Sequential Linear Complementarity (SLCP) Method

• The SLCP method is an advanced framework that traces piecewise-affine solution paths for LCPs with parameters varying in an affine manner.
• It employs a tracing algorithm that updates the solution basis at breakpoints, ensuring robust convergence with complexity typically around O(h^3) per interval.
• Extensions of SLCP include stochastic and multi-agent models, where techniques like progressive hedging facilitate efficient, parallel decomposition for equilibrium problems.

The Sequential Linear Complementarity Problem (SLCP) method is an advanced framework for analyzing and solving linear complementarity problems (LCPs) and their parametric, stochastic, and multi-agent generalizations. SLCP encompasses a suite of discrete and continuous methods for cases where problem data vary sequentially—often affinely—according to a parameter, random variable, or scenario structure. The approach exploits piecewise-affine and finite-dimensional structure, enabling efficient tracing of solution paths, rigorous convergence, and robust application to models in optimization, game theory, and stochastic programming.

1. Formalization and Piecewise-Affine Solution Structure

The SLCP arises when the input data of an LCP depend affinely on a scalar parameter $\lambda\in[\alpha,\beta]$, i.e., $M(\lambda)=M_0+\lambda M_1$ and $q(\lambda)=q_0+\lambda q_1$, with $h$-dimensional constraint and variable spaces. The SLCP seeks a function $z:[\alpha,\beta]\to\mathbb{R}^h$ satisfying $\forall\lambda\in[\alpha,\beta]$:

$$0\leq z(\lambda)\perp M(\lambda)z(\lambda)+q(\lambda)\geq 0,$$

subject to matrix sufficiency (all principal minors of $G(\lambda)=[I\ |\ -M(\lambda)]$ do not change sign) and feasibility for all $\lambda$.

Under these hypotheses, the solution $z(\lambda)$ is piecewise-affine in $\lambda$ within intervals defined by breakpoints where some basic variable becomes zero. For fixed complementary basis $B$ (one of each pair $\{i, i+h\}$ per variable), the solution in each region,

$y_B(\lambda) = G_B(\lambda)^{-1} q(\lambda),$

is affine, as $G_B(\lambda)$ is nonsingular and its determinant's sign is preserved. Breakpoints arise when $y_{B,i}(\lambda)=0$, which are simple roots (solving linear equations). Sorting these roots partitions $[\alpha,\beta]$ into intervals where a particular basis is valid (Adelgren, 2022).

2. Tracing Algorithm and Computational Complexity

The SLCP_Trace algorithm advances from $\lambda=\alpha$ to $\lambda=\beta$, maintaining current basis $B$, interval $[\lambda_L,\lambda_R]$, and the affine representation:

1. Initialization: Solve LCP for $\lambda=\alpha$, recover basis $B^0$, and compute affine formula $y_{B^0}(\lambda)$.
2. Breakpoint Computation: Find all simple roots of $y_{B^0,i}(\lambda) = 0$ in $(\alpha,\beta]$.
3. Tracing: For each root in sequence, update partition, swap basis index with its complement at a zero crossing, and recompute $y_{B}(\lambda)$.
4. Termination: Proceed until $\lambda=\beta$.

Each interval requires $O(h^3)$ for symbolic inversion; total basis regions $K$ are at most $O(h^2)$. Practical implementations (Python, criss-cross initialization, symbolic algebra) demonstrate tractability up to $h\sim175$, with intervals and CPU time growing $O(h^2)$–$O(h^3)$ overall, especially for denser problem classes (e.g., biobjective QP reduction) (Adelgren, 2022).

3. Stochastic and Distributionally Robust Extensions

SLCP methodology generalizes to two-stage stochastic models where data are random ($\xi$ in $\Xi$):

• First-stage: $0\le x\perp Ax+\mathbb{E}[B(\xi)y(\xi)]+q_1\ge 0$.
• Second-stage: $$0\le y(\xi)\perp M(\xi)y(\xi)+N(\xi)x+q_2(\xi)\ge 0$$ for a.e. $\xi$.

