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This paper proposes a novel symplectic pseudospectral iteration framework for the nonlinear optimal control problem (OCP) of a pantograph delay system with inequality constraints. Traditional numerical algorithms for OCPs with pantograph delay systems have predominantly focused on unconstrained scenarios, neglecting constrained problems. Therefore, we propose a novel symplectic pseudospectral iteration framework for constrained problems. First, we employ the successive convexification (SCvx) method to transform the original problem into a series of linear quadratic (LQ) problems, addressing nonlinearity. Then, leveraging time-scale transformations and parametric variational principles, we derive the first-order necessary conditions (FONCs) for these convexified problems, including a Hamiltonian two-point boundary value problem (HTBVP) with stretching terms and a linear complementarity problem. To handle pantograph time delays effectively, we employ a local Legendre-Gauss-Lobatto (LGL) pseudospectral method with proportional grid discretization. Finally, employing a symplectic indirect algorithm, we develop a symplectic pseudospectral method (SPM) for solving the transformed LQ problems, converting them into a system of sparse linear algebraic equations and a linear complementarity problem. Validation of the framework is performed using diverse illustrative examples, demonstrating strong agreement between SCvx and SPM in computational efficiency. Numerical experiments confirm the sparsity of the core matrix and robustness to initial guesses. [ABSTRACT FROM AUTHOR]

In this paper, we propose a smoothing penalty approach for solving the second-order cone complementarity problem (SOCCP). The SOCCP is approximated by a smooth nonlinear equation with penalization parameter. We show that any solution sequence of the approximating equations converges to the solution of the SOCCP under the assumption that the associated function of the SOCCP satisfies a uniform Cartesian-type property. We present a corresponding algorithm for solving the SOCCP based on this smoothing penalty approach, and we demonstrate the efficiency of our method for solving linear, nonlinear and tensor complementarity problems in the second-order cone setting. [ABSTRACT FROM AUTHOR]

The choice of relaxation parameter in the projected successive overrelaxation (PSOR) method for nonnegative quadratic programming problems is problem-dependent. We present novel adaptive PSOR algorithms that adaptively control the relaxation parameter using the Wolfe conditions. The method and its variants can be applied to various problems without requiring additional assumptions, barring the positive semidefiniteness concerning the matrix that defines the objective function, and the cost for updating the parameter is negligible in the whole iteration. Numerical experiments show that the proposed methods often perform comparably to (or sometimes superior to) the PSOR method with a nearly optimal relaxation parameter. [ABSTRACT FROM AUTHOR]

This paper aims at a predictor-corrector interior-point algorithm for solving weighted linear complementarity problem with $ P_{*}(\kappa) $-matrices, which is a variant of weighted complementarity problem and has wide applications in science, engineering, and economics. We first apply the algebraic equivalent transformation technique, and then use the identity function to determine the new search directions. Under suitable conditions, the feasibility and convergence of the algorithm are established. Moreover, we show that the proposed algorithm has polynomial-time complexity. As far as we know, this is the first predictor-corrector interior-point algorithm for $ P_{*}(\kappa) $-weighted linear complementarity problem based on the above-mentioned search directions. Preliminary numerical results demonstrate that our algorithm performs well and efficiently on the test problems. [ABSTRACT FROM AUTHOR]

The infinity norm bounds for the inverse of Nekrasov matrices play an important role in scientific computing. We in this paper propose a triangulation-based approach that can easily be implemented to seek sharper infinity norm bounds for the inverse of Nekrasov matrices. With the help of such sharper bounds, new error estimates for the linear complementarity problem of Nekrasov matrices are presented, and a new infinity norm estimate of the iterative matrix of parallel-in-time methods for an all-at-once system from Volterra partial integral-differential problems is given. Finally, these new bounds are compared with other state-of-the-art results so that the effectiveness of our proposed results is verified. [ABSTRACT FROM AUTHOR]

