Extended Horizontal LCP (EHLCP)

• EHLCP is a multi-block generalization of LCP, featuring a chain structure that couples a linear balance equation with sequential complementarity relations.
• The framework extends classic LCP theory by introducing generalized properties (R0-W, SSM-W, and cS-W) that guarantee boundedness, existence, and convexity of the solution set.
• Iterative methods reformulate EHLCP into a one-variable piecewise-linear system, enabling rapid convergence and efficient error estimation in large-scale computational settings.

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" (Yadav et al., 2023), 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 $q\in\mathbb{R}^n$0 for all $q\in\mathbb{R}^n$1 (Yadav et al., 2023). 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” $q\in\mathbb{R}^n$2, is the feature that distinguishes EHLCP from ordinary LCPs and makes both its structural theory and its algorithmics more intricate (Wu et al., 17 Sep 2025).

1. Definition, notation, and reduction to classical problems

In the notation of (Yadav et al., 2023), the solution set is denoted

$q\in\mathbb{R}^n$3

The defining system consists of three layers: a linear identity involving all blocks $q\in\mathbb{R}^n$4; a complementarity relation between the first two variables $q\in\mathbb{R}^n$5 and $q\in\mathbb{R}^n$6; and a sequence of complementarity relations between $q\in\mathbb{R}^n$7 and $q\in\mathbb{R}^n$8. The vectors $q\in\mathbb{R}^n$9 are required to lie in $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$0, so the intermediate variables are naturally interpreted as upper-bounded by positive thresholds.

A notational variant appears in the 2025 iterative study (Wu et al., 17 Sep 2025), which writes the EHLCP data as

$d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$1

and seeks $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$2 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 (Wu et al., 17 Sep 2025).

The principal reductions are standard. If $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$3, EHLCP reduces to the HLCP

$d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$4

If, further, $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$5, HLCP reduces to the standard LCP (Yadav et al., 2023). The same reduction is stated in the 2025 convexity paper: for $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$6, one recovers HLCP, and for $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$7 with $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$8, the standard LCP (Yadav et al., 29 Apr 2025). In this sense, EHLCP extends the familiar two-block complementarity architecture to a finite ordered chain.

A useful equivalence, recalled in (Yadav et al., 29 Apr 2025), is

$d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$9

This allows the EHLCP constraints to be written either in min-map form $x_0,x_1,\dots,x_k\in\mathbb{R}^n$0 or in standard nonnegative complementarity form $x_0,x_1,\dots,x_k\in\mathbb{R}^n$1, $x_0,x_1,\dots,x_k\in\mathbb{R}^n$2, $x_0,x_1,\dots,x_k\in\mathbb{R}^n$3.

2. Matrix-set properties governing solvability

A central contribution of (Yadav et al., 2023) is the extension of classical $x_0,x_1,\dots,x_k\in\mathbb{R}^n$4 and strictly semimonotone notions from single-matrix LCP theory to ordered tuples of matrices. These generalized properties are formulated directly for $x_0,x_1,\dots,x_k\in\mathbb{R}^n$5.

The $x_0,x_1,\dots,x_k\in\mathbb{R}^n$6 property is defined by requiring that the system

$x_0,x_1,\dots,x_k\in\mathbb{R}^n$7

have only the zero solution (Yadav et al., 2023). When $x_0,x_1,\dots,x_k\in\mathbb{R}^n$8 and $x_0,x_1,\dots,x_k\in\mathbb{R}^n$9, this coincides with the usual $$C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_1,\qquad (d_j-x_j)\perp x_{j+1},\quad 1\le j\le k-1.$$0 condition for a matrix. The paper also uses Lemma 3.1 to show that any EHLCP solution satisfies the homogeneous complementarity pattern

$$C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_1,\qquad (d_j-x_j)\perp x_{j+1},\quad 1\le j\le k-1.$$1

which is crucial in the boundedness argument (Yadav et al., 2023).

The $$C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_1,\qquad (d_j-x_j)\perp x_{j+1},\quad 1\le j\le k-1.$$2 property generalizes strict semimonotonicity. The classical condition for a single matrix $$C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_1,\qquad (d_j-x_j)\perp x_{j+1},\quad 1\le j\le k-1.$$3 is

$$C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_1,\qquad (d_j-x_j)\perp x_{j+1},\quad 1\le j\le k-1.$$4

For a tuple $$C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_1,\qquad (d_j-x_j)\perp x_{j+1},\quad 1\le j\le k-1.$$5, the generalized condition is that

$$C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_1,\qquad (d_j-x_j)\perp x_{j+1},\quad 1\le j\le k-1.$$6

implies

$$C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_1,\qquad (d_j-x_j)\perp x_{j+1},\quad 1\le j\le k-1.$$7

(Yadav et al., 2023). Within the paper’s framework, $$C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_1,\qquad (d_j-x_j)\perp x_{j+1},\quad 1\le j\le k-1.$$8 is stronger than $$C_0x_0=q+\sum_{i=1}^k C_ix_i,\qquad x_0\perp x_1,\qquad (d_j-x_j)\perp x_{j+1},\quad 1\le j\le k-1.$$9; the former leads to full existence results, while the latter primarily yields boundedness and, with a degree condition, existence.

