--- title: Extended Horizontal LCP (EHLCP) url: https://www.emergentmind.com/topics/extended-horizontal-linear-complementarity-problem-ehlcp type: topic --- # Extended Horizontal LCP (EHLCP) The extended horizontal linear complementarity problem (EHLCP) is a multi-block generalization of the horizontal linear complementarity problem (HLCP) and, through HLCP, of the standard linear complementarity problem (LCP). In the formulation studied in "Generalizations of R0 and SSM properties; Extended Horizontal Linear Complementarity Problem" [2301.01479], one is given an ordered family of matrices \(C=(C_0,C_1,\dots,C_k)\), a vector \(q\in\mathbb{R}^n\), and positive vectors \(d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}\), and seeks vectors \(x_0,x_1,\dots,x_k\in\mathbb{R}^n\) such that \[ C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_1,\qquad (d_j-x_j)\perp x_{j+1},\quad 1\le j\le k-1. \] Here \(a\perp b\) means \(a\ge 0\), \(b\ge 0\), and \(a^\top b=0\), equivalently \(a\wedge b=0\), or componentwise \(a_ib_i=0\) for all \(i\) [2301.01479]. The problem couples a single linear balance equation with a chain of complementarity relations. This chain structure, later described as a “horizontal chain-like dependence” \( \mathrm{w}\leftrightarrow \mathrm{x}_1\leftrightarrow\cdots\leftrightarrow \mathrm{x}_m \), is the feature that distinguishes EHLCP from ordinary LCPs and makes both its structural theory and its algorithmics more intricate [2509.14491]. ## 1. Definition, notation, and reduction to classical problems In the notation of [2301.01479], the solution set is denoted \[ \mathrm{SOL}(C,d,q). \] The defining system consists of three layers: a linear identity involving all blocks \(C_0,\dots,C_k\); a complementarity relation between the first two variables \(x_0\) and \(x_1\); and a sequence of complementarity relations between \(d_j-x_j\) and \(x_{j+1}\). The vectors \(d_j\) are required to lie in \(\mathbb{R}^n_{++}\), so the intermediate variables are naturally interpreted as upper-bounded by positive thresholds. A notational variant appears in the 2025 iterative study [2509.14491], which writes the EHLCP data as \[ \mathrm{H}=(\mathrm{M},\mathrm{H}_1,\ldots,\mathrm{H}_m),\qquad \mathrm{d}=(\mathrm{q},\mathrm{d}_1,\ldots,\mathrm{d}_{m-1}), \] and seeks \(\mathrm{w},\mathrm{x}_1,\ldots,\mathrm{x}_m\) satisfying \begin{align} \mathrm{M}\mathrm{w} &= \mathrm{q}+\sum_{i=1}^m \mathrm{H}_i\mathrm{x}_i, \\ \mathrm{w},\mathrm{x}_i &\ge 0,\qquad i=1,\ldots,m, \\ \mathrm{w}^T\mathrm{x}_1 &= 0, \\ \mathrm{x}_i\le \mathrm{d}_i,\qquad (\mathrm{d}_i-\mathrm{x}_i)^T\mathrm{x}_{i+1}=0,\qquad i=1,\ldots,m-1. \end{align} This is the same chain-patterned problem written with a different block naming convention [2509.14491]. The principal reductions are standard. If \(k=1\), EHLCP reduces to the HLCP \[ C_0x_0-C_1x_1=q,\qquad x_0\perp x_1. \] If, further, \(C_0=I\), HLCP reduces to the standard LCP [2301.01479]. The same reduction is stated in the 2025 convexity paper: for \(k=1\), one recovers HLCP, and for \(k=1\) with \(C_0=I\), the standard LCP [2504.20377]. In this sense, EHLCP extends the familiar two-block complementarity architecture to a finite ordered chain. A useful equivalence, recalled in [2504.20377], is \[ x\wedge y = 0 \quad\Longleftrightarrow\quad x,y\ge 0 \text{ and } x*y=0 \quad\Longleftrightarrow\quad x,y\ge 0 \text{ and } (x,y)=0. \] This allows the EHLCP constraints to be written either in min-map form \(x\wedge y=0\) or in standard nonnegative complementarity form \(x\ge 0\), \(y\ge 0\), \(x*y=0\). ## 2. Matrix-set properties governing solvability A central contribution of [2301.01479] is the extension of classical \(R_0\) and strictly semimonotone notions from single-matrix LCP theory to ordered tuples of matrices. These generalized properties are formulated directly for \(C=(C_0,\dots,C_k)\). The \(R0\text{-}W\) property is defined by requiring that the system \[ C_0x_0=\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_j\ \ \forall j\in[k] \] have only the zero solution [2301.01479]. When \(k=1\) and \(C_0=I\), this coincides with the usual \(R_0\) condition for a matrix. The paper also uses Lemma 3.1 to show that any EHLCP solution satisfies the homogeneous complementarity pattern \[ C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_j\ \ \forall j\in[k], \] which is crucial in the boundedness argument [2301.01479]. The \(SSM\text{-}W\) property generalizes strict semimonotonicity. The classical condition for a single matrix \(M\) is \[ x\ge 0,\quad x\circ Mx\le 0 \quad \Longrightarrow\quad x=0. \] For a tuple \(C=(C_0,\dots,C_k)\), the generalized condition is that \[ C_0x_0=\sum_{i=1}^k C_ix_i,\qquad x_i\ge 0,\qquad x_0\circ x_i<0\ \ \forall i\in[k] \] implies \[ x=(x_0,x_1,\dots,x_k)=0 \] [2301.01479]. Within the paper’s framework, \(SSM\text{-}W\) is stronger than \(R0\text{-}W\); the former leads to full existence results, while the latter primarily yields boundedness and, with a degree condition, existence. The structural consequences of \(SSM\text{-}W\) are explicit. Proposition 4.1 of [2301.01479] shows that if \(C\) has \(SSM\text{-}W\), then \(C_0\) is invertible, each \(C_0^{-1}C_i\) is strictly semimonotone, \((I,C_0^{-1}C_1,\dots,C_0^{-1}C_k)\) also has the \(SSM\text{-}W\) property, and the property is invariant under simultaneous permutation similarity \[ (P^\top C_0P,\;P^\top C_1P,\;\dots,\;P^\top C_kP). \] This places the generalized property within the same invariance pattern familiar from matrix complementarity theory. A further relation connects these notions to the column \(W\)-property. Theorem 4.3 states that if \(C\) has the column \(W\)-property, then \(C\) has the \(SSM\text{-}W\) property [2301.01479]. The converse fails in general: Example 4.4 provides a \(2\times 2\) example where \(SSM\text{-}W\) holds but the column \(W\)-property fails because \(\det(C_1)=0\) [2301.01479]. This separates generalized semimonotonicity from determinant-sign-based representative conditions. ## 3. Existence, compactness, and uniqueness The basic existence theorem under \(R0\text{-}W\) is Theorem 3.3 of [2301.01479]. If \(C\) has the \(R0\text{-}W\) property and the EHLCP-degree satisfies \[ \deg(C,0)\neq 0, \] then for every \(q\in\mathbb{R}^n\) and every \(d\in(\mathbb{R}^n_{++})^{k-1}\), the set \(\mathrm{SOL}(C,d,q)\) is nonempty and compact [2301.01479]. The argument is degree-theoretic: boundedness is first obtained from \(R0\text{-}W\), then a homotopy between the homogeneous and shifted maps is used, and a nonzero degree yields a zero of the shifted system. The existence theory becomes cleaner