Online 中打开

syms x111 x114 x121 x124 x211 x214 x221 x224
min = @(A,B) piecewise(A<=B, A, B)
max = @(A,B) piecewise(A>=B, A, B)
eq1 = min((min(3*x111,8*x111)+min(2*x211,11*x211)),(6*min(3*x111,8*x111)+6*min(2*x211,11*x211)))+min((7*min(3*x121,8*x121)+7*min(2*x221,11*x221)),(-3*max(3*x124,8*x124)-3*max(2*x224,11*x224)))+min((2*min(x111,7*x111)+2*min(3*x211,7*x211)),(-5*max(x114,7*x114)-5*max(3*x214,7*x214)))+min((min(x121,7*x121)+min(3*x221,7*x221)),(5*min(x121,7*x121)+5*min(2*x221,7*x221)))==-964;
eq2 = min ((6* min (3* x111, 8 *x111)+6* min (2* x211, 11 *x211)),(-max (3*x114, 8* x114) -max (2* x214, 11* x214)))+min ((min (3 *x121, 8* x121)+min (2* x221, 11* x221)),(7* min (3* x121, 8* x121)+7 *min (2 *x221, 11 *x221)))+ min ((3 *min (x111, 7* x111)+3 *min (3* x211, 7* x211)), (6 *min (x111, 7* x111)+6* min (3 *x211, 7* x211)))+min ((4 *min (x121, 7 *x121) +4 *min (3* x221, 7* x221)),(-3 *max (x124, 7 *x124) -3* max (3* x224, 7 *x224)))== -793;
eq3 = min ((min (-8 *x114, -x114) +min (-4 *x214, 6* x211)), (6 *min (-8* x114, -x114) +6 *min (-4* x214, 6* x211)))+min ((7* min (-8* x124, -x124) +7* min (-4* x224, 6* x221)), (-3* max (-x121, -8* x121) -3* max (6* x224, -4* x221)))+min ((2* min (-5* x114, -x114) +2* min (-6 *x214, 2* x211)), (-5* max (-x111, -5* x111) -5* max (2* x214, -6* x211)))+min ((min (-5* x124, -x124) +min (-6* x224, 2* x221)), (5* min (-5* x124, -x124) +5 *min (-6* x224, 2* x221))) == -1331;
eq4 = min ((6* min (-8* x114, -x114) +6* min (-4 *x214, 6* x211)), (-max (-x111, -8* x111) -max (6* x214, -4* x211)))+min ((min (-8* x124, -x124) +min (-4 *x224, 6* x221)), (7* min (-8* x124, -x124) +7 *min (-4* x224, 6* x221)))+min ((3* min (-5* x114, -x114) +3* min (-6* x214, 2* x211)), (6* min (-5* x114, -x114) +6* min (-6* x214, 2* x211)))+min ((4* min (-5 *x124, -x124) +4* min (-6* x224, 2* x221)), (-3* max (-x121, -5* x121) -3* max (2* x224, -6* x221))) == -1476;
eq5 = max ((max (3* x114, 8* x114) +max (2* x214, 11* x214)), (6* max (3* x114, 8* x114) +6 *max (2 *x214, 11 *x214)))+max ((-3 *min (3 *x121, 8* x121) -3* min (2 *x221, 11* x221)), (7 *max (3* x124, 8* x124) +7* max (2* x224, 11* x224)))+max ((-5* min (x111, 7* x111) -5 *min (3* x211, 7* x211)), (2 *max (x114, 7 *x114) + 2* max (3* x214, 7* x214)))+max ((max (x124, 7 *x124) +max (3 *x224, 7* x224)), (5 *max (x124, 7* x124) +5 *max (3 *x224, 7* x224))) == 1816;
eq6 = max ((-min (3* x111, 8* x111) -min (2* x211, 11* x211)), (6* max (3* x114, 8* x114) +6 *max (2* x214, 11* x214)))+max ((max (3* x124, 8* x124) +max (2* x224, 11* x224)), (7 *max (3 *x124, 8* x124) +7* max (2* x224, 11* x224)))+max ((3* max (x114, 7* x114) +3 *max (3 *x214, 7* x214)), (6* max (x114, 7* x114) +6* max (3 *x214, 7* x214)))+max ((-3* min (x121, 7* x121) -3* min (3 *x221, 7* x221)), (4* max (x124, 7* x124) +4* max (3 *x224, 7 *x224))) == 2019;
eq7 = max ((max (-x111, -8* x111) +max (6 *x214, -4 *x211)), (6* max (-x111, -8* x111) +6* max (6 *x214, -4 *x211)))+max ((-3* min (-8* x124, -x124) -3 *min (-4* x224, 6 *x221)), (7 *max (-x121, -8* x121) +7 *max (6 *x224, -4* x221)))+max ((-5* min (-5* x114, -x114) -5 *min (-6 *x214, 2* x211)), (2* max (-x111, -5* x111) +2 *max (2* x214, -6* x211)))+max ((max (-x121, -5 *x121) +max (2* x224, -6 *x221)), (5* max (-x121, -5 *x121) +5 *max (2 *x224, -6* x221))) == 968;
eq8 = max ((-min (-8 *x114, -x114) -min (-4 *x214, 6 *x211)), (6* max (-x111, -8* x111) +6* max (6* x214, -4 *x211)))+max ((max (-x121, -8 *x121) +max (6* x224, -4* x221)), (7 *max (-x121, -8 *x121) +7 *max (6* x224, -4* x221)))+max ((3 *max (-x111, -5 *x111) +3 *max (2* x214, -6 *x211)), (6 *max (-x111, -5* x111) +6* max (2 *x214, -6 *x211)))+max ((-3* min (-5* x124, -x124) -3 *min (-6* x224, 2* x221)), (4* max (-x121, -5 *x121) +4 *max (2 *x224, -6 *x221))) == 843;
eqns = [eq1 eq2 eq3 eq4 eq5 eq6 eq7 eq8]
EQFUN = matlabFunction(eqns, 'vars', {[x111 x114 x121 x124 x211 x214 x221 x224]}, 'file', 'q744767_eqns.m', 'optimize', false)
!cat q744767_eqns.m
sol = fsolve(EQFUN, [x111 x114 x121 x124 x211 x214 x221 x224]);
x1Sol = sol.x111;
x2Sol = sol.x114;
x3Sol = sol.x121;
x4Sol = sol.x124;
x5Sol = sol.x211;
x6Sol = sol.x214;
x7Sol = sol.x221;
x8Sol = sol.x224;

Execution might take a while... A long while. Just to build the function properly :(

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Walter Roberson 2021-2-14

Note: when you use piecewise() with matlabFunction then you must send the output to a file, where it will be turned into if/else . matlabFunction() cannot deal with piecewise() in a pure anonymous function.

Walter Roberson 2021-2-14

Interesting, but I can't say that I am astonished.

piecewise() resolves out most (or all) of the branches, so it has a fairly high fan-out.

Your system is not suitable for fsolve(), by the way. fsolve() expects equations with continuous derivatives, but min() and max() do not have continuous derivatives.

Perhaps fminmax would help? Though you might need to rewrite some of the max() in terms of the negative of the min() of the negative of the expressions; https://www.mathworks.com/help/optim/ug/fminimax.html

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