In this manuscript, we proposed the stability and error analysis for the backward difference operator splitting (BDF-OS) methods to solve the linear complementarity problem (LCP) for pricing the American option under the Black–Scholes framework. The OS schemes have been successfully applied to a variety of Black–Scholes models. It is easy to apply on LCP because the complementarity conditions and the differential equation are segregated and examined separately. We provided an error estimate for these methods and the priori stability estimates for operator splitting strategies based on the BDF1 and BDF2 approaches. We performed numerical experiments and illustrated the order and efficiency of the BDF1 and BDF2 approaches for the test problems to emphasize the convergence behavior of the proposed methods. We have also verified the numerical results with the existing methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel neural network (NN) for solving optimal control problems associated with American options in financial markets. American options provide holders with the flexibility to exercise the option before expiration, thereby affecting potential profitability. This paper focuses on determining the optimal exercise strategy and option price to maximize the payoff by solving a class of American option optimal control problems. We reformulate the optimal control problem into a linear complementarity problem(LCP). Subsequently, we employ the penalty approach and smoothing method to convert the LCP into a bi-nonlinear system with a set of partial differential equations (PDEs). By solving the reformulated PDE equations with the proposed method, we obtain numerical solutions that yield the optimal exercise strategy and option price. Numerical examples of American call and put options demonstrate the efficiency and usefulness of the proposed methods. [ABSTRACT FROM AUTHOR]

The growing importance of the monotone linear complementarity problems (LCP) lies in the different applications it covers, both in mathematics and in practice. In this paper, based on the optimization techniques, we propose a descent logarithmic barrier interior-point method for solving the (LCP). The idea is to transform the LCP into an equivalent convex quadratic optimization problem, denoted by (CQO). Then, the associated barrier problem to CQO is formulated. The existence and the uniqueness of optimal solution of the barrier problem is showed. For its numerical aspects, the descent direction is computed by using the classical Newton's method. However, to determine the displacement step along this direction, guaranteeing the maintenance of the new iterates inside the domain during the algorithm process and the improvement of the value of the objective function, we apply a new approach using approximation functions known as "minorant and majorant approximating functions". The numerical results obtained are very promising and show the effectiveness of this new strategy. [ABSTRACT FROM AUTHOR]

Motivated by the study of column competent and column adequate matrices in the standard linear complementarity problems, we define column competent tensor, column adequate tensor and study the underlying properties of these tensors. We establish equivalent conditions for a column adequate tensor within the class of symmetric tensors and Z tensors. We then introduce the $ {\mathbf{w}} $ w -solution set for the tensor complementarity problem and examine the uniqueness and finiteness of the $ {\mathbf{w}} $ w -solution set. At last, we define column competent property and prove that under some additional condition, column competent property gives the finiteness of the $ \mathbf{w} $ w -solution set of the corresponding tensor complementarity problem. [ABSTRACT FROM AUTHOR]

We prove global convergence of classical projection algorithms for feasibility problems involving union convex sets, which refer to sets expressible as the union of a finite number of closed convex sets. We present a unified strategy for analyzing global convergence by means of studying fixed-point iterations of a set-valued operator that is the union of a finite number of compact-valued upper semicontinuous maps. Such a generalized framework permits the analysis of a class of proximal algorithms for minimizing the sum of a piecewise smooth function and the difference between the pointwise minimum of finitely many weakly convex functions and a piecewise smooth convex function. When realized on two-set feasibility problems, this algorithm class recovers alternating projections and averaged projections as special cases, and thus we obtain global convergence criterion for these projection algorithms. Using these general results, we derive sufficient conditions to guarantee global convergence for several projection algorithms for solving the sparse affine feasibility problem and a feasibility reformulation of the linear complementarity problem. Notably, we obtain global convergence of both the alternating and the averaged projection methods to the solution set for linear complementarity problems involving P-matrices. By leveraging the structures of the classes of problems we consider, we also propose acceleration algorithms with guaranteed global convergence. Numerical results further exemplify that the proposed acceleration schemes significantly improve upon their non-accelerated counterparts in efficiency. [ABSTRACT FROM AUTHOR]

Based on a new equivalent reformulation, a scalable modulus-based matrix splitting (SMMS) method is proposed to solve the vertical linear complementarity problem (VLCP). By introducing a relaxation parameter and employing the two-sweep technique, we further enhance the scalability of the method, leading to a framework of the scalable relaxation two-sweep modulus-based matrix splitting (SRTMMS) method. To theoretically demonstrate the acceleration of the convergence provided by the SMMS method, we present a comparison theorem for the case of s = 2 . Furthermore, we establish the convergence of the SRTMMS method for arbitrary s. Preliminary numerical results indicate promising performance of the SRTMMS method. [ABSTRACT FROM AUTHOR]