under \(SSM\text{-}W\). Theorem 4.9 states that if \(C\) has the \(SSM\text{-}W\) property, then for every \(q\in\mathbb{R}^n\) and every \(d\in(\mathbb{R}^n_{++})^{k-1}\), \[ \mathrm{SOL}(C,d,q)\neq\emptyset \] [2301.01479]. The proof again uses degree theory, now with the additional fact that \(SSM\text{-}W\Rightarrow R0\text{-}W\). Uniqueness requires additional structure. Theorem 4.10 of [2301.01479] shows that if \(C\) has the \(SSM\text{-}W\) property and \(C_0\) is an \(M\)-matrix, then for every \(q\in\mathbb{R}^n\) and every \(d\in(\mathbb{R}^n_{++})^{k-1}\), \(\mathrm{EHLCP}(C,d,q)\) has a unique solution. The proof identifies the candidate \[ (C_0^{-1}q,0,\dots,0) \] as a solution and then uses the \(SSM\text{-}W\) condition to show that any other solution must coincide with it [2301.01479]. A sharper equivalence emerges under a \(Z\)-matrix hypothesis. Theorem 4.7 states that if each \(C_0^{-1}C_i\) is a \(Z\)-matrix, then the following are equivalent: \(C\) has the column \(W\)-property, and \(C\) has the \(SSM\text{-}W\) property [2301.01479]. Corollary 4.8 then gives the corresponding problem-theoretic characterization: \[ C \text{ has } SSM\text{-}W \quad\Longleftrightarrow\quad \text{for all } q\in\mathbb{R}^n,\ d\in(\mathbb{R}^n_{++})^{k-1},\ \mathrm{EHLCP}(C,d,q)\text{ has a unique solution.} \] Under this additional sign structure, \(SSM\text{-}W\) becomes exactly the uniqueness condition. The column \(W\)-property itself admits an equivalent uniqueness statement. The 2025 cS-W paper recalls that, for \(C=(C_0,\dots,C_k)\), the following are equivalent: \(C\) has the column \(W\)-property; for any nonnegative diagonal matrices \(D_0,\dots,D_k\) with \(\operatorname{diag}(D_0+\cdots+D_k)>0\), one has \(\det(C_0D_0+\cdots+C_kD_k)\ne 0\); \(C_0\) is invertible and \((I,C_0^{-1}C_1,\dots,C_0^{-1}C_k)\) has the column \(W\)-property; and, for all \(q\) and \(d\), \(\operatorname{EHLCP}(C,d,q)\) has a unique solution [2504.20377]. This is the determinant-representative formulation most directly analogous to classical \(P\)-matrix theory. ## 4. Convexity theory and column sufficient-\(W\) A major later development is the introduction of the column sufficient-\(W\) property, abbreviated cS-\(W\), in "On Column sufficiency and Extended Horizontal Linear Complementarity Problem" [2504.20377]. For \(C=(C_0,\dots,C_k)\), cS-\(W\) is defined by the implication \[ \Big( x_i*x_j\ge 0\ \forall\,1\le i0\), together with degree stability under small perturbations and the isolation of a bounded connected component by an open bounded neighborhood. The contradiction argument relies on nonvanishing degree and uniqueness for the perturbed problem. This connectedness theory generalizes a classical result of Jones and Gowda for standard LCPs [2301.01479]. In the EHLCP setting, the semi-algebraic character of the solution set is not merely a technical observation; it is the bridge that allows topological connectedness to be handled through path arguments. ## 6. Computational reformulations and iterative methods A 2025 paper states that, since the EHLCP was first introduced and studied by Kaneko in 1977, no iterative methods or error analysis had been developed for it because of the interdependence of its multiple unknowns in a “chain-like” structure [2509.14491]. That work addresses