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I'm considering the following minimization problem: $$ \begin{cases} min f(x) \\ s.t.\ \ \ \ g_{i}(x)\le0 & x \in R^n, i \in I={1,..m} & f,g_{i} \in C^1 \end{cases} \\ $$

Given $x \in S $ where S is the feasible region

$ Set: \\ D1(\bar x)=\{d \in R^n : \exists \alpha>0 : x+\alpha d \in S \}\\ I(\bar x)=\{i \in I: g_{i}(\bar x)=0 \} \\ D2(\bar x)=\{d \in R^n : \nabla^tg_{i}d \le0 \hspace{0.2 cm} \forall i \in I(\bar x) \} \\ $

We say that the qualification constraint (QC) holds at $\bar x$ if $ D1(\bar x)=D2(\bar x) $.

Now, according to my notes, if the $g_{i}$'s are convex and there exixsts a point $\alpha$ s.t. $g_{i}(\alpha)<0$ $ \forall i$ then the QC holds. I'm having some trouble with this last part. If a function is convex then the lower contours are convex sets so if, for istance, I take only one constraint, say: $g=x^2 + y^2-3, $ which is convex, then the feasible region is a disk and for each point of the circle the tangent t belongs to D2 but not to D1. So I'm thinking that maybe the $g_{i}$ must be concave or maybe I'm missing something.

Giacomo
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  • The feasible region in your example is the entire disc, not just the circle. Are you dealing with affine $g_i$ by any chance? The condition GC will not hold on the boundary of the disc because $D_1$ is fairly restrictive. – copper.hat Jun 09 '18 at 17:03
  • I've corrected the mistake, thanks. No, the $ g_{i} $ are generic $ C^1 $ functions. – Giacomo Jun 09 '18 at 18:09
  • Are you sure that the inequality in the definition of $D_2$ is not strict? – Misha Lavrov Jun 09 '18 at 18:15
  • Yes, I'm sure. Indeed my notes give the following example of a case in which $D_1 \subset D_2$. The constraints are $g_1=-x_1$, $g_2=-x_2$, $g_3=-(1-x_1)^3+x_2$. and the problem is in (1,0) where the gradient of $g_3$ is orthogonal to the $(\alpha,0)$ direction which is not feasible. – Giacomo Jun 09 '18 at 18:25
  • Ok, the QC was wrong. QC holds if $ \bar{D_1} (\bar x) =D_2(\bar x)$. This should solve the issue. – Giacomo Jun 09 '18 at 18:47
  • Do you have a reference for this? Are you dealing with convex functions? – copper.hat Jun 09 '18 at 19:10

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