The structural consequences of $a\perp b$0 are explicit. Proposition 4.1 of (Yadav et al., 2023) shows that if $a\perp b$1 has $a\perp b$2, then $a\perp b$3 is invertible, each $a\perp b$4 is strictly semimonotone, $a\perp b$5 also has the $a\perp b$6 property, and the property is invariant under simultaneous permutation similarity

$a\perp b$7

This places the generalized property within the same invariance pattern familiar from matrix complementarity theory.

A further relation connects these notions to the column $a\perp b$8-property. Theorem 4.3 states that if $a\perp b$9 has the column $a\ge 0$0-property, then $a\ge 0$1 has the $a\ge 0$2 property (Yadav et al., 2023). The converse fails in general: Example 4.4 provides a $a\ge 0$3 example where $a\ge 0$4 holds but the column $a\ge 0$5-property fails because $a\ge 0$6 (Yadav et al., 2023). This separates generalized semimonotonicity from determinant-sign-based representative conditions.

3. Existence, compactness, and uniqueness

The basic existence theorem under $a\ge 0$7 is Theorem 3.3 of (Yadav et al., 2023). If $a\ge 0$8 has the $a\ge 0$9 property and the EHLCP-degree satisfies

$b\ge 0$0

then for every $b\ge 0$1 and every $b\ge 0$2, the set $b\ge 0$3 is nonempty and compact (Yadav et al., 2023). The argument is degree-theoretic: boundedness is first obtained from $b\ge 0$4, 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 $b\ge 0$5. Theorem 4.9 states that if $b\ge 0$6 has the $b\ge 0$7 property, then for every $b\ge 0$8 and every $b\ge 0$9,

$a^\top b=0$0

(Yadav et al., 2023). The proof again uses degree theory, now with the additional fact that $a^\top b=0$1.

Uniqueness requires additional structure. Theorem 4.10 of (Yadav et al., 2023) shows that if $a^\top b=0$2 has the $a^\top b=0$3 property and $a^\top b=0$4 is an $a^\top b=0$5-matrix, then for every $a^\top b=0$6 and every $a^\top b=0$7, $a^\top b=0$8 has a unique solution. The proof identifies the candidate

$a^\top b=0$9

as a solution and then uses the $a\wedge b=0$0 condition to show that any other solution must coincide with it (Yadav et al., 2023).

A sharper equivalence emerges under a $a\wedge b=0$1-matrix hypothesis. Theorem 4.7 states that if each $a\wedge b=0$2 is a $a\wedge b=0$3-matrix, then the following are equivalent: $a\wedge b=0$4 has the column $a\wedge b=0$5-property, and $a\wedge b=0$6 has the $a\wedge b=0$7 property (Yadav et al., 2023). Corollary 4.8 then gives the corresponding problem-theoretic characterization: $a\wedge b=0$8 Under this additional sign structure, $a\wedge b=0$9 becomes exactly the uniqueness condition.

The column $q\in\mathbb{R}^n$00-property itself admits an equivalent uniqueness statement. The 2025 cS-W paper recalls that, for $q\in\mathbb{R}^n$01, the following are equivalent: $q\in\mathbb{R}^n$02 has the column $q\in\mathbb{R}^n$03-property; for any nonnegative diagonal matrices $q\in\mathbb{R}^n$04 with $q\in\mathbb{R}^n$05, one has $q\in\mathbb{R}^n$06; $q\in\mathbb{R}^n$07 is invertible and $q\in\mathbb{R}^n$08 has the column $q\in\mathbb{R}^n$09-property; and, for all $q\in\mathbb{R}^n$10 and $q\in\mathbb{R}^n$11, $q\in\mathbb{R}^n$12 has a unique solution (Yadav et al., 29 Apr 2025). This is the determinant-representative formulation most directly analogous to classical $q\in\mathbb{R}^n$13-matrix theory.

