Beginner Explanation
Imagine you’re building a LEGO tower. The first time you build it, you might take a while to find all the right pieces and figure out how to stack them. But if you decide to build a similar tower again, you can use the first one as a base. You already know which pieces fit well together, so you can build it much faster. Warm starting is like using that first LEGO tower to help you build the next one quicker, by starting from where you left off instead of starting from scratch.Technical Explanation
Warm starting in optimization involves initializing an algorithm with the solution from a previous related problem. This is particularly useful in iterative methods, such as gradient descent, where previous solutions can inform the current search direction. For example, in Python using SciPy, you can warm start a linear programming problem by passing the previous solution as the initial guess for the next problem: “`python from scipy.optimize import linprog # Previous solution x0 = [1, 2] # New problem with similar structure c = [1, 1] A = [[-1, -2], [1, 1]] b = [-2, 2] # Warm start res = linprog(c, A_ub=A, b_ub=b, x0=x0) print(res.x) “` This approach can significantly reduce computation time and improve convergence in optimization tasks.Academic Context
Warm starting is a well-studied concept in optimization, particularly in the context of convex problems and iterative solvers. Theoretical foundations can be traced back to the works of Nesterov and Polyak on gradient methods, which highlight the benefits of using previous iterates to enhance convergence rates. Key papers include ‘Gradient Methods for Minimizing Composite Functions’ (Nesterov, 2013) and ‘Convergence of the Gradient Method for Convex Optimization’ (Polyak, 1964). These studies provide a mathematical framework for understanding how warm starts can lead to faster convergence in various optimization scenarios, particularly in machine learning and operations research.Code Examples
Example 1:
from scipy.optimize import linprog
# Previous solution
x0 = [1, 2]
# New problem with similar structure
c = [1, 1]
A = [[-1, -2], [1, 1]]
b = [-2, 2]
# Warm start
res = linprog(c, A_ub=A, b_ub=b, x0=x0)
print(res.x)Example 2:
from scipy.optimize import linprog
# Previous solution
x0 = [1, 2]
View Source: https://arxiv.org/abs/2511.16340v1