Minimization problem with integral constraint

Question

Esteban Garcia 2022-5-9

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1715140-minimization-problem-with-integral-constraint

评论： Esteban Garcia 2022-5-11

采纳的回答： Matt J

Hello, I'm working with a 2D numerical density profile.

. I have a set of radius

a maximum radius R and I want to find the best fit to the data r. I know that I can use maximum likelihood or another method, but I have problems with the constraints for

, because I require that

At first I tried with bins and adjusted the curve with cftool, but I need more precision. So I want to use minimization with that constraint.

Thank you so much.

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Answer 1

Matt J 2022-5-9

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1715140-minimization-problem-with-integral-constraint#answer_960610

编辑：Matt J 2022-5-9

Perhaps you could reparametrize the curve as,

which automatically satisfies the constraint for any b and c. Moreoever, since this form has only two unknown parameters, it should be relatively easy to do a parameter sweep to find at least a good initial guess of b and c.

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Esteban Garcia 2022-5-9

编辑：Esteban Garcia 2022-5-9

在 MATLAB Online 中打开

Hello Matt, your answer was extremely helpful!.. I already did the sweep and found c=-1.295 and b=0.76 and it makes sense.

Is there a method to get the minimization faster?, because I need to apply it to a lot of galaxy clusters. I've seen and tried some methods in the documentation but none of them seemed to accept variables as exponents..

