Find maximum of function with 3 variables

How to find maximum of function with 3 variables?

-1<= x1 x2 x3 <=1

function =78.4802 + 5.9976*x1 - 1.94506*x2- 1.7211*x3 - 6.89185*x1^2 - 1.9625*x1*x3 - 5.97065*x2^2 - 1.5375*x2*x3 - 7.51189*x3^2;

[John D'Errico]

That is literally impossible to say, since you do not show only 3 variables. I see:

x1, x2, x3, x12, x1*x3, x22, x2*x3, x32

So there are some products of variables. Then there are others that you may want to be interpreted as such, but you have named them with two numbers. Should we be able to read your mind here? Hey, maybe x32 was intended to represent the square of x3, thus x3^2. The crystal ball is so fuzzy today.

Anyway, if you think that x32 should really be seen as x3*x2, then exactly how is that different from x2*x3? I'm pretty sure that multiplication is a commutative operator, ok, as long as x2 and x3 are real numbers. Admittedly, it has been many years since I worried about that, so the new math may have changed since then. Is Pi now 3? ;-)

[ly]

I editted x12, x22, and x32 to x1^2, x2^2 and x3^2.

Answer 1

[Walter Roberson]

See https://www.mathworks.com/matlabcentral/answers/898147-how-can-i-caculate-the-range-of-value-of-a-function-by-matlab#answer_766752

However, you have the problem that x12, x22, and x32 are not defined.

[ly]

I editted x12, x22, and x32 to x1^2, x2^2 and x3^2.

[Walter Roberson]

The calculus approach works fine for that revised equation.

[ly]

this link: https://www.mathworks.com/matlabcentral/answers/898147-how-can-i-caculate-the-range-of-value-of-a-function-by-matlab#answer_766752

is to find the inflection points.

But I want to find maximum of a function:

-1<= x1 x2 x3 <=1

function =78.4802 + 5.9976*x1 - 1.94506*x2- 1.7211*x3 - 6.89185*x1^2 - 1.9625*x1*x3 - 5.97065*x2^2 - 1.5375*x2*x3 - 7.51189*x3^2;

max(function)=??? x1 x2 x3=???

[Walter Roberson]

The maximum of a continuous function over the infinite domain occurs has a positive slope on the left (increasing towards the maximum) and a negative slope towards the right (decreasing away from the maximum). It follows that the maximum on an infinite domain must be at a location where the slope is 0. The discussion I pointed to shows finding the points with slope 0. You can then assess each to determine whether it is a maximum or minimum or saddle point.

Over a finite domain, the maximum must be either at an inflection point or at a boundary.

So... catalog the inflection points and the boundary points and evaluate the function at end of the locations and whichever one is the largest is the maximum over the finite domain.