To the theory of extremum for abnormal problems (Q5954766)

The article considers the problem of minimization with constraints in the form of equations \[ f(x) \to \min, \quad x \in X, \quad F(x) = 0, \quad \tag{1} \] where \(X\) is a vector space, \(f: X \to E\), \(F: X \to E^k\) are smooth mappings and \(E^k\) is \(k\)-dimensional arithmetic space. Let \(x_0\) be a solution to the problem (1). If \(x_0\) is an abnormal point, then the rule of Lagrange multipliers gives for a multiplier such a value that classical necessary conditions of the second order are, generally speaking, not true. In the article the necessary conditions are received for abnormal extremum problems (1). They strengthen the earlier received results.