For example, for the first galaxy I minimized this function for a and b

F=-(b^(706*c)*(1/b^2 + 1)^c*(4/b^2 + 1)^(2*c)*(1/(4*b^2) + 1)^(2*c)*(9/b^2 + 1)^(2*c)*(9/(4*b^2) + 1)^c*(1/(16*b^2) + 1)^c*(9/(16*b^2) + 1)^(5*c)*(25/(4*b^2) + 1)^c*(9/(25*b^2) + 1)^(4*c)*(16/(25*b^2) + 1)^(2*c)*(25/(16*b^2) + 1)^c*(49/(16*b^2) + 1)^c*(49/(25*b^2) + 1)^c*(64/(25*b^2) + 1)^(2*c)*(81/(25*b^2) + 1)^c*(9/(100*b^2) + 1)^c*(121/(16*b^2) + 1)^c*(121/(25*b^2) + 1)^(3*c)*(49/(100*b^2) + 1)^(3*c)*(169/(16*b^2) + 1)^(2*c)*(169/(25*b^2) + 1)^c*(121/(100*b^2) + 1)^c*(169/(100*b^2) + 1)^(2*c)*(324/(25*b^2) + 1)^c*(9/(400*b^2) + 1)^c*(49/(400*b^2) + 1)^c*(361/(100*b^2) + 1)^(5*c)*(81/(400*b^2) + 1)^(2*c)*(121/(400*b^2) + 1)^(4*c)*(169/(400*b^2) + 1)^(3*c)*(4/(625*b^2) + 1)^c*(529/(100*b^2) + 1)^(3*c)*(9/(625*b^2) + 1)^(2*c)*(16/(625*b^2) + 1)^c*(36/(625*b^2) + 1)^c*(49/(625*b^2) + 1)^c*(64/(625*b^2) + 1)^(2*c)*(289/(400*b^2) + 1)^(3*c)*(81/(625*b^2) + 1)^c*(121/(625*b^2) + 1)^c*(361/(400*b^2) + 1)^c*(144/(625*b^2) + 1)^(3*c)*(196/(625*b^2) + 1)^(2*c)*(729/(100*b^2) + 1)^c*(441/(400*b^2) + 1)^c*(256/(625*b^2) + 1)^(2*c)*(289/(625*b^2) + 1)^(2*c)*(529/(400*b^2) + 1)^(2*c)*(324/(625*b^2) + 1)^(2*c)*(361/(625*b^2) + 1)^c*(961/(100*b^2) + 1)^c*(441/(625*b^2) + 1)^(4*c)*(484/(625*b^2) + 1)^c*(529/(625*b^2) + 1)^(3*c)*(729/(625*b^2) + 1)^c*(961/(400*b^2) + 1)^(2*c)*(784/(625*b^2) + 1)^(2*c)*(841/(625*b^2) + 1)^(2*c)*(1089/(400*b^2) + 1)^(2*c)*(1024/(625*b^2) + 1)^(3*c)*(1369/(400*b^2) + 1)^c*(1156/(625*b^2) + 1)^(2*c)*(1296/(625*b^2) + 1)^(4*c)*(1521/(400*b^2) + 1)^c*(1521/(625*b^2) + 1)^c*(1849/(400*b^2) + 1)^c*(1764/(625*b^2) + 1)^(2*c)*(1849/(625*b^2) + 1)^c*(1/(2500*b^2) + 1)^c*(9/(2500*b^2) + 1)^c*(49/(2500*b^2) + 1)^(2*c)*(1936/(625*b^2) + 1)^c*(121/(2500*b^2) + 1)^c*(169/(2500*b^2) + 1)^c*(2116/(625*b^2) + 1)^(3*c)*(289/(2500*b^2) + 1)^(2*c)*(2401/(400*b^2) + 1)^c*(361/(2500*b^2) + 1)^c*(2304/(625*b^2) + 1)^c*(441/(2500*b^2) + 1)^c*(2601/(400*b^2) + 1)^c*(2401/(625*b^2) + 1)^(3*c)*(529/(2500*b^2) + 1)^c*(2601/(625*b^2) + 1)^c*(729/(2500*b^2) + 1)^(4*c)*(2704/(625*b^2) + 1)^(2*c)*(841/(2500*b^2) + 1)^(2*c)*(961/(2500*b^2) + 1)^c*(2916/(625*b^2) + 1)^(3*c)*(1089/(2500*b^2) + 1)^(4*c)*(3249/(400*b^2) + 1)^c*(1369/(2500*b^2) + 1)^c*(3249/(625*b^2) + 1)^(2*c)*(1521/(2500*b^2) + 1)^(3*c)*(3481/(625*b^2) + 1)^(3*c)*(1849/(2500*b^2) + 1)^c*(3844/(625*b^2) + 1)^c*(4096/(625*b^2) + 1)^c*(4356/(625*b^2) + 1)^c*(2601/(2500*b^2) + 1)^c*(4489/(625*b^2) + 1)^c*(4624/(625*b^2) + 1)^c*(5041/(625*b^2) + 1)^c*(3249/(2500*b^2) + 1)^c*(5184/(625*b^2) + 1)^c*(3481/(2500*b^2) + 1)^c*(3721/(2500*b^2) + 1)^c*(4489/(2500*b^2) + 1)^c*(5041/(2500*b^2) + 1)^(2*c)*(5329/(2500*b^2) + 1)^c*(7396/(625*b^2) + 1)^(2*c)*(5929/(2500*b^2) + 1)^c*(1/(10000*b^2) + 1)^c*(9/(10000*b^2) + 1)^(2*c)*(7569/(2500*b^2) + 