Verification of High Order Optimality Conditions Using the Polyadic Representation of Derivatives

Conditions involving derivatives higher than the second must be checked if the first and second optimality conditions fail to determine whether or not a point locally minimizes an unconstrained function. An implementable inductive algorithm for doing this is developed based on the presentation of higher derivatives by sums of symmetric polyads (generalized outer product matrices). A set of equations is developed which must be satisfied for the odd order conditions to hold. Checking the even order conditions involves the solution of successive nonconvex programming problems. An example is included.