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Solve a polynomial optimization problem

Create a random polynomial of degree k using the Dot operator and find its minimum using the Nonlinear solver.

Further explanation of this example: 'Xpress Python Reference Manual'[download all files]

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# Minimize a polynomial constructed with the Dot product
# (C) Fair Isaac Corp., 1983-2024

from __future__ import print_function

import xpress as xp
import numpy as np

# Generate a random coefficient tensor T of dimension k + 1 and sizes
# n+1 for each dimension except for the first, which is h, then use it
# to create h polynomial constraints. The lhs of each constraint has a
# polynomial of degree k, and not homogeneous as we amend the vector
# of variable with the constant 1. This is accomplished via a single
# dot product.

n = 10  # dimension of variable space
h = 3   # number of polynomial constraints
k = 4   # degree of each polynomial

p = xp.problem()

# Vector of n elements: (1, x1, ..., x_{n-1}), declared with NumPy's
# dtype notation for Xpress expressions (to guarantee Xpress
# operations will be used).
x = np.array([1] + [p.addVariable(lb=-10, ub=10) for _ in range(n-1)], dtype=xp.npexpr)

sizes = [n]*k  # creates list [n,n,...,n] of k elements

# Operator * before a list translates the list into its
# (unparenthesized) tuple, i.e., the result is a reshape list of
# argument that looks like (h, n, n, ..., n)

T = np.random.random(h * n ** k).reshape(h, *sizes)


T2list = [x]*k

compact = xp.Dot(T, *T2list) <= 0


# Solve this problem with a local nonlinear solver
p.controls.nlpsolver = 1

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