Complementarity Problem Research Articles

Equilibria of large random Lotka-Volterra systems with vanishing species: a mathematical approach.

Ecosystems with a large number of species are often modelled as Lotka-Volterra dynamical systems built around a large interaction matrix with random part. Under some known conditions, a global equilibrium exists and is unique. In this article, we rigorously study its statistical properties in the large dimensional regime. Such an equilibrium vector is known to be the solution of a so-called Linear Complementarity Problem. We describe its statistical properties by designing an Approximate Message Passing (AMP) algorithm, a technique that has recently aroused an intense research effort in the fields of statistical physics, machine learning, or communication theory. Interaction matrices based on the Gaussian Orthogonal Ensemble, or following a Wishart distribution are considered. Beyond these models, the AMP approach developed in this article has the potential to describe the statistical properties of equilibria associated to more involved interaction matrix models.

A note on the numerical approximation of Greeks for American-style options

In this note, we consider the approximation of the Greeks Delta and Gamma of American-style options through the numerical solution of time-dependent partial differential complementarity problems (PDCPs). This approach is very attractive as it can yield accurate approximations to these Greeks at essentially no additional computational cost during the numerical solution of the PDCP for the pertinent option value function. For the temporal discretization, the Crank–Nicolson method is arguably the most popular method in computational finance. It is well-known, however, that this method can have an undesirable convergence behaviour in the approximation of the Greeks Delta and Gamma for American-style options, even when backward Euler damping (Rannacher smoothing) is employed.In this note, for the temporal discretization of the PDCP, we study an interesting family of diagonally implicit Runge–Kutta (DIRK) methods together with the two-stage Lobatto IIIC method. Through ample numerical experiments for one- and two-asset American-style options, it is shown that these methods can yield a regular second-order convergence behaviour for the option value as well as for the Greeks Delta and Gamma. A mutual comparison reveals that the DIRK method with suitably chosen parameter θ is preferable.

Improved modulus-based matrix splitting iteration method for horizontal implicit complementarity problems

Improved modulus-based matrix splitting iteration method for horizontal implicit complementarity problems

An interior-point derivative-free algorithm for nonlinear complementarity problems

An interior-point derivative-free algorithm for nonlinear complementarity problems

More on Scarf's Complementarity Problem and Its Error Bounds

More on Scarf's Complementarity Problem and Its Error Bounds

The weighted vertical linear complementarity problem on Euclidean Jordan algebra

The weighted vertical linear complementarity problem on Euclidean Jordan algebra

A recursive algorithm for dynamics of planar multibody systems with frictional unilateral constraints

The contact impact problem in planar multibody systems can be efficiently solved by formulating it as the linear complementarity problem, which requires a complex modeling process. To simplify the process, a recursive algorithm for the dynamics of planar multibody systems with frictional unilateral constraints is proposed based on the reduced multibody system transfer matrix method. Firstly, the contact forces of frictional unilateral constraints are integrated into the recurrence relations of system components using Lagrange multipliers. Subsequently, the relative motion equations of the contact positions are discretized in time, which are then utilized to describe the linear complementarity problem for systems. The dynamics of the system is solved by the Moreau time-stepping method with the recursive method. Finally, the proposed algorithm was validated using the woodpecker toy and used to model a slider-crank mechanism with clearance, which shows its characteristics of facilitating modeling, universal, and highly programmable. This recursive algorithm provides an effective tool for solving non-smooth planar multibody systems while extending the application of the multibody system transfer matrix method.

Harnessing the Power of Complementarity Between Smart Tracking Technology and Associated Health Information Technologies: Longitudinal Study.