the gap by deriving an equivalent one-variable piecewise-linear system. The key transformation introduces a single vector \(\mathrm{y}\in\mathbb{R}^n\). In the scalar case, Lemma 2.1 of [2509.14491] shows that the complementarity-and-bounds chain \[ w,x_i\ge0,\quad wx_1=0,\quad x_i\le d_i,\quad (d_i-x_i)x_{i+1}=0 \] is equivalent to the existence of a scalar \(y\) such that \[ w=\max\{0,-y\}, \] \[ x_i=\max\left\{0,\min\left\{y-\sum_{j=0}^{i-1}d_j,d_i\right\}\right\},\qquad i=1,\ldots,m-1, \] \[ x_m=\max\left\{0,y-\sum_{i=0}^{m-1}d_i\right\}, \] with \(d_0=0\). Extended componentwise, this yields Proposition 2.1: \begin{align} \mathrm{M}\max\{0,-\mathrm{y}\} &=\mathrm{q} +\sum_{i=1}^{m-1}\mathrm{H}_i\max\left\{0,\min\left\{\mathrm{y}-\sum_{j=0}^{i-1}\mathrm{d}_j,\mathrm{d}_i\right\}\right\} \\ &\quad +\mathrm{H}_m\max\left\{0,\mathrm{y}-\sum_{i=0}^{m-1}\mathrm{d}_i\right\}. \end{align} The original variables are then recovered explicitly from \(\mathrm{y}\) [2509.14491]. This reformulation supports a general fixed-point iteration: \[ \mathrm{M}\mathrm{y}^{k+1} = \mathrm{M}\max\{0,\mathrm{y}^k\}-\mathrm{q}-\phi(\mathrm{y}^k), \] where \[ \phi(\mathrm{y}^k)= \sum_{i=1}^{m-1}\mathrm{H}_i\max\left\{0,\min\left\{\mathrm{y}^k-\sum_{j=0}^{i-1}\mathrm{d}_j,\mathrm{d}_i\right\}\right\} +\mathrm{H}_m\max\left\{0,\mathrm{y}^k-\sum_{i=0}^{m-1}\mathrm{d}_i\right\} \] [2509.14491]. The paper emphasizes that this method avoids pivoting, preserves the original matrices \(\mathrm{M},\mathrm{H}_i\), and is suitable for large sparse problems. Its convergence theory is based on the column \(W\)-property. Theorem 3.1 states that if \(\mathrm{H}\) has the column \(W\)-property, then the piecewise-linear system has a unique solution [2509.14491]. Theorem 3.3 then gives a global convergence condition for the iteration: if \(\mathrm{H}\) has the column \(W\)-property and \[ \rho(\mathrm{L})<1, \qquad \mathrm{L} = \mathrm{I}-\mathrm{M}^{-1}(\mathrm{MD}_0+\mathrm{H}_1\mathrm{D}_1+\cdots+\mathrm{H}_m\mathrm{D}_m), \] for the relevant diagonal partition, then \(\mathrm{y}^k\to \mathrm{y}^*\) for any initial \(\mathrm{y}^0\) [2509.14491]. Corollary 3.1 provides the more directly checkable sufficient conditions \[ \rho\!\left(\sum_{i=1}^m \left|\mathrm{I}-\mathrm{M}^{-1}\mathrm{H}_i\right|\right)<1 \quad\text{or}\quad \sum_{i=1}^m \left\|\mathrm{I}-\mathrm{M}^{-1}\mathrm{H}_i\right\|<1. \] The same work derives global error bounds. If \((\mathrm{w}(\mathrm{y}),\mathrm{x}_1(\mathrm{y}),\ldots,\mathrm{x}_m(\mathrm{y}))\) is the point induced by \(\mathrm{y}\), and \(\mathrm{y}^*\) is the exact solution, then Theorem 4.1 gives the two-sided estimate \[ \frac{1}{\underline{\alpha}(\mathrm{H})} \left\|\mathrm{r}(\mathrm{w}(\mathrm{y}),\mathrm{x}_1(\mathrm{y}),\ldots,\mathrm{x}_m(\mathrm{y}))\right\| \le \|\mathrm{y}-\mathrm{y}^*\| \le \overline{\alpha}(\mathrm{H}) \left\|\mathrm{r}(\mathrm{w}(\mathrm{y}),\mathrm{x}_1(\mathrm{y}),\ldots,\mathrm{x}_m(\mathrm{y}))\right\| \] [2509.14491]. The paper further gives computable bounds under diagonal-splitting and strict diagonal dominance assumptions, and reports that its specialized method converges in only 3–5 iterations in tested large-scale problems, while an older projection-type method needs 16 iterations consistently and much more CPU time [2509.14491]. ## 