I am struggling a bit with the second order conditions of a constrained maximization problem with [itex]n[/itex] variables and [itex]k[/itex] constraints (with [itex]k>n[/itex]).(adsbygoogle = window.adsbygoogle || []).push({});

In the equality constraints case we have to check if the [itex](n-k)[/itex] leading principal minors of the bordered Hessian alternate in sign, starting from the sign given by [itex](-1)^{n}[/itex].

In the inequality constraints case we have to check if the [itex][n-(e+k)][/itex] leading principal minors of the bordered Hessian alternate in sign, starting from the sign given by [itex](-1)^{n}[/itex] (with [itex]e[/itex] equals to the number of binding constraints and [itex]k[/itex] equals to the number of not-binding constraints).

Fair enough, but how to I behave when I have [itex]n-k<0[/itex] or [itex]n-(e+k)<0[/itex] (e.g. 2 variables in the objective function and 4 equality constraints)?

Do I have to focus only on the number I get in order to know which minors I have to check, without focusing on the sign?

Thanks a lot.

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# Leading Principal Minors of Bordered Hessian in Constrained Max Problems

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