Complementarity Form of the Schumaker Strang Conjecture

Definitions and mathematical statement

The supplied record leaves the dimension $n$ and the matrix $M$ unspecified. To interpret its displayed expressions, one needs a fixed finite dimension $n$ and a fixed real matrix $M\in{\mathbb{R}}^{n\times n}$. These are typing requirements; the record supplies no construction of $M$, hypotheses on its entries, or intended class of matrices.

For vectors $u,v\in{\mathbb{R}}^n$, write $u\geq0$ when $u_i\geq0$ for every coordinate, and define the Euclidean inner product by $\langle u,v\rangle=\sum_{i=1}^n u_iv_i$. Conditional on the intended fixed choices of $n$ and $M$, the supplied question is whether \[ \forall q\in{\mathbb{R}}^n\;\exists x\in{\mathbb{R}}^n:\qquad x\geq0,\qquad Mx+q\geq0,\qquad \langle Mx+q,x\rangle=0. \] Equivalently, the last condition, together with the two coordinatewise inequalities, requires $x_i(Mx+q)_i=0$ for every coordinate $i$.

The record calls this the complementarity form of the Schumaker--Strang conjecture, but supplies neither the matrix arising from that conjecture nor a statement establishing the correspondence. Thus the formula records the stated complementarity question, while the intended mathematical conjecture remains underdetermined. In particular, no universal quantifier over matrices $M$ is supplied.

Short English statement

For an intended fixed matrix $M$, the question asks whether every vector $q$ admits a nonnegative vector $x$ such that $Mx+q$ is nonnegative and each coordinate of $x$ has zero product with the corresponding coordinate of $Mx+q$. The matrix and its hypotheses are missing, so the intended conjecture cannot yet be stated precisely.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30002862$, source problem code catalogue entry, titled "Complementarity Form of the Schumaker Strang Conjecture." The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2015; no individual proposer is identified.
• Supplied report citation: Multivariate Splines and Algebraic Geometry (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/21, pp. 1139--1200. https://doi.org/10.4171/owr/2015/21. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied literature-source label for this link is "Oberwolfach Report 21."
• Supplied literature source: "Generalized Splines on Graphs, EJC 2024." https://www.combinatorics.org/ojs/index.php/eljc/article/download/v31i1p29/pdf.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the Schumaker--Strang lower-bound conjecture and its complementarity formulation remain active in general, with important degree and genericity cases proved but without a proof of the stated full conjecture. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.