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A matrix $A$ is called totally positive (or totally non-negative) of order$k$, denoted by TP_k (or TN_k), if all minors of size at most $k$ are positive(or non-negative). These matrices have featured in diverse areas inmathematics, including algebra, analysis, combinatorics, and probabilitytheory. The goal of this article is to provide a novel connection between totalpositivity and optimization/game theory. Specifically, we draw a relationshipbetween TP matrices and the Linear Complementarity Problem (LCP), whichgeneralizes and unifies linear and quadratic programming problems and bimatrixgames - this connection is unexplored, to the best of our knowledge. We showthat $A$ is $TP_k$ if and only if for every contiguous square submatrix $A_r$of $A$, $LCP(A_r,q)$ has a unique solution for each vector $q<0$. In fact thiscan be strengthened to check the solution set of LCP at a single vector foreach such square submatrix. These novel characterizations are in the spirit ofclassical results characterizing $TP$ matrices by Gantmacher-Krein [Compos.Math. 1937] and P-matrices by Ingleton [Proc. London Math. Soc. 1966].  Our work contains two other contributions, both of which characterize TPusing single test vectors. First, we improve on one of the main results inrecent joint work [Bull. London Math. Soc., 2021], which provided a novelcharacterization of TP_k matrices using sign non-reversal phenomena. We furtherimprove on a classical characterization of TP by Brown-Johnstone-MacGibbon [J.Amer. Statist. Assoc. 1981] (following Gantmacher-Krein, 1950) involving thevariation diminishing property. Finally, we use a P\'olya frequency function ofKarlin [Trans. Amer. Math. Soc. 1964] to show that our aforementionedcharacterizations of TP, involving test-vectors drawn from the `alternating'bi-orthant, do not work if these vectors are drawn from any other open orthant.

Characterizing total positivity: single vector tests via Linear Complementarity, sign non-reversal, and variation diminution