Minimizing a function over the sphere

Using HomotopyContinuation.jl for a problem in optimization

We want to solve following optimization problem

$$ \text{minimize} \; 3x^3y+y^2z^2-2xy-4xz^3 \quad \text{s.t.} \quad x^2+y^2+z^2=1 $$

The strategy to find the global optimum is to use the method of Lagrange multipliers to find all critical points of the objective function such that the equality constraint is satisfied. We start with defining our Lagrangian.

using HomotopyContinuation, DynamicPolynomials
@polyvar x y z
J = 3x^3*y+y^2*z^2-2x*y-x*4z^3
g = x^2+y^2+z^2-1
# Introduce auxillary variable for Lagrangian
@polyvar λ
# define Lagrangian
L = J - λ * g
3x³y - 4xz³ + y²z² - x²λ - y²λ - z²λ - 2xy + λ

In order to compute all critical points we have to solve the square system of equations

$$ \nabla_{(x,y.z,\lambda)}L = 0 $$

For this we first compute the gradient of $L$ and then use the solve routine to find all critical points.

# compute the gradient
∇L = differentiate(L, [x, y, z, λ])
# Now we solve the polynomial system
result = solve(∇L)
Result with 26 solutions
==================================
• 26 non-singular solutions (22 real)
• 0 singular solutions (0 real)
• 54 paths tracked
• random seed: 556761

We see that from the theoretical 54 possible (complex) critical points there are only 26. Also we check the number of real critical points by

nreal(result)
22

and see that there are 22.

In order to find the global minimum we now have to evaluate all real solutions and find the value where the minimum is attained.

reals = real_solutions(result);
# Now we simply evaluate the objective J and find the minimum
minval, minindex = findmin(map(s -> J(s[1:3]), reals))
(-1.32842129069426, 21)
minarg = reals[minindex][1:3]
3-element Array{Float64,1}:
 -0.4968755205596932
 -0.09460829635048905
 -0.8626494000056988

We found that the minimum of $J$ over the unit sphere is attained at $(0.496876, 0.0946083, 0.862649)$ with objective value $-1.32842$.

Cite this example:
@Misc{ optimization2019 ,
    author =  { Sascha Timme },
    title = { Minimizing a function over the sphere },
    howpublished = { \url{ https://www.JuliaHomotopyContinuation.org/examples/optimization/ } },
    note = { Accessed: September 18, 2019 }
}

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