Polynomial Optimization and Algebraic Geometry

Polynomial optimization concerns the problem of finding global minima or maxima of multivariate polynomial functions subject to polynomial constraints. Such problems are inherently nonconvex and often NP-hard, yet they admit systematic relaxations based on semidefinite programming and sums-of-squares representations. Algebraic geometry contributes foundational certificates, such as Positivstellensatz results, that guarantee nonnegativity of polynomials on semialgebraic sets and yield exact or convergent hierarchies of convex relaxations. Through spectrahedral descriptions of feasible regions and moment-based dual formulations, this interplay has led to powerful algorithms with applications ranging from control and signal processing to quantum information and data-driven scientific discovery. Recent advances have focused on improving convergence rates of hierarchies, exploiting sparsity and symmetry, and unifying data-driven inference with algebraic certificates to derive and validate governing equations in the physical sciences.

Research from Nature Portfolio

Recent studies have demonstrated that when all underlying scientific laws and axioms can be encoded as polynomials, mixed-integer and semidefinite relaxations can be used to discover compact mathematical models directly from experimental data. By introducing binary variables to enforce minimal complexity and employing Positivstellensatz certificates to ensure validity, researchers have shown that classical laws—such as those governing planetary motion and radiated gravitational power—can be rederived in a principled manner. This work highlights the potential of polynomial-based frameworks to unify background theory with empirical observations in a rigorous optimisation setting.

Research from all publishers

In recent years, significant progress has been made beyond symmetric cone settings. One line of work has extended conic optimisation to homogeneous but non-self-dual matrix cones, characterising their automorphism groups, establishing spectrahedral representations via algebraic theory, and analysing interior-point methods adapted to these structures. Another strand has developed chordal and factor-width decomposition techniques for semidefinite and polynomial optimisation, exploiting sparsity patterns to decompose large semidefinite constraints into tractable subproblems, thereby enabling the efficient analysis and control of high-dimensional dynamical systems. Improvements in convergence analysis of measure-based hierarchies have also been obtained: for polynomial minimisation on compact domains such as the sphere, upper and lower bounds based on sum-of-squares densities now achieve quadratic rates of convergence, with proofs of tightness for broad classes of functions and extensions to general moment problems.

Polynomial Optimization and Algebraic Geometry publication trend

The graph below shows the total number of articles in polynomial optimization and algebraic geometry across all publications each year (not limited to Nature Index journals).

Technical terms

Polynomial optimisation: The task of minimizing or maximizing a polynomial function subject to polynomial equality and inequality constraints.

Semidefinite programming: A class of convex optimisation problems in which a linear function is optimized over the intersection of the cone of positive semidefinite matrices with an affine space.

Sum-of-squares: A certificate for polynomial nonnegativity expressing a given polynomial as a sum of squares of other polynomials, which can be enforced via semidefinite constraints.

Positivstellensatz certificate: An algebraic certificate that establishes nonnegativity of a polynomial on a semialgebraic set through combinations of sums-of-squares and the defining polynomials.

Spectrahedral representation: A description of a convex semialgebraic set as the solution set of a linear matrix inequality, yielding a feasible region of a semidefinite program.

Chordal decomposition: A technique that exploits sparsity in semidefinite constraints by decomposing large matrix variables according to a chordal graph structure, reducing computational complexity.

References

1. Evolving scientific discovery by unifying data and background knowledge with AI Hilbert. Nature Communications (2024).
2. Linear optimization over homogeneous matrix cones. Acta Numerica (2023).
3. Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization. Annual Reviews in Control (2021).
4. Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere. Mathematical Programming (2020).
5. Improved convergence analysis of Lasserre’s measure-based upper bounds for polynomial minimization on compact sets. Mathematical Programming (2020).
6. Nonnegative polynomials and sums of squares. Journal of the American Mathematical Society (2012).