1)^c*(81/(10000*b^2) + 1)^(2*c)*(289/(10000*b^2) + 1)^(2*c)*(361/(10000*b^2) + 1)^c*(529/(10000*b^2) + 1)^c*(961/(10000*b^2) + 1)^(3*c)*(1089/(10000*b^2) + 1)^(2*c)*(8649/(2500*b^2) + 1)^(2*c)*(1369/(10000*b^2) + 1)^c*(1521/(10000*b^2) + 1)^(3*c)*(1681/(10000*b^2) + 1)^c*(1849/(10000*b^2) + 1)^c*(9409/(2500*b^2) + 1)^(3*c)*(2209/(10000*b^2) + 1)^(4*c)*(9801/(2500*b^2) + 1)^c*(2401/(10000*b^2) + 1)^(2*c)*(2601/(10000*b^2) + 1)^(2*c)*(10201/(2500*b^2) + 1)^c*(2809/(10000*b^2) + 1)^(5*c)*(10609/(2500*b^2) + 1)^(2*c)*(3481/(10000*b^2) + 1)^c*(3721/(10000*b^2) + 1)^c*(3969/(10000*b^2) + 1)^(4*c)*(11881/(2500*b^2) + 1)^(6*c)*(4489/(10000*b^2) + 1)^c*(4761/(10000*b^2) + 1)^c*(12321/(2500*b^2) + 1)^(2*c)*(5041/(10000*b^2) + 1)^c*(12769/(2500*b^2) + 1)^(2*c)*(5329/(10000*b^2) + 1)^c*(13689/(2500*b^2) + 1)^c*(6561/(10000*b^2) + 1)^c*(6889/(10000*b^2) + 1)^c*(14641/(2500*b^2) + 1)^c*(7569/(10000*b^2) + 1)^(3*c)*(7921/(10000*b^2) + 1)^(3*c)*(8649/(10000*b^2) + 1)^c*(16641/(2500*b^2) + 1)^(2*c)*(9409/(10000*b^2) + 1)^c*(17161/(2500*b^2) + 1)^c*(17689/(2500*b^2) + 1)^c*(10201/(10000*b^2) + 1)^(2*c)*(10609/(10000*b^2) + 1)^(2*c)*(11449/(10000*b^2) + 1)^c*(11881/(10000*b^2) + 1)^c*(12321/(10000*b^2) + 1)^c*(12769/(10000*b^2) + 1)^c*(20449/(2500*b^2) + 1)^c*(13689/(10000*b^2) + 1)^c*(14161/(10000*b^2) + 1)^(3*c)*(14641/(10000*b^2) + 1)^(2*c)*(22801/(2500*b^2) + 1)^c*(16129/(10000*b^2) + 1)^c*(16641/(10000*b^2) + 1)^(2*c)*(17161/(10000*b^2) + 1)^(3*c)*(17689/(10000*b^2) + 1)^c*(19321/(10000*b^2) + 1)^c*(19881/(10000*b^2) + 1)^c*(20449/(10000*b^2) + 1)^(2*c)*(28561/(2500*b^2) + 1)^c*(21609/(10000*b^2) + 1)^(2*c)*(23409/(10000*b^2) + 1)^c*(24649/(10000*b^2) + 1)^c*(25281/(10000*b^2) + 1)^(3*c)*(25921/(10000*b^2) + 1)^c*(26569/(10000*b^2) + 1)^c*(27889/(10000*b^2) + 1)^(2*c)*(28561/(10000*b^2) + 1)^c*(29241/(10000*b^2) + 1)^c*(31329/(10000*b^2) + 1)^c*(32041/(10000*b^2) + 1)^c*(33489/(10000*b^2) + 1)^c*(35721/(10000*b^2) + 1)^c*(36481/(10000*b^2) + 1)^c*(37249/(10000*b^2) + 1)^c*(39601/(10000*b^2) + 1)^c*(40401/(10000*b^2) + 1)^(2*c)*(41209/(10000*b^2) + 1)^c*(42849/(10000*b^2) + 1)^(2*c)*(43681/(10000*b^2) + 1)^c*(44521/(10000*b^2) + 1)^c*(49729/(10000*b^2) + 1)^c*(51529/(10000*b^2) + 1)^(3*c)*(56169/(10000*b^2) + 1)^c*(57121/(10000*b^2) + 1)^c*(58081/(10000*b^2) + 1)^c*(59049/(10000*b^2) + 1)^(2*c)*(61009/(10000*b^2) + 1)^c*(63001/(10000*b^2) + 1)^(2*c)*(66049/(10000*b^2) + 1)^c*(69169/(10000*b^2) + 1)^c*(72361/(10000*b^2) + 1)^c*(73441/(10000*b^2) + 1)^c*(77841/(10000*b^2) + 1)^c*(80089/(10000*b^2) + 1)^(3*c)*(85849/(10000*b^2) + 1)^c*(95481/(10000*b^2) + 1)^c*(114921/(10000*b^2) + 1)^c*(116281/(10000*b^2) + 1)^c*(124609/(10000*b^2) + 1)^c*(137641/(10000*b^2) + 1)^c*(c + 1)^353)/(pi^353*((b^2 + 968562431735785/70368744177664)^(c + 1) - b^(2*c + 2))^353)

