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What's the best way to check if a system of one equation and one inequality has any solution?

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Richárd Tóth
Richárd Tóth 2019-8-9
评论： Torsten 2019-8-9
Hello
I've tried the solve function, but it gave me empty result:
syms x y
>> cond1 = x^2/25 + y^2/25 == 1;
>> cond2 = y <= x-4;
>> conds = [cond1 cond2];
>> sol = solve(conds, [x y], 'ReturnConditions', true);
Tried it on wolframalpha which shows that there is solution : https://www.wolframalpha.com/input/?i=x%5E2%2F25+%2B+y%5E2%2F25+%3D+1+and+y+%3C%3D+x-4
So what do you recommend if I'm not interested in the solutions only want to know if there is any solution or not?

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采纳的回答

Torsten
Torsten 2019-8-9
Use "fmincon" with an arbitrary objective function (e.g. 0).

更多回答（1 个）

Bruno Luong
Bruno Luong 2019-8-9
编辑：Bruno Luong 2019-8-9
In your case use first Euler-Lagrange condition gives one solution if it exists
meaning find lambda, x, y, st
[2*x/25, 2*y/25] = lambda * [-1,1]
x^2/25 + y^2/25 = 1
That gives 2 solutuions
(x,y) = sqrt(1/2)*5*[1,-1];
and
(x,y) = sqrt(1/2)*5*[-1,1];
Then check for the inequalities. The first solution meets the inequalities
y <= x-4
This solution does not require any toolbox or numerical calculation. Just pencil and paper. If you take some care with Kuhn Turker condition with the sign of lambda, you can even discard one of the two solutions before the last step.

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