4. Convexity theory and column sufficient-$q\in\mathbb{R}^n$14

A major later development is the introduction of the column sufficient-$q\in\mathbb{R}^n$15 property, abbreviated cS-$q\in\mathbb{R}^n$16, in "On Column sufficiency and Extended Horizontal Linear Complementarity Problem" (Yadav et al., 29 Apr 2025). For $q\in\mathbb{R}^n$17, cS-$q\in\mathbb{R}^n$18 is defined by the implication

$q\in\mathbb{R}^n$19

For $q\in\mathbb{R}^n$20, cS-$q\in\mathbb{R}^n$21 becomes the $q\in\mathbb{R}^n$22-column-sufficiency property used for HLCP, and for $q\in\mathbb{R}^n$23 with $q\in\mathbb{R}^n$24, it reduces to the classical column-sufficient matrix property (Yadav et al., 29 Apr 2025).

The principal consequence is geometric. Theorem 3.2 states that if $q\in\mathbb{R}^n$25 has the cS-$q\in\mathbb{R}^n$26 property, then $q\in\mathbb{R}^n$27 is convex for every $q\in\mathbb{R}^n$28 and every $q\in\mathbb{R}^n$29 (Yadav et al., 29 Apr 2025). The proof compares two solutions $q\in\mathbb{R}^n$30 and $q\in\mathbb{R}^n$31, derives

$q\in\mathbb{R}^n$32

together with the sign relations

$q\in\mathbb{R}^n$33

and then applies cS-$q\in\mathbb{R}^n$34 to obtain

$q\in\mathbb{R}^n$35

From this, the complementarity relations are shown to persist under convex combinations.

The same paper proves a uniqueness theorem under positivity of the right-hand side. If $q\in\mathbb{R}^n$36 has the cS-$q\in\mathbb{R}^n$37 property and $q\in\mathbb{R}^n$38 is an $q\in\mathbb{R}^n$39-matrix, then for every $q\in\mathbb{R}^n$40 and every $q\in\mathbb{R}^n$41, the EHLCP has a unique solution (Yadav et al., 29 Apr 2025). The canonical solution is again

$q\in\mathbb{R}^n$42

and the positivity of $q\in\mathbb{R}^n$43 under the $q\in\mathbb{R}^n$44-matrix hypothesis drives the uniqueness argument.

The cS-$q\in\mathbb{R}^n$45 property is positioned between earlier $q\in\mathbb{R}^n$46-type notions. Theorem 4.2 of (Yadav et al., 29 Apr 2025) states that the following are equivalent: $q\in\mathbb{R}^n$47 has the column $q\in\mathbb{R}^n$48-property; $q\in\mathbb{R}^n$49 has the cS-$q\in\mathbb{R}^n$50 property and the column ND-$q\in\mathbb{R}^n$51 property; and $q\in\mathbb{R}^n$52 has the column $q\in\mathbb{R}^n$53-property and the column ND-$q\in\mathbb{R}^n$54 property. Theorem 4.3 summarizes the implication chain

$q\in\mathbb{R}^n$55

The reverse implications fail in general. Example 4.4 shows that cS-$q\in\mathbb{R}^n$56 does not imply column $q\in\mathbb{R}^n$57, and Remark 2 gives an example where column $q\in\mathbb{R}^n$58 does not imply cS-$q\in\mathbb{R}^n$59 (Yadav et al., 29 Apr 2025). This clarifies that convexity and uniqueness for positive $q\in\mathbb{R}^n$60 can be obtained under a weaker condition than full determinant-sign coherence.

Under a $q\in\mathbb{R}^n$61-matrix assumption, (Yadav et al., 29 Apr 2025) also introduces a cone cS-$q\in\mathbb{R}^n$62 property and proves that if each $q\in\mathbb{R}^n$63 is a $q\in\mathbb{R}^n$64-matrix, then cS-$q\in\mathbb{R}^n$65 is equivalent to cone cS-$q\in\mathbb{R}^n$66. Corollary 4.6 then yields convexity of $q\in\mathbb{R}^n$67 for every $q\in\mathbb{R}^n$68 and positive $q\in\mathbb{R}^n$69 under the cone version. This extends the classical passage from signed sufficient conditions to cone-sufficient ones.

5. Connectedness and semi-algebraic structure

Beyond nonemptiness, compactness, convexity, and uniqueness, (Yadav et al., 2023) studies connectedness of the solution set. The tuple $q\in\mathbb{R}^n$70 is called connected if $q\in\mathbb{R}^n$71 is connected for all $q\in\mathbb{R}^n$72 and all positive $q\in\mathbb{R}^n$73. A key preliminary fact is that $q\in\mathbb{R}^n$74 is semi-algebraic; by Theorem 5.1 cited there, connectedness is therefore equivalent to path-connectedness (Yadav et al., 2023).

