Beginner Explanation
Imagine you have a big box of different colored balls, and you want to find the best way to organize them to make a beautiful display. The Sum-of-Squares Framework is like a special set of rules that helps you figure out the best arrangement by breaking down the problem into smaller, easier parts. Instead of just guessing, it uses math to create a plan that helps you understand how to group the balls so that you get the most beautiful display possible, even if you can’t see all the combinations at once.Technical Explanation
The Sum-of-Squares (SOS) Framework is a powerful tool in optimization that leverages polynomial equations to derive relaxations for combinatorial optimization problems. It involves expressing the objective function and constraints as polynomials, and then using SOS polynomials to create lower bounds on the original problem. A polynomial is SOS if it can be represented as a sum of squares of other polynomials. For instance, consider maximizing a function subject to certain constraints. We can formulate this as an SOS program, where we seek to find a polynomial that minimizes the difference between the original function and the SOS relaxation. This can be implemented using libraries like ‘SOSTools’ in Python. Here’s a simple example: “`python import numpy as np from SOSTools import * # Define variables x = sdpvar(1) # Define a polynomial p = x**2 + 1 # Check if p is SOS is_sos = sos(p) print(f’Is the polynomial SOS? {is_sos}’) “`Academic Context
The Sum-of-Squares Framework is rooted in algebraic geometry and optimization theory. It has gained prominence through seminal works by researchers like Lasserre (2001), who introduced SOS programming as a method for polynomial optimization. The mathematical foundation rests on the concept that a non-negative polynomial can be expressed as a sum of squares of other polynomials, which ensures that the polynomial remains non-negative over its domain. Key papers include ‘Global Optimization with Polynomials and the Problem of Moments’ by Lasserre and ‘Sum of Squares and the Polynomial Hierarchy’ by Parrilo. The framework has applications in various fields, including control theory, statistics, and machine learning, particularly in deriving tight relaxations for NP-hard problems.Code Examples
Example 1:
import numpy as np
from SOSTools import *
# Define variables
x = sdpvar(1)
# Define a polynomial
p = x**2 + 1
# Check if p is SOS
is_sos = sos(p)
print(f'Is the polynomial SOS? {is_sos}')Example 2:
import numpy as np
from SOSTools import *
# Define variables
x = sdpvar(1)Example 3:
from SOSTools import *
# Define variables
x = sdpvar(1)
View Source: https://arxiv.org/abs/2511.16613v1