Again, thank you!

Esteban Garcia 2022-5-10

编辑：Esteban Garcia 2022-5-10

在 MATLAB Online 中打开

Hello Matt, thank you for answer

I see, but evaluating directly didn't give me a result. When I evaluated the function into b=0.1:0.01:2, c=-3.005:0.01:-0.015

-(b^(706*c)*(1/b^2 + 1)^c*(4/b^2 + 1)^(2*c)*(1/(4*b^2) + 1)^(2*c)*(9/b^2 + 1)^(2*c)*(9/(4*b^2) + 1)^c*(1/(16*b^2) + 1)^c*(9/(16*b^2) + 1)^(5*c)*(25/(4*b^2) + 1)^c*(9/(25*b^2) + 1)^(4*c)*(16/(25*b^2) + 1)^(2*c)*(25/(16*b^2) + 1)^c*(49/(16*b^2) + 1)^c*(49/(25*b^2) + 1)^c*(64/(25*b^2) + 1)^(2*c)*(81/(25*b^2) + 1)^c*(9/(100*b^2) + 1)^c*(121/(16*b^2) + 1)^c*(121/(25*b^2) + 1)^(3*c)*(49/(100*b^2) + 1)^(3*c)*(169/(16*b^2) + 1)^(2*c)*(169/(25*b^2) + 1)^c*(121/(100*b^2) + 1)^c*(169/(100*b^2) + 1)^(2*c)*(324/(25*b^2) + 1)^c*(9/(400*b^2) + 1)^c*(49/(400*b^2) + 1)^c*(361/(100*b^2) + 1)^(5*c)*(81/(400*b^2) + 1)^(2*c)*(121/(400*b^2) + 1)^(4*c)*(169/(400*b^2) + 1)^(3*c)*(4/(625*b^2) + 1)^c*(529/(100*b^2) + 1)^(3*c)*(9/(625*b^2) + 1)^(2*c)*(16/(625*b^2) + 1)^c*(36/(625*b^2) + 1)^c*(49/(625*b^2) + 1)^c*(64/(625*b^2) + 1)^(2*c)*(289/(400*b^2) + 1)^(3*c)*(81/(625*b^2) + 1)^c*(121/(625*b^2) + 1)^c*(361/(400*b^2) + 1)^c*(144/(625*b^2) + 1)^(3*c)*(196/(625*b^2) + 1)^(2*c)*(729/(100*b^2) + 1)^c*(441/(400*b^2) + 1)^c*(256/(625*b^2) + 1)^(2*c)*(289/(625*b^2) + 1)^(2*c)*(529/(400*b^2) + 1)^(2*c)*(324/(625*b^2) + 1)^(2*c)*(361/(625*b^2) + 1)^c*(961/(100*b^2) + 1)^c*(441/(625*b^2) + 1)^(4*c)*(484/(625*b^2) + 1)^c*(529/(625*b^2) + 1)^(3*c)*(729/(625*b^2) + 1)^c*(961/(400*b^2) + 1)^(2*c)*(784/(625*b^2) + 1)^(2*c)*(841/(625*b^2) + 1)^(2*c)*(1089/(400*b^2) + 1)^(2*c)*(1024/(625*b^2) + 1)^(3*c)*(1369/(400*b^2) + 1)^c*(1156/(625*b^2) + 1)^(2*c)*(1296/(625*b^2) + 1)^(4*c)*(1521/(400*b^2) + 1)^c*(1521/(625*b^2) + 1)^c*(1849/(400*b^2) + 1)^c*(1764/(625*b^2) + 1)^(2*c)*(1849/(625*b^2) + 1)^c*(1/(2500*b^2) + 1)^c*(9/(2500*b^2) + 1)^c*(49/(2500*b^2) + 1)^(2*c)*(1936/(625*b^2) + 1)^c*(121/(2500*b^2) + 1)^c*(169/(2500*b^2) + 1)^c*(2116/(625*b^2) + 1)^(3*c)*(289/(2500*b^2) + 1)^(2*c)*(2401/(400*b^2) + 1)^c*(361/(2500*b^2) + 1)^c*(2304/(625*b^2) + 1)^c*(441/(2500*b^2) + 1)^c*(2601/(400*b^2) + 1)^c*(2401/(625*b^2) + 1)^(3*c)*(529/(2500*b^2) + 1)^c*(2601/(625*b^2) + 1)^c*(729/(2500*b^2) + 1)^(4*c)*(2704/(625*b^2) + 1)^(2*c)*(841/(2500*b^2) + 1)^(2*c)*(961/(2500*b^2) + 1)^c*(2916/(625*b^2) + 1)^(3*c)*(1089/(2500*b^2) + 1)^(4*c)*(3249/(400*b^2) + 1)^c*(1369/(2500*b^2) + 1)^c*(3249/(625*b^2) + 1)^(2*c)*(1521/(2500*b^2) + 1)^(3*c)*(3481/(625*b^2) + 1)^(3*c)*(1849/(2500*b^2) + 1)^c*(3844/(625*b^2) + 1)^c*(4096/(625*b^2) + 1)^c*(4356/(625*b^2) + 1)^c*(2601/(2500*b^2) + 1)^c*(4489/(625*b^2) + 1)^c*(4624/(625*b^2) + 1)^c*(5041/(625*b^2) + 1)^c*(3249/(2500*b^2) + 1)^c*(5184/(625*b^2) + 1)^c*(3481/(2500*b^2) + 1)^c*(3721/(2500*b^2) + 1)^c*(4489/(2500*b^2) + 1)^c*(5041/(2500*b^2) + 1)^(2*c)*(5329/(2500*b^2) + 1)^c*(7396/(625*b^2) + 