Theorem 5.3 gives a strong restriction under an $q\in\mathbb{R}^n$75-matrix assumption: if $q\in\mathbb{R}^n$76 is an $q\in\mathbb{R}^n$77-matrix and $q\in\mathbb{R}^n$78 is connected, then for every $q\in\mathbb{R}^n$79 and every $q\in\mathbb{R}^n$80,

$q\in\mathbb{R}^n$81

(Yadav et al., 2023). Thus, under this hypothesis, connectedness forces the entire solution set to collapse to the single canonical solution. A plausible implication is that connectedness is a very restrictive property in the presence of monotonicity-type sign structure.

Theorem 5.4 supplies a connectedness criterion. If $q\in\mathbb{R}^n$82 has the column $q\in\mathbb{R}^n$83-property and $q\in\mathbb{R}^n$84 has a bounded connected component, then $q\in\mathbb{R}^n$85 is connected (Yadav et al., 2023). The proof uses the $q\in\mathbb{R}^n$86-perturbation $q\in\mathbb{R}^n$87, for which column $q\in\mathbb{R}^n$88 holds for every $q\in\mathbb{R}^n$89, 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 (Yadav et al., 2023). 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 (Wu et al., 17 Sep 2025). That work addresses the gap by deriving an equivalent one-variable piecewise-linear system.

The key transformation introduces a single vector $q\in\mathbb{R}^n$90. In the scalar case, Lemma 2.1 of (Wu et al., 17 Sep 2025) shows that the complementarity-and-bounds chain

$q\in\mathbb{R}^n$91

is equivalent to the existence of a scalar $q\in\mathbb{R}^n$92 such that

$q\in\mathbb{R}^n$93

$q\in\mathbb{R}^n$94

$q\in\mathbb{R}^n$95

with $q\in\mathbb{R}^n$96. 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 $q\in\mathbb{R}^n$97 (Wu et al., 17 Sep 2025).

This reformulation supports a general fixed-point iteration: $q\in\mathbb{R}^n$98 where

$q\in\mathbb{R}^n$99

(Wu et al., 17 Sep 2025). The paper emphasizes that this method avoids pivoting, preserves the original matrices $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$00, and is suitable for large sparse problems.

Its convergence theory is based on the column $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$01-property. Theorem 3.1 states that if $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$02 has the column $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$03-property, then the piecewise-linear system has a unique solution (Wu et al., 17 Sep 2025). Theorem 3.3 then gives a global convergence condition for the iteration: if $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$04 has the column $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$05-property and

$d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$06

for the relevant diagonal partition, then $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$07 for any initial $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$08 (Wu et al., 17 Sep 2025). Corollary 3.1 provides the more directly checkable sufficient conditions

$d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$09

The same work derives global error bounds. If $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$10 is the point induced by $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$11, and $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$12 is the exact solution, then Theorem 4.1 gives the two-sided estimate

$d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$13

(Wu et al., 17 Sep 2025). 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 (Wu et al., 17 Sep 2025).

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 $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$14, a weight $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$15, and $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$16 by

$d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$17

When $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$18, this becomes the symmetric-cone HLCP, and in the special case $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$19, $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$20, it becomes the standard LCP on $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$21 (Chi et al., 2017). The paper "The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra" develops $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$22-pairs, $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$23-pairs, and $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$24-pairs, together with degree-based solvability theory, thereby providing a symmetric-cone analogue of the horizontal framework from which EHLCP arises (Chi et al., 2017).

That paper’s main solvability theorem states that if $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$25 is an $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$26-pair and $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$27, then for every $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$28, the problem $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$29 has a nonempty compact solution set (Chi et al., 2017). In $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$30, it also proves the equivalence between the $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$31-pair property, unique solvability of $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$32 for every $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$33, and unique solvability of $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$34 for every $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$35 (Chi et al., 2017). 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 $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$36 over the extended second order cone $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$37, converts it into a mixed complementarity problem on $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$38, and then reformulates that problem through a Fischer–Burmeister equation system (Németh et al., 2017). Its main contribution is the chain

$d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$39

which supports Newton and Levenberg–Marquardt algorithms (Németh et al., 2017). 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 $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$40 as the boundedness condition and $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$41 as the stronger property yielding existence and, under additional structure, uniqueness (Yadav et al., 2023). The 2025 convexity paper identifies cS-$d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$42 as the property ensuring convexity of the EHLCP solution set and uniqueness for $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$43 when $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$44 is an $d=(d_1,\dots,d_{k-1})\in(\mathbb{R}^n_{++})^{k-1}$45-matrix (Yadav et al., 29 Apr 2025). The 2025 iterative paper turns the chained complementarity relations into a one-variable piecewise-linear equation, enabling fixed-point algorithms and global error bounds (Wu et al., 17 Sep 2025). 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.