# Constrained Optimization with Multiple Constraints: Do multiple strictly positive multipliers imply a solution at a vertex?

This might be a bit of a silly question but I am interested in solving standard economic problems with many constraints and am wondering if there are any shortcuts.

To preface suppose we have the following generic utility maximization problem with $$k$$ many constraints which hold with equality.

$$\max U(x_1,...,x_n)$$ subject to $$m_1\geq\sum_{i=1}^nr_i^1 x_i \tag{1}$$ $$...$$ $$m_k\ge\sum_{i=1}^n r_i^k x_i \tag{k}$$

The traditional way of solving these sort of problems would be to identify possible optimum considering one constraint at a time and then seeing if it violates any constraints. Its possible however that a corner solution exists where in this case we would seek the values at the vertices on our feasible set as defined by our set of constraints.

This is a tedious problem however I'm wondering if just looking at the values of the Lagrange multipliers associated with each one of these constraints (checking if multiple are positive) to infer if a vertex on our feasible region is indeed the optimum.

In short if I identify a case where say two multipliers $$\lambda_i$$ and $$\lambda_j$$ are strictly positive, does that mean the optimum is at a vertex for this problem?

• A positive Lagrange multiplier means the corresponding constraint holds with equality at the optimum, which you seem to already assume. Or do you also have nonnegativity constraints for the $x$'s and the $\lambda$'s are for those constraints? Commented Feb 21, 2022 at 1:30
• Perhaps you meant to write $$m_1 \geq \sum_{i=1}^nr_i^1 x_i$$ instead of $$m_1=\sum_{i=1}^nr_i^1 x_i$$? Commented Feb 21, 2022 at 6:57
• @Giskard correct. Edited accordingly
– EconJohn
Commented Feb 21, 2022 at 15:46
• @HerrK. The salient assumption is that $x\in\mathbb{R}_+^l$
– EconJohn
Commented Feb 21, 2022 at 15:48
• I added tags to the constraints for easier reference. Commented Feb 21, 2022 at 19:30

If by $$\lambda_i$$ you mean the multiplier belonging to constraint ($$i$$), then $$\lambda_i$$ and $$\lambda_j$$ being positive do mean that these constraints are active/effective/realized as equalities.

Now it is not quite clear to me what you mean by "corner solution". The constraints ($$1$$),($$2$$),...,($$k$$) usually define a polyhedron in $$\mathbb{R}^n$$, where $$(x_1,x_2,\dots,x_n)$$ is such that all constraints are fulfilled, so this polyhedron is the set/region of feasible solutions. If a solution is in the interior of this set, it is definitely not a corner solution; otherwise I am not sure. E.g., is a solution on the edge (but not in any corner/vertex) of a cube a corner solution? Or perhaps you meant at least one $$x_i$$ is zero?

If less then $$n$$ multipliers are positive, then it is likely your solution is not in a vertex of the polyhedron, but merely on a face of it.

For an example of this, consider $$U(x_1,x_2,x_3) = x_1x_2x_3,$$ and the constraints are $$9 \geq 2x_1 + x_2 + x_3 \tag{1}$$ $$9 \geq x_1 + 2x_2 + x_3 \tag{2}$$ and the usual non-negativity constraints.

The optimal solution in this case is $$(x_1, x_2, x_3) = (2,2,3)$$, and the multipliers are $$\lambda_1 = \lambda_2 = 2 > 0$$. The optimal solution is not a vertex of the feasible region, it is a convex combination of the feasible solutions $$(3,3,0)$$ and $$(0,0,9)$$. (These two solutions are vertices.)

• I see that the use of the term corner solution is probably inappropriate here as I'm interested in also seeing if the optimum is defined by one of the vertices from the feasible region of the two constraints. I guess a more appropriate edit of the question would be if one can immediately jump to the conclusion from inspection of the multipliers values that our optimum lies on a vertex?
– EconJohn
Commented Feb 21, 2022 at 19:57
• @EconJohn I gave a 3D example with two positive multipliers. Commented Feb 22, 2022 at 6:54
• Thank you so much for your efforts. Just to confirm, the presence of multiple positive multipliers which indicate multiple binding constraints does not necessarily mean the optimum is a vertex?
– EconJohn
Commented Feb 22, 2022 at 16:05
• @EconJohn I have answered this exact question above: "The optimal solution is not a vertex of the feasible region" Commented Feb 22, 2022 at 16:40