In a given constrained optimization problem, the objective is convex and the constraints are strictly convex. I know that at least one of the constraints is binding. The Karush-Kuhn-Tucker multipliers are, by definition, non-negative, but can I claim their sum is strictly positive?
Edit
Let $\alpha>1$, $\mathbf{q}, \mathbf{k},\mathbf{1}, \mathbf{0}\in \mathbb R_+^n$, with $\mathbf{1}$ and $\mathbf{0}$ vectors of ones and zeros respectively. Let also $\mathbf{X}$ be a $n\times n$ non singular matrix, whose elements are non negative and such that for every row and column there is at least one strictly positive value. $F(\mathbf{k})=q:\Bbb R_+^m\to\Bbb R_+$ associates to each $\mathbf{k}$ the maximum value of \begin{align} \text{maximize}&\quad\mathbf{q}^\top\mathbf{1}\\ \text{subject to}&\quad\left(\mathbf{q}^\alpha\right)^\top\mathbf{X}\leq\mathbf{k}\\ &\quad\mathbf{q}\geq\mathbf{0} \end{align}
This function can be shown to be homogeneous, so that $$\forall \mu>0\quad F(\mathbf{k})=q\Longleftrightarrow F(\mu\mathbf{k})=\mu^{1/\alpha}q.$$
If $F(\mathbf{k})>0$, at the optimum at least one of the first set of inequalities needs to be binding. KKT multipliers are non negative by construction. Let $\gamma_i$ denote the multipliers associated with non negativity constraints and $\lambda_j$ those with resource constraints. I believe I can argue that $\sum \lambda_j$ needs to be strictly positive. Otherwise, as multipliers measure the marginal effect of the constraints on the objective, relaxing or tightening all of them should not have any impact on the objective. In particular, choosing $\mu\approx 1$ we'd have $F(\mu\mathbf{k})=F(\mathbf{k})$ which contradicts homogeneity.
Is this correct?