Necessary and sufficient conditions for a minimum of mixed order (Q915661)

We consider the problem (1) f(x)\(\to \min\), \(x\in H\), where H is a Hilbert space, f is a real functional on H, Fréchet differentiable p times. As a rule, the existing optimality criteria in problem (1) make use of the fact that either \(p=2\), or (for \(p>2)\) all the derivatives up to and including order p-1 of the functional f at the minimum point \(\bar x\in H\) are equal to zero. However, there exists a large class of problems (1), in which the operator \(f^{(2)}(\bar x)\), where \(f^{(\ell)}(x)\) is the \(\ell\)-th Fréchet derivative at the point x, is degenerate, but not identically equal to zero, i.e., the solution x of the problem (1) may be degenerate [see \textit{B. T. Polyak}, ``Introduction to optimization'' (Russian) (1983; Zbl 0652.49002)]. For the case \(H=R^ n\), rank \(f^{(2)}(\bar x)=n-1\), \(p=4\), necessary and sufficient conditions for a minimum were indicated by \textit{A. Griewank} and \textit{M. R. Osborne} [SIAM J. Numer. Anal. 20, 747-773 (1983; Zbl 0525.65025)] and then used for the investigation of the rate of convergence of the Newton method for the problem (1) in the neighborhood of the minimum point with a rank-one degeneracy of the Hessian. We present in a somewhat different form the necessary and sufficient conditions for a minimum of mixed order for the degenerate problem (1), generalizing the known optimality conditions of both second and p-th order.