1.2.2.2 Second-order conditions

For the sake of completeness, we quickly state the second-order conditions for constrained optimality; they will not be used in the sequel. For the necessary condition, suppose that $x^*$ is a regular point of $D$ and a local minimum of $f$ over $D$ , where $D$ is defined by the equality constraints (1.18) as before. We let $\lambda^*$ be the vector of Lagrange multipliers provided by the first-order necessary condition, and define the augmented cost $\ell$ as in (1.27). We also assume that $f$ is $\mathcal C^2$ . Consider the Hessian of $\ell$ with respect to $x$ evaluated at $(x^*,\lambda^*)$ :

$${\ell}_{{x}{x}}(x^*,\lambda^*)=\nabla^2 f(x^*)+\sum_{i=1}^m \lambda_i^* \nabla^2 h_i(x^*).$$

The second-order necessary condition says that this Hessian matrix must be positive semidefinite on the tangent space to $D$ at $x^*$ , i.e., we must have $d^T{\ell}_{{x}{x}}(x^*,\lambda^*) d\ge 0$ for all $d\in T_{x^*}D$ . Note that this is weaker than asking the above Hessian matrix to be positive semidefinite in the usual sense (on the entire $\mathbb{R}^n$ ).

The second-order sufficient condition says that a point $x^*\in D$ is a strict constrained local minimum of $f$ if the first-order necessary condition for optimality (1.25) holds and, in addition, we have

$$d^T{\ell}_{{x}{x}}(x^*,\lambda^*)d>0\qquad \forall\,d \ \ \nabla h_i(x^*)\cdot d=0,\ i=1,\dots,m.$$ (1.29)

Again, here $\lambda^*$ is the vector of Lagrange multipliers and $\ell$ is the corresponding augmented cost. Note that regularity of $x^*$ is not needed for this sufficient condition to be true. If $x^*$ is in fact a regular point, then we know (from our derivation of the first-order necessary condition for constrained optimality) that the condition imposed on $d$ in (1.29) describes exactly the tangent vectors to $D$ at $x^*$ . In other words, in this case (1.29) is equivalent to saying that ${\ell}_{{x}{x}}(x^*,\lambda^*)$ is positive definite on the tangent space $T_{x^*}D$ .