Mathematics, an age-old discipline, continues to surprise and evolve with discoveries that push back the limits of human knowledge. In 2025, a major breakthrough was achieved in the application of Newton’s method, a fundamental pillar of numerical calculation since the 17th century.
This new approach promises to transform the way complex equations are solved, offering unprecedented prospects for scientific and technological research. The potential impact of this breakthrough is immense, affecting fields from engineering to artificial intelligence. Find out how this innovation could redefine the horizons of modern mathematics.
Improvement and evolution of Newton’s method
A team of researchers has recently pushed the limits of Newton’s method, making it more powerful than ever. For over three centuries, this ingenious algorithm, devised by Isaac Newton, has been a mainstay for solving complex problems in diverse fields such as logistics, finance and computer vision.
However, its scope was limited, as it did not apply efficiently to all functions. This obstacle could now be overcome thanks to the work of Amir Ali Ahmadi of Princeton University and his former students, Abraar Chaudhry and Jeffrey Zhang. Their breakthrough enables Newton’s method to handle a wider class of functions, opening up new perspectives in optimization.
A new perspective on Newton’s method
Ahmadi, Chaudhry and Zhang have revolutionized Newton’s method by developing an algorithm capable of efficiently handling an unlimited number of variables and derivatives. Their approach is based on two essential properties: convexity and expression as a sum of squares.
These features simplify the minimization of complex functions. Using semidefinite programming, they adjusted the Taylor expansion to meet these conditions, making the algorithm more efficient. Although each iteration is more computationally expensive, future technological advances could make this method indispensable in fields such as machine learning. Ahmadi hopes that within 10 to 20 years, this technique will be as fast in practice as it is in theory.
Future prospects and potential applications
The improved algorithm of Ahmadi, Chaudhry and Zhang promises to transform the landscape of mathematical optimization. Although its current implementation is computationally expensive, anticipated technological advances could reduce these financial hurdles.
In the next 10 to 20 years, this method could become an essential tool in fields such as machine learning, where speed and efficiency are crucial. Ahmadi expresses cautious optimism about the practical applicability of their algorithm, anticipating that it will outperform current methods not only in theory but also in practice, paving the way for significant innovations in various sectors.
