Is duality in UMP and EMP (economics) same as duality in convex optimization?

Is duality between Utility Maximization Problem (UMP) and Expenditure Minimization Problem (EMP) same as duality in convex optimization?

Most economics textbooks say UMP and EMP are dual so I’ve tried to derive EMP from UMP by applying duality approach in convex optimization, which ended up in failure.

UMP: max $U(x)$ subject to $p.x leq I$

EMP: min $p.x$ subject to $U(x) geq u$

Is it possible to get EMP from UMP by applying Lagrangian duality approach in convex optimization?