Necessary higher-order conditions in extremum problems (Q1802609)

Consider the optimization problem (P): \(\varphi(x)\to\min\), \(F(x)=0\), where \(\varphi:X\to\mathbb{R}\), \(F:X\to Y\) and \(X,Y\) are Banach spaces. The Lagrangian \(L:X\times Y\to\mathbb{R}\) is defined by \(L(x,\lambda)=\varphi(x)+\lambda(F(x))\) for \(x\in X\), \(\lambda\in Y^*\). If \(\varphi\) and \(F\) are twice Fréchet differentiable at the point \(x^*\in X\) and \(\text{Im} F'(x^*)=Y\), then the known first and second order necessary conditions for the problem (P) can be expressed as follows: If \(x^*\) is optimal for (P), then (1) there exists \(\lambda^*\in Y^*\) such that \(L_ x(x^*,\lambda^*)=0\); (2) the set \(H=\{h:F'(x^*)h=0\), \(L_{xx}(x^*,\lambda^*)[h,h]=0\}\) is a closed subspace of \(X\) and there exists a uniquely defined mapping \(h\to\alpha^*(h):H\to Y^*\) such that \(L_{xx}(x^*,\lambda^*)[h]+F'{}^*(x^*)\alpha^*(h)=0\) for \(h\in H\). If the data are three times Fréchet differentiable at \(x^*\), then the author obtains the following new third order necessary condition: \[ L_{xx}(x^*,\lambda^*)[h,h]+{1\over 3}L_{xxx}(x^*,\lambda^*)[h,h,h]+\alpha^*(h)(F''(x^*)[h,h])=0. \tag{3} \] The author also explains how to obtain fourth and higher order necessary conditions of similar type.