1)^(2*c)*(5929/(2500*b^2) + 1)^c*(1/(10000*b^2) + 1)^c*(9/(10000*b^2) + 1)^(2*c)*(7569/(2500*b^2) + 1)^c*(81/(10000*b^2) + 1)^(2*c)*(289/(10000*b^2) + 1)^(2*c)*(361/(10000*b^2) + 1)^c*(529/(10000*b^2) + 1)^c*(961/(10000*b^2) + 1)^(3*c)*(1089/(10000*b^2) + 1)^(2*c)*(8649/(2500*b^2) + 1)^(2*c)*(1369/(10000*b^2) + 1)^c*(1521/(10000*b^2) + 1)^(3*c)*(1681/(10000*b^2) + 1)^c*(1849/(10000*b^2) + 1)^c*(9409/(2500*b^2) + 1)^(3*c)*(2209/(10000*b^2) + 1)^(4*c)*(9801/(2500*b^2) + 1)^c*(2401/(10000*b^2) + 1)^(2*c)*(2601/(10000*b^2) + 1)^(2*c)*(10201/(2500*b^2) + 1)^c*(2809/(10000*b^2) + 1)^(5*c)*(10609/(2500*b^2) + 1)^(2*c)*(3481/(10000*b^2) + 1)^c*(3721/(10000*b^2) + 1)^c*(3969/(10000*b^2) + 1)^(4*c)*(11881/(2500*b^2) + 1)^(6*c)*(4489/(10000*b^2) + 1)^c*(4761/(10000*b^2) + 1)^c*(12321/(2500*b^2) + 1)^(2*c)*(5041/(10000*b^2) + 1)^c*(12769/(2500*b^2) + 1)^(2*c)*(5329/(10000*b^2) + 1)^c*(13689/(2500*b^2) + 1)^c*(6561/(10000*b^2) + 1)^c*(6889/(10000*b^2) + 1)^c*(14641/(2500*b^2) + 1)^c*(7569/(10000*b^2) + 1)^(3*c)*(7921/(10000*b^2) + 1)^(3*c)*(8649/(10000*b^2) + 1)^c*(16641/(2500*b^2) + 1)^(2*c)*(9409/(10000*b^2) + 1)^c*(17161/(2500*b^2) + 1)^c*(17689/(2500*b^2) + 1)^c*(10201/(10000*b^2) + 1)^(2*c)*(10609/(10000*b^2) + 1)^(2*c)*(11449/(10000*b^2) + 1)^c*(11881/(10000*b^2) + 1)^c*(12321/(10000*b^2) + 1)^c*(12769/(10000*b^2) + 1)^c*(20449/(2500*b^2) + 1)^c*(13689/(10000*b^2) + 1)^c*(14161/(10000*b^2) + 1)^(3*c)*(14641/(10000*b^2) + 1)^(2*c)*(22801/(2500*b^2) + 1)^c*(16129/(10000*b^2) + 1)^c*(16641/(10000*b^2) + 1)^(2*c)*(17161/(10000*b^2) + 1)^(3*c)*(17689/(10000*b^2) + 1)^c*(19321/(10000*b^2) + 1)^c*(19881/(10000*b^2) + 1)^c*(20449/(10000*b^2) + 1)^(2*c)*(28561/(2500*b^2) + 1)^c*(21609/(10000*b^2) + 1)^(2*c)*(23409/(10000*b^2) + 1)^c*(24649/(10000*b^2) + 1)^c*(25281/(10000*b^2) + 1)^(3*c)*(25921/(10000*b^2) + 1)^c*(26569/(10000*b^2) + 1)^c*(27889/(10000*b^2) + 1)^(2*c)*(28561/(10000*b^2) + 1)^c*(29241/(10000*b^2) + 1)^c*(31329/(10000*b^2) + 1)^c*(32041/(10000*b^2) + 1)^c*(33489/(10000*b^2) + 1)^c*(35721/(10000*b^2) + 1)^c*(36481/(10000*b^2) + 1)^c*(37249/(10000*b^2) + 1)^c*(39601/(10000*b^2) + 1)^c*(40401/(10000*b^2) + 1)^(2*c)*(41209/(10000*b^2) + 1)^c*(42849/(10000*b^2) + 1)^(2*c)*(43681/(10000*b^2) + 1)^c*(44521/(10000*b^2) + 1)^c*(49729/(10000*b^2) + 1)^c*(51529/(10000*b^2) + 1)^(3*c)*(56169/(10000*b^2) + 1)^c*(57121/(10000*b^2) + 1)^c*(58081/(10000*b^2) + 1)^c*(59049/(10000*b^2) + 1)^(2*c)*(61009/(10000*b^2) + 1)^c*(63001/(10000*b^2) + 1)^(2*c)*(66049/(10000*b^2) + 1)^c*(69169/(10000*b^2) + 1)^c*(72361/(10000*b^2) + 1)^c*(73441/(10000*b^2) + 1)^c*(77841/(10000*b^2) + 1)^c*(80089/(10000*b^2) + 1)^(3*c)*(85849/(10000*b^2) + 1)^c*(95481/(10000*b^2) + 1)^c*(114921/(10000*b^2) + 1)^c*(116281/(10000*b^2) + 1)^c*(124609/(10000*b^2) + 1)^c*(137641/(10000*b^2) + 1)^c*(c + 1)^353)/(pi^353*((b^2 + 968562431735785/70368744177664)^(c + 1) - b^(2*c + 2))^353)

