MATH.10:4.1 - Fix the comparison

State the candidate, the conditions to retain, the scalar quantity and the conclusion sought. Write the candidate as x and the quantity as J(x) when that notation helps. For minimization, smaller values of J are preferred; for maximization, reverse the comparisons below.

Keep the meaning of J visible. It might be a supplied allocation cost, an approximation error or a physical action functional, which assigns a number to a history. A physical principle can require stationarity of an action without selecting its minimum. Recover that principle and its conditions before using the variational calculation to describe physical behavior.

Distinguish a search for one improving change from a claim of local or global optimality. Local optimality concerns a specified neighborhood of the candidate; global optimality concerns all candidates admitted by the problem.