Discretization partitions $\Xi$ into $K$ cells; conditional averages yield finite LCPs per scenario. Scenario-wise solutions are combined for first-stage complementarity, yielding unique convergence to the true solution as $K\to\infty$, with explicit $O(\max_i\operatorname{diam}\Xi_i^K)$ error bounds under Lipschitz continuity and strong monotonicity (Chen et al., 2017).

For ambiguity in $\xi$'s distribution (distributionally robust LCP), first-stage conditions are replaced by supremum expectations over sets of plausible measures specified by moment constraints. Dualization and discretization transform this into a finite deterministic LCP with explicit moment enforcement via auxiliary variables.

4. SLCP in Multi-Agent Game and Generalized Nash Equilibrium Problems (GNEPs)

SLCP methods have been extended to solve GNEPs, wherein $N$ agents each optimize subject to convex (possibly coupled) constraints:

$$\min_{x^\nu} \theta^\nu(x^\nu,x^{-\nu}),\quad \text{s.t. } g_i^\nu(x^\nu,x^{-\nu})\le 0.$$

By concatenating KKT conditions across agents, the equilibrium conditions form a mixed complementarity system:

$F(x,\lambda)=0,\quad 0\le\lambda\perp -G(x)\ge 0.$

At each iterate, a Newton-type linearization yields a mixed LC subproblem, which is reduced to a standard LCP in the multipliers. The sequential method iterates by solving these LCPs, updating primal-dual directions, and employing a merit function:

$$\Phi_\rho(x,\lambda)=(-\lambda^\top G(x))_+ + \frac{\rho}{2}\|F(x,\lambda)\|^2 + \sum_{\nu,i}(g_i^\nu(x))_+,$$

to guarantee descent and globalization. Under uniform monotonicity and Slater-type feasibility, global convergence to a GNE is established. Local superlinear (quadratic) rates are shown under standard second-order sufficient conditions and regularity (Diao et al., 22 Jan 2026).

5. Scenario Decomposition and Progressive Hedging

For discretized two-stage SLCPs, scenario structure enables parallelization and operator splitting. The Progressive Hedging Method (PHM) decomposes into scenario-specific LCPs with added proximal terms and consensus multipliers.

• Step 1: Each scenario $i$ solves its subproblem with current iterates plus consensus and penalty terms.
• Step 2: The common variable is updated via weighted averages.
• Step 3: Scenario iterates are replaced with consensus; multipliers are updated.

PHM converges robustly under monotonicity conditions and is compatible with discretized stochastic SLCPs and distributionally robust formulations. For two-stage duopoly and robust equilibrium models, the approach produces global solution bounds and recovers theoretical optima precisely as discretization refines (Chen et al., 2017).

6. Numerical Experiments and Practical Performance

Benchmarking SLCP methods on uni-parametric LCPs, biobjective QPs, stochastic two-stage, and GNEPs demonstrates:

• Reliable partitioning of parameter space into affine regions, with tractable scaling for moderate dimensions ($h\le175$).
• Superior performance of SLCP algorithms over interior-point, semismooth Newton, and augmented Lagrangian methods for high-accuracy GNEP solutions; fastest method in $\sim85\%$ of tests.
• Notable reduction in Hessian/Jacobian evaluations due to piecewise-affine formula computation per basis/region.
• Practical convergence rates match theoretical predictions in scenarios such as Cournot-duopoly and supply–demand equilibrium models (Adelgren, 2022, Diao et al., 22 Jan 2026, Chen et al., 2017).

7. Modeling Pipeline and Applications

SLCP methodology supports a modeling-to-solution pipeline:

• Complex equilibrium models (stochastic or multi-agent).
• Reduction to uni-parametric or scenario-based LCP.
• Application of tracing, decomposition or progressive hedging algorithms.
• Quantitative error bounds and rigorous convergence theory.

Applications include weighted scalarizations of multiobjective QP, stochastic or robust game equilibria, supply–demand markets, and internet-switching models, with extension to ambiguity sets via moment constraints.

The method's tractability, generality, and rigorous convergence properties establish SLCP as a cornerstone for parametric, stochastic, and multi-agent linear complementarity analysis.