or

vpa(-(b^(706*c)*(1/b^2 + 1)^c*(4/b^2 + 1)^(2*c)*(1/(4*b^2) + 1)^(2*c)*(9/b^2 + 1)^(2*c)*(9/(4*b^2) + 1)^c*(1/(16*b^2) + 1)^c*(9/(16*b^2) + 1)^(5*c)*(25/(4*b^2) + 1)^c*(9/(25*b^2) + 1)^(4*c)*(16/(25*b^2) + 1)^(2*c)*(25/(16*b^2) + 1)^c*(49/(16*b^2) + 1)^c*(49/(25*b^2) + 1)^c*(64/(25*b^2) + 1)^(2*c)*(81/(25*b^2) + 1)^c*(9/(100*b^2) + 1)^c*(121/(16*b^2) + 1)^c*(121/(25*b^2) + 1)^(3*c)*(49/(100*b^2) + 1)^(3*c)*(169/(16*b^2) + 1)^(2*c)*(169/(25*b^2) + 1)^c*(121/(100*b^2) + 1)^c*(169/(100*b^2) + 1)^(2*c)*(324/(25*b^2) + 1)^c*(9/(400*b^2) + 1)^c*(49/(400*b^2) + 1)^c*(361/(100*b^2) + 1)^(5*c)*(81/(400*b^2) + 1)^(2*c)*(121/(400*b^2) + 1)^(4*c)*(169/(400*b^2) + 1)^(3*c)*(4/(625*b^2) + 1)^c*(529/(100*b^2) + 1)^(3*c)*(9/(625*b^2) + 1)^(2*c)*(16/(625*b^2) + 1)^c*(36/(625*b^2) + 1)^c*(49/(625*b^2) + 1)^c*(64/(625*b^2) + 1)^(2*c)*(289/(400*b^2) + 1)^(3*c)*(81/(625*b^2) + 1)^c*(121/(625*b^2) + 1)^c*(361/(400*b^2) + 1)^c*(144/(625*b^2) + 1)^(3*c)*(196/(625*b^2) + 1)^(2*c)*(729/(100*b^2) + 1)^c*(441/(400*b^2) + 1)^c*(256/(625*b^2) + 1)^(2*c)*(289/(625*b^2) + 1)^(2*c)*(529/(400*b^2) + 1)^(2*c)*(324/(625*b^2) + 1)^(2*c)*(361/(625*b^2) + 1)^c*(961/(100*b^2) + 1)^c*(441/(625*b^2) + 1)^(4*c)*(484/(625*b^2) + 1)^c*(529/(625*b^2) + 1)^(3*c)*(729/(625*b^2) + 1)^c*(961/(400*b^2) + 1)^(2*c)*(784/(625*b^2) + 1)^(2*c)*(841/(625*b^2) + 1)^(2*c)*(1089/(400*b^2) + 1)^(2*c)*(1024/(625*b^2) + 1)^(3*c)*(1369/(400*b^2) + 1)^c*(1156/(625*b^2) + 1)^(2*c)*(1296/(625*b^2) + 1)^(4*c)*(1521/(400*b^2) + 1)^c*(1521/(625*b^2) + 1)^c*(1849/(400*b^2) + 1)^c*(1764/(625*b^2) + 1)^(2*c)*(1849/(625*b^2) + 1)^c*(1/(2500*b^2) + 1)^c*(9/(2500*b^2) + 1)^c*(49/(2500*b^2) + 1)^(2*c)*(1936/(625*b^2) + 1)^c*(121/(2500*b^2) + 1)^c*(169/(2500*b^2) + 1)^c*(2116/(625*b^2) + 1)^(3*c)*(289/(2500*b^2) + 1)^(2*c)*(2401/(400*b^2) + 1)^c*(361/(2500*b^2) + 1)^c*(2304/(625*b^2) + 1)^c*(441/(2500*b^2) + 1)^c*(2601/(400*b^2) + 1)^c*(2401/(625*b^2) + 1)^(3*c)*(529/(2500*b^2) + 1)^c*(2601/(625*b^2) + 1)^c*(729/(2500*b^2) + 1)^(4*c)*(2704/(625*b^2) + 1)^(2*c)*(841/(2500*b^2) + 1)^(2*c)*(961/(2500*b^2) + 1)^c*(2916/(625*b^2) + 1)^(3*c)*(1089/(2500*b^2) + 1)^(4*c)*(3249/(400*b^2) + 1)^c*(1369/(2500*b^2) + 1)^c*(3249/(625*b^2) + 1)^(2*c)*(1521/(2500*b^2) + 1)^(3*c)*(3481/(625*b^2) + 1)^(3*c)*(1849/(2500*b^2) + 1)^c*(3844/(625*b^2) + 1)^c*(4096/(625*b^2) + 1)^c*(4356/(625*b^2) + 1)^c*(2601/(2500*b^2) + 1)^c*(4489/(625*b^2) + 1)^c*(4624/(625*b^2) + 1)^c*(5041/(625*b^2) + 1)^c*(3249/(2500*b^2) + 1)^c*(5184/(625*b^2) + 1)^c*(3481/(2500*b^2) + 1)^c*(3721/(2500*b^2) + 1)^c*(4489/(2500*b^2) + 1)^c*(5041/(2500*b^2) + 1)^(2*c)*(5329/(2500*b^2) + 1)^c*(7396/(625*b^2) + 1)^(2*c)*(5929/(2500*b^2) + 1)^c*(1/(10000*b^2) + 1)^c*(9/(10000*b^2) + 1)^(2*c)*(7569/(2500*b^2) + 1)^c*(81/(10000*b^2) + 