Smart tracking technology (STT) that was applied for clinical use has the potential to reduce 30-day all-cause readmission risk through streamlining clinical workflows with improved accuracy, mobility, and efficiency. However, previously published literature has inadequately addressed the joint effects of STT for clinical use and its complementary health ITs (HITs) in this context. Furthermore, while previous studies have discussed the symbiotic and pooled complementarity effects among different HITs, there is a lack of evidence-based research specifically examining the complementarity effects between STT for clinical use and other relevant HITs. Through a complementarity theory lens, this study aims to examine the joint effects of STT for clinical use and 3 relevant HITs on 30-day all-cause readmission risk. These HITs are STT for supply chain management, mobile IT, and health information exchange (HIE). Specifically, this study examines whether the pooled complementarity effect exists between STT for clinical use and STT for supply chain management, and whether symbiotic complementarity effects exist between STT for clinical use and mobile IT and between STT for clinical use and HIE. This study uses a longitudinal in-patient dataset, including 879,122 in-patient hospital admissions for 347,949 patients in 61 hospitals located in Florida and New York in the United States, from 2014 to 2015. Logistic regression was applied to assess the effect of HITs on readmission risks. Time and hospital fixed effects were controlled in the regression model. Robust standard errors (SEs) were used to account for potential heteroskedasticity. These errors were further clustered at the patient level to consider possible correlations within the patient groups. The interaction between STT for clinical use and STT for supply chain management, mobile IT, and HIE was negatively associated with 30-day readmission risk, with coefficients of -0.0352 (P=.003), -0.0520 (P<.001), and -0.0216 (P=.04), respectively. These results indicate that the pooled complementarity effect exists between STT for clinical use and STT for supply chain management, and symbiotic complementarity effects exist between STT for clinical use and mobile IT and between STT for clinical use and HIE. Furthermore, the joint effects of these HITs varied depending on the hospital affiliation and patients' disease types. Our results reveal that while individual HIT implementations have varying impacts on 30-day readmission risk, their joint effects are often associated with a reduction in 30-day readmission risk. This study substantially contributes to HIT value literature by quantifying the complementarity effects among 4 different types of HITs: STT for clinical use, STT for supply chain management, mobile IT, and HIE. It further offers practical implications for hospitals to maximize the benefits of their complementary HITs in reducing the 30-day readmission risk in their respective care scenarios.

An RBF Method for Time Fractional Jump-Diffusion Option Pricing Model under Temporal Graded Meshes

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described by a linear complementarity problem. For European option, we present nonuniform discretization along time and the radial basis function (RBF) method for spatial discretization. The stability and convergence analysis of the discrete scheme are carried out in the case of European options. For American option, the operator splitting method is adopted which split linear complementary problem into two simple equations. The numerical results confirm the accuracy of the proposed method.

A multi-commodity partial equilibrium model of imperfect competition in future global hydrogen markets

Techno-economic studies are investigating procurement costs of hydrogen and related derivatives across various international trade routes. However, the strategic behavior of exporters is rarely considered in this context, despite similar behavior frequently observed in the fossil fuel world and market characteristics indicating some potential. This work introduces a novel techno-economic model of oligopolistic trade tailored around the value chain of potential future global hydrogen markets. It is formulated as a mixed complementarity problem with a convex reformulation, allowing to solve both competitive and oligopolistic equilibria much beyond the capabilities of traditional complementarity models. Illustrative results show that assuming perfectly competitive price formation in early phases of future global hydrogen markets may lead to overly optimistic hydrogen price projections. This is especially the case within Europe, Japan, and South Korea. In contrast, strategically withholding quantities in export may decrease prices for some domestic markets.

An interior penalty method for a parabolic complementarity problem involving a fractional Black-Scholes operator

In this paper, an interior penalty method is proposed to solve a parabolic complementarity problem involving fractional Black–Scholes operator arising in pricing American options under a geometric Lévy process. The complementarity problem is first reformulated as a fractional partial differential variational inequality problem using the representations of fractional order operators and appropriate mathematical techniques. A penalty equation is then proposed to approximate the variational inequality problem by introducing a novel interior-point based penalty term. The existence and uniqueness of the solution to the penalized problem are proved, and an upper bound on the distance between the solutions to the penalty equation and the variational inequality problem is established. To test our method, we discretize the penalty equation by a finite difference method in space and the Crank–Nicolson method in time. We then present numerical experimental results to demonstrate the usefulness and effectiveness for the interior penalty method.

Globalization of convergence of the constrained piecewise Levenberg–Marquardt method

We develop linesearch algorithms intended for globalization of convergence of the piecewise Levenberg–Marquardt method for constrained piecewise smooth equation. Conditions ensuring global convergence properties and asymptotic superlinear convergence rate are proposed. The peculiarities of the global convergence results in the piecewise smooth case are discussed and illustrated by examples. We also provide numerical results for unconstrained and constrained reformulations of nonlinear complementarity problems, comparing the performance of the globalized piecewise Levenberg–Marquardt algorithm with some relevant alternatives.