In this paper we investigate a class of stochastic absolute value equations (SAVE). After establishing the relationship between the stochastic linear complementarity problem and SAVE, we study the expected residual minimization (ERM) formulation for SAVE and its Monte Carlo sample average approximation. In particular, we show that the ERM problem and its sample average approximation have optimal solutions under the condition of R 0 pair, and the optimal value of the sample average approximation has uniform exponential convergence. Furthermore, we prove that the solutions to the ERM problem are robust for SAVE. For a class of SAVE problems, we use its special structure to construct a smooth residual and further study the convergence of the stationary points. Finally, a smoothing gradient method is proposed by simultaneously considering sample sampling and smooth techniques for solving SAVE. Numerical experiments exhibit the effectiveness of the method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we examine almost (strictly) semimonotone matrices which are also either Z -matrices or matrices having Property (+ +) and show that both parts of Wendler's conjecture [22] (2019) hold for these matrices. Another important result of this paper is the characterization of copositive matrices beyond the matrices with non-positive off-diagonal elements. We discuss some interesting properties related to the structure of almost (strictly) semimonotone matrices and also present results pertaining to the existence and multiplicity of solutions to the linear complementarity problem associated with an almost strictly semimonotone matrix. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A two-step simplified modulus-based matrix splitting iteration method is presented for solving the linear complementarity problem. According to general matrix splitting and special matrix splitting, a general convergence analysis and a specific convergence analysis are described, respectively. Numerical experiments show that the iteration method is effective and that the convergence theories are valid. [ABSTRACT FROM AUTHOR]

Robotic manipulation and locomotion often entail nearly-simultaneous collisions—such as heel and toe strikes during a foot step—with outcomes that are extremely sensitive to the order in which impacts occur. Robotic simulators and state estimation commonly lack the fidelity and accuracy to predict this ordering, and instead pick one with a heuristic. This discrepancy degrades performance when model-based controllers and policies learned in simulation are placed on a real robot. We reconcile this issue with a set-valued rigid-body model which generates a broad set of outcomes to simultaneous frictional impacts with any impact ordering. We first extend Routh's impact model to multiple impacts by reformulating it as a differential inclusion (DI), and show that any solution will resolve all impacts in finite time. By considering time as a state, we embed this model into another DI which captures the continuous-time evolution of rigid-body dynamics, and guarantee existence of solutions. We finally cast simulation of simultaneous impacts as a linear complementarity problem (LCP), and develop an algorithm for tight approximation of the post-impact velocity set with probabilistic guarantees. We demonstrate our approach on several examples drawn from manipulation and legged locomotion, and compare the predictions to other models of rigid and compliant collisions. [ABSTRACT FROM AUTHOR]

We consider how to solve a class of non-Lipschitz mathematical programs with equilibrium constraints (MPEC) where the objective function involves a non-Lipschitz sparsity-inducing function and other functions are smooth. Solving the non-Lipschitz MPEC is highly challenging since the standard constraint qualifications fail due to the existence of equilibrium constraints and the subdifferential of the objective function is unbounded due to the existence of the non-Lipschitz function. On the one hand, for tackling the non-Lipschitzness of the objective function, we introduce a novel class of locally Lipschitz approximation functions that consolidate and unify a diverse range of existing smoothing techniques for the non-Lipschitz function. On the other hand, we use the Kanzow and Schwartz regularization scheme to approximate the equilibrium constraints since this regularization can preserve certain perpendicular structure as in equilibrium constraints, which can induce better convergence results. Then an approximation method is proposed for solving the non-Lipschitz MPEC and its convergence is established under weak conditions. In contrast with existing results, the proposed method can converge to a better stationary point under weaker qualification conditions. Finally, a computational study on the sparse solutions of linear complementarity problems is presented. The numerical results demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Solution of sparse linear complementarity problem (LCP) has been widely discussed in many applications. In this paper, we consider the ℓ p regularization problem with nonnegative constraint for