1)^(2*c)*(289/(10000*b^2) + 1)^(2*c)*(361/(10000*b^2) + 1)^c*(529/(10000*b^2) + 1)^c*(961/(10000*b^2) + 1)^(3*c)*(1089/(10000*b^2) + 1)^(2*c)*(8649/(2500*b^2) + 1)^(2*c)*(1369/(10000*b^2) + 1)^c*(1521/(10000*b^2) + 1)^(3*c)*(1681/(10000*b^2) + 1)^c*(1849/(10000*b^2) + 1)^c*(9409/(2500*b^2) + 1)^(3*c)*(2209/(10000*b^2) + 1)^(4*c)*(9801/(2500*b^2) + 1)^c*(2401/(10000*b^2) + 1)^(2*c)*(2601/(10000*b^2) + 1)^(2*c)*(10201/(2500*b^2) + 1)^c*(2809/(10000*b^2) + 1)^(5*c)*(10609/(2500*b^2) + 1)^(2*c)*(3481/(10000*b^2) + 1)^c*(3721/(10000*b^2) + 1)^c*(3969/(10000*b^2) + 1)^(4*c)*(11881/(2500*b^2) + 1)^(6*c)*(4489/(10000*b^2) + 1)^c*(4761/(10000*b^2) + 1)^c*(12321/(2500*b^2) + 1)^(2*c)*(5041/(10000*b^2) + 1)^c*(12769/(2500*b^2) + 1)^(2*c)*(5329/(10000*b^2) + 1)^c*(13689/(2500*b^2) + 1)^c*(6561/(10000*b^2) + 1)^c*(6889/(10000*b^2) + 1)^c*(14641/(2500*b^2) + 1)^c*(7569/(10000*b^2) + 1)^(3*c)*(7921/(10000*b^2) + 1)^(3*c)*(8649/(10000*b^2) + 1)^c*(16641/(2500*b^2) + 1)^(2*c)*(9409/(10000*b^2) + 1)^c*(17161/(2500*b^2) + 1)^c*(17689/(2500*b^2) + 1)^c*(10201/(10000*b^2) + 1)^(2*c)*(10609/(10000*b^2) + 1)^(2*c)*(11449/(10000*b^2) + 1)^c*(11881/(10000*b^2) + 1)^c*(12321/(10000*b^2) + 1)^c*(12769/(10000*b^2) + 1)^c*(20449/(2500*b^2) + 1)^c*(13689/(10000*b^2) + 1)^c*(14161/(10000*b^2) + 1)^(3*c)*(14641/(10000*b^2) + 1)^(2*c)*(22801/(2500*b^2) + 1)^c*(16129/(10000*b^2) + 1)^c*(16641/(10000*b^2) + 1)^(2*c)*(17161/(10000*b^2) + 1)^(3*c)*(17689/(10000*b^2) + 1)^c*(19321/(10000*b^2) + 1)^c*(19881/(10000*b^2) + 1)^c*(20449/(10000*b^2) + 1)^(2*c)*(28561/(2500*b^2) + 1)^c*(21609/(10000*b^2) + 1)^(2*c)*(23409/(10000*b^2) + 1)^c*(24649/(10000*b^2) + 1)^c*(25281/(10000*b^2) + 1)^(3*c)*(25921/(10000*b^2) + 1)^c*(26569/(10000*b^2) + 1)^c*(27889/(10000*b^2) + 1)^(2*c)*(28561/(10000*b^2) + 1)^c*(29241/(10000*b^2) + 1)^c*(31329/(10000*b^2) + 1)^c*(32041/(10000*b^2) + 1)^c*(33489/(10000*b^2) + 1)^c*(35721/(10000*b^2) + 1)^c*(36481/(10000*b^2) + 1)^c*(37249/(10000*b^2) + 1)^c*(39601/(10000*b^2) + 1)^c*(40401/(10000*b^2) + 1)^(2*c)*(41209/(10000*b^2) + 1)^c*(42849/(10000*b^2) + 1)^(2*c)*(43681/(10000*b^2) + 1)^c*(44521/(10000*b^2) + 1)^c*(49729/(10000*b^2) + 1)^c*(51529/(10000*b^2) + 1)^(3*c)*(56169/(10000*b^2) + 1)^c*(57121/(10000*b^2) + 1)^c*(58081/(10000*b^2) + 1)^c*(59049/(10000*b^2) + 1)^(2*c)*(61009/(10000*b^2) + 1)^c*(63001/(10000*b^2) + 1)^(2*c)*(66049/(10000*b^2) + 1)^c*(69169/(10000*b^2) + 1)^c*(72361/(10000*b^2) + 1)^c*(73441/(10000*b^2) + 1)^c*(77841/(10000*b^2) + 1)^c*(80089/(10000*b^2) + 1)^(3*c)*(85849/(10000*b^2) + 1)^c*(95481/(10000*b^2) + 1)^c*(114921/(10000*b^2) + 1)^c*(116281/(10000*b^2) + 1)^c*(124609/(10000*b^2) + 1)^c*(137641/(10000*b^2) + 1)^c*(c + 1)^353)/(pi^353*((b^2 + 968562431735785/70368744177664)^(c + 1) - b^(2*c + 2))^353)) 