Improvement of Convergence of One- and Two-Step MSM Iteration Methods for Nondifferentiable Nonlinear Complementarity Problems

Improvement of Convergence of One- and Two-Step MSM Iteration Methods for Nondifferentiable Nonlinear Complementarity Problems

Smoothing penalty approach for solving second-order cone complementarity problems

Smoothing penalty approach for solving second-order cone complementarity problems

Cross‐border cooperation and tax avoidance of U.S. cross‐listed firms

AbstractWe investigate how cross‐border cooperation formed by the Multilateral Memorandum of Understanding (MMoU) affects tax avoidance of foreign firms cross‐listed in the U.S. Prior studies show that the MMoU enhances U.S. SEC's regulatory oversight of foreign firms and increases such firms' corporate governance. We utilize data from 27 jurisdictions from 2000 to 2009 and find that U.S. cross‐listed firms mitigate tax avoidance more than do their home‐listed peers after their home jurisdictions become signatories of the MMoU. Our further analyses reveal that improved information environment and enhanced foreign institutional investor monitoring are two possible underlying mechanisms, consistent with the complementarity theory of tax avoidance that self‐serving managers can employ tax avoidance to divert assets when corporate governance is ineffective. We also show that the effect of the MMoU increases with U.S. IRS monitoring and the strength of institutional regimes in foreign firms' home jurisdictions. Collectively, our findings elucidate unintended consequences of the MMoU and provide implications for practitioners and regulators.

Two-Step Simplified Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

Two-Step Simplified Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

Superlinear convergence of an interior point algorithm on linear semi-definite feasibility problems

In the literature, besides the assumption of strict complementarity, superlinear convergence of implementable polynomial-time interior point algorithms using known search directions, namely, the HKM direction, its dual or the NT direction, to solve semi-definite programs (SDPs) is shown by (i) assuming that the given SDP is nondegenerate and making modifications to these algorithms [Kojima et al. Local convergence of predictor–corrector infeasible-interior-point algorithms for SDPs and SDLCPs, Math. Program. Ser. A 80 (1998), pp. 129–160], or (ii) considering special classes of SDPs, such as the class of linear semi-definite feasibility problems (LSDFPs) and requiring the initial iterate to the algorithm to satisfy certain conditions [Sim, Superlinear convergence of an infeasible predictor–corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg–Kojima–Monteiro direction, SIAM J. Optim. 21 (2011), pp. 102–126], [Sim, Interior point method on semi-definite linear complementarity problems using the Nesterov-Todd (NT) search direction: Polynomial complexity and local convergence, Comput. Optim. Appl. 74 (2019), pp. 583–621]. Otherwise, these algorithms are not easy to implement even though they are shown to have polynomial iteration complexities and superlinear convergence [Luo et al. Superlinear convergence of a symmetric primal–dual path following algorithm for semidefinite programming, SIAM J. Optim. 8 (1998), pp. 59–81]. The conditions in [Sim, Superlinear convergence of an infeasible predictor–corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg–Kojima–Monteiro direction, SIAM J. Optim. 21 (2011), pp. 102–126], [Sim, Interior point method on semi-definite linear complementarity problems using the Nesterov-Todd (NT) search direction: Polynomial complexity and local convergence, Comput. Optim. Appl. 74 (2019), pp. 583–621] that the initial iterate to the algorithm is required to satisfy to have superlinear convergence when solving LSDFPs however are not practical. In this paper, we propose a practical initial iterate to an implementable infeasible interior point algorithm that guarantees superlinear convergence when the algorithm is used to solve the homogeneous feasibility model of an LSDFP.

A Neural Network Approach for Solving Weighted Nonlinear Complementarity Problems

In this paper, we consider the weighted nonlinear complementarity problem which is the extension of the general complementarity problems and contains a wide class of optimization problems. Based on the weighted complementarity function, we reformulate the weighted nonlinear complementarity problem as an unconstrained minimization problem and present the steepest descent-based neural network to solve it. Under mild conditions, we derive some results related to the asymptotic stability of the neural network. Numerical experiments indicate that the proposed algorithm is quite effective.

On the modulus-based methods without auxiliary variable for vertical linear complementarity problems

On the modulus-based methods without auxiliary variable for vertical linear complementarity problems

Large-step algorithm for linear complementarity problem with new search direction

Recently, [Darvay Zs, Takács PR: Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood. Cent Eur J Oper Res. 2018;26(3):551–563] introduced a new large-step interior-point algorithm (IPA) for linear optimization (LO). The purpose of this paper is to extend this method to the class of P ∗ ( κ ) -linear complementarity problems (LCPs). First, we propose a new wide neighbourhood of the central path for LCPs, which differs from the one presented by [Ai W, Zhang S: An O ( n L ) iteration primal–dual path-following method, based on wide neighbourhoods and large updates, for monotone LCP. SIAM J Optim. 2005;16(2):400–417]. Our method uses new search directions obtained by an algebraically equivalent transformation of the system which defines the central path. We present a large-step algorithm, but we demonstrate that it has the same iteration complexity as the best short-step algorithms. Finally, we give some numerical results to prove the efficiency of the new method.