7. Related extensions and broader context The EHLCP sits within a broader family of horizontal and weighted complementarity formulations. In Euclidean Jordan algebras, the weighted horizontal linear complementarity problem (wHLCP) is defined for linear transformations \(A,B\), a weight \(w\in V_+\), and \(q\in V\) by \[ x\ge 0,\qquad y\ge 0,\qquad x\circ y=w,\qquad Ax+By=q. \] When \(w=0\), this becomes the symmetric-cone HLCP, and in the special case \(A=I\), \(B=-M\), it becomes the standard LCP on \(V\) [1710.03162]. The paper "The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra" develops \(R_0\)-pairs, \(R\)-pairs, and \(P\)-pairs, together with degree-based solvability theory, thereby providing a symmetric-cone analogue of the horizontal framework from which EHLCP arises [1710.03162]. That paper’s main solvability theorem states that if \(\{A,B\}\) is an \(R_0\)-pair and \(\deg(A,B)\neq 0\), then for every \((w,q)\in V_+\times V\), the problem \(wHLCP(A,B,w,q)\) has a nonempty compact solution set [1710.03162]. In \(\mathbb{R}^n\), it also proves the equivalence between the \(P\)-pair property, unique solvability of \(wHLCP(A,B,w,q)\) for every \((w,q)\), and unique solvability of \(HLCP(A,B,q)\) for every \(q\) [1710.03162]. This suggests that the EHLCP literature extends a wider topological-degree and monotonicity program already present in weighted and cone-based horizontal complementarity. A different but related generalization appears in complementarity problems on extended second order cones. The paper "Linear complementarity problems on extended second order cones" studies \(\LCP(T,r,L)\) over the extended second order cone \(L(k,\ell)\), converts it into a mixed complementarity problem on \(\mathbb{R}^k_+\), and then reformulates that problem through a Fischer–Burmeister equation system [1707.04268]. Its main contribution is the chain \[ \LCP(T,r,L) \;\Longrightarrow\; \MixCP(\widetilde F_1,\widetilde F_2,\mathbb{R}^k_+) \;\Longrightarrow\; \mathbb F^{\MixCP}_{FB}(z)=0, \] which supports Newton and Levenberg–Marquardt algorithms [1707.04268]. While this is not an EHLCP formulation, it is part of the same broader pattern: complementarity structures beyond the standard nonnegative orthant are frequently analyzed by converting them into orthant-based mixed systems and then applying nonlinear equation methods. Within EHLCP proper, several themes now stand out. The 2023 theory identifies \(R0\text{-}W\) as the boundedness condition and \(SSM\text{-}W\) as the stronger property yielding existence and, under additional structure, uniqueness [2301.01479]. The 2025 convexity paper identifies cS-\(W\) as the property ensuring convexity of the EHLCP solution set and uniqueness for \(q\ge 0\) when \(C_0\) is an \(M\)-matrix [2504.20377]. The 2025 iterative paper turns the chained complementarity relations into a one-variable piecewise-linear equation, enabling fixed-point algorithms and global error bounds [2509.14491]. Taken together, these developments show that EHLCP has evolved from a structural extension of HLCP into a problem class with a distinct theory of matrix-set properties, topological solvability, geometric solution-set analysis, and numerical computation. Source: https://www.emergentmind.com/topics/extended-horizontal-linear-complementarity-problem-ehlcp