The result is always -inf...

But using the symbolyc function I got numbers like,

-2.966948035e-508

I think that is because the precision

Matt J 2022-5-10

Maybe fitting log(sigma) will behave better.

Esteban Garcia 2022-5-11

在 MATLAB Online 中打开

Hello Matt it is much faster now without symbolyc and with log. Thank you

N=length(Rproj);
R=max(Rproj);
A=1000000
B=0
C=0
syms b c
for j=1:N
F(j)=log(2*Rproj(j)*(1+(Rproj(j)/b)^2)^c/(((b^2+R^2)^(c+1)-b^(2*c+2))/(b^(2*c)*(c+1))));
end
E=-sum(F);
fstr=string(E);
fstr=replace(fstr,'b','b(k)');
fstr=replace(fstr,'c','c(j)');
b=0.01:0.001:0.8
c=-1.405:0.001:-0.705
for k=1:length(b)
for j=1:length(c)
n=eval(fstr);
if n<A
A=n;
B=b(k);
C=c(j);
end
end
end

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Answer 2

Mitch Lautigar 2022-5-9

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1715140-minimization-problem-with-integral-constraint#answer_960445

My suggestion is to use a smaller step size for <a,b,c> if you know what they are. Typically when you are trying to fix the curve, the only thing you can do is try to add in more datapoints. If you can provide some code, I can provide more feedback.