Get the analytical solution of inequalities in Matlab.

Question

zhenyu zeng 2023-9-3

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https://ww2.mathworks.cn/matlabcentral/answers/2016171-get-the-analytical-solution-of-inequalities-in-matlab

编辑： Bruno Luong 2023-9-3

Hello,

x + 2 y + 3 c + 4 d == 3/2 && x + y + c + d == 2 && x >= 2 && 
  y < 2 && c < 3 && d > 5

How to solve this inequations to get the solution of x, y, c, and d in Matlab?

Best regards.

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Answer 1

Walter Roberson 2023-9-3

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Most of the time, MATLAB is not able to solve systems of inequalities.

See the recent discussion at https://www.mathworks.com/matlabcentral/answers/2007972-why-do-solutions-get-lost-when-i-add-a-assumption-where-i-know-that-the-correct-solution-would-sati#answer_1287092

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Walter Roberson 2023-9-3

在 MATLAB Online 中打开

syms x y c d

eqns = [x + 2 * y + 3 * c + 4 * d == 3/2

x + y + c + d == 2

x >= 2

y < 2

c < 3

d > 5]

eqns =

sol = solve(eqns, 'returnconditions', true)

sol = struct with fields:

c: 13/2 - 2*z1 - 3*z d: 2*z + z1 - 9/2 x: z y: z1 parameters: [z z1] conditions: 7 < 6*z + 4*z1 & 19 < 4*z + 2*z1 & z1 < 2 & 2 <= z

%now try to find the boundaries

parts = children(sol.conditions)

parts = 1×4 cell array

{[7 < 6*z + 4*z1]} {[19 < 4*z + 2*z1]} {[z1 < 2]} {[2 <= z]}

sol2 = solve([lhs(parts{1})==rhs(parts{1}), lhs(parts{2})==rhs(parts{2})])

sol2 = struct with fields:

z: 31/2 z1: -43/2

subs(sol, sol2)

ans = struct with fields:

c: 3 d: 5 x: 31/2 y: -43/2 parameters: [31/2 -43/2] conditions: 7 < 7 & 19 < 19 & -43/2 < 2 & 2 <= 31/2

After that you have to explore which side of the bounds the values need to be on

Note that z effectively stands in for x and z1 effectively stands in for y -- that once you have particular x and y values then the c and d are pinned down.

zhenyu zeng 2023-9-3

在 MATLAB Online 中打开

Or this one

{[{31/2 < x}, {y < 2, 7/4 - (3*x)/2 < y}, {c = 13/2 - 3*x - 2*y}, {d = -9/2 + 2*x + y}], [{x <= 31/2, 15/4 < x}, {y < 2, 19/2 - 2*x < y}, {c = 13/2 - 3*x - 2*y}, {d = -9/2 + 2*x + y}]}

Walter Roberson 2023-9-3

The Maple output looks like this

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Answer 2

Bruno Luong 2023-9-3

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编辑：Bruno Luong 2023-9-3

在 MATLAB Online 中打开

The real question is how would you get as formal decription of the so called "solution" of the equality linear system?

There is no mathematical semantic AFAIK to describe the solution beside what is similar to you question, a set S that satisfies

A*x <= b
Aeq*x = beq
lo <= x <= up

One can show to linear transform the variables x to y so as the above can be reduced to the so called canonical form of

C*y = d
y <= 0

but there is no way to express it shorter.

Another way is if the region S is bounded, it is a polytope and one can express as a "sub-convex" combination of a finite set of vertexes.

S = sum wi * Vi
sum wi <= 1 

In other word a convex hull where the vetices are {Vi}. There is a numerical method in FEX made by Matt J. It works OK for toy dimension of 2-3, but I wouldn't trust it for dimension >= 10.

In short the most convenient to "solve" linear inequalitues is to let it as it is. You want to know a point is a solution? Just replace and check it.