Polynomial optimization, also called polynomial programming, is the problem of minimizing or maximizing a polynomial subject to polynomial equality and inequality constraints. A minimization problem can be written
 |
where the 
and 
are polynomials. The feasible region is a basic semialgebraic set.
The special case in which every polynomial has polynomial degree at most 1 is linear programming. In general, polynomial optimization need not be convex, and exact solution can be difficult. Exact approaches include quantifier elimination and cylindrical algebraic decomposition, while sum-of-squares optimization gives hierarchies of semidefinite programming relaxations.
Some optimization problems containing expressions with radicals can be reformulated as polynomial optimization problems. Maaz and Strzeboński (2025) construct a defining polynomial for each expression using resultants, then add polynomial inequalities that select the intended algebraic branch. This can replace one expression with radicals by one auxiliary variable instead of introducing one variable for every nested radical.
