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The code of minimize_constrained starts checking if the first argument is a symbolic expression. If this were the case, it gets the variables on which such an expression depends. Then the code generates a function whose argument is a vector, needed for any subsequent computation; each variable of the expression is replaced by a component of the vector. For example, if the expression is x^2+y, the variables are x and y. The code defines a function whose argument is a vector, say p, which yields p[0]^2+p[1].

When $f(x,y)=x$, the expression f(x,y) has only one variable, which is x, and so the function defined inside minimize_constrained does not depend on a vector with two components, as required, raising errors.

To use minimize_constrained, I would advise to directly define Python functions depending on vectors, either with def or lambda. For example, for $f(x,y)=x+y$, we can write

def f(x): return x[0] + x[1]
def c(x): return 1 - x[0]^2 - x[1]^2

or also

f = lambda x: x[0] + x[1]
c = lambda x: 1 - x[0]^2 - x[1]^2

In both cases, we get:

sage: minimize_constrained(f, c, [0,0])
(-0.7071774954004159, -0.7070360740443554)

Likewise,

sage: f = lambda x: x[0]
sage: c = lambda x: 1 - x[0]^2 - x[1]^2
sage: minimize_constrained(f, c, [0,0])
(-1.0000000049989504, 9.999999987505248e-05)

which is the expected result.

If, for clarity, you prefer to explicitly define functions in terms of x and y, you can still do that:

sage: f = lambda x,y: x 
sage: c = lambda x,y: 1 - x^2 - y^2
sage: ff = lambda p: f(*p)
sage: cc = lambda p: c(*p)
sage: minimize_constrained(ff, cc, [0,0])
(-1.0000000049989504, 9.999999987505248e-05)

The code of minimize_constrained starts checking if the first argument is a symbolic expression. If this were the case, it gets the variables on which such an expression depends. Then the code generates a function whose argument is a vector, needed for any subsequent computation; each variable of the expression is replaced by a component of the vector. For example, if the expression is x^2+y, the variables are x and y. The code defines a function whose argument is a vector, say p, which yields p[0]^2+p[1].

When $f(x,y)=x$, the expression f(x,y) has only one variable, which is x, and so the function defined inside minimize_constrained does not depend on a vector with two components, as required, raising errors.

To use minimize_constrained, I would advise to directly define Python functions depending on vectors, either with def or lambda. For example, for $f(x,y)=x+y$, we can write

def f(x): return x[0] + x[1]
def c(x): return 1 - x[0]^2 - x[1]^2

or also

f = lambda x: x[0] + x[1]
c = lambda x: 1 - x[0]^2 - x[1]^2

In both cases, we get:

sage: minimize_constrained(f, c, [0,0])
(-0.7071774954004159, -0.7070360740443554)

Likewise, for $f(x,y)=x$, we have:

sage: f = lambda x: x[0]
sage: c = lambda x: 1 - x[0]^2 - x[1]^2
sage: minimize_constrained(f, c, [0,0])
(-1.0000000049989504, 9.999999987505248e-05)

which is the expected result.

If, for clarity, you prefer to explicitly define functions in terms of x and y, you can still do that:

sage: f = lambda x,y: x 
sage: c = lambda x,y: 1 - x^2 - y^2
sage: ff = lambda p: f(*p)
sage: cc = lambda p: c(*p)
sage: minimize_constrained(ff, cc, [0,0])
(-1.0000000049989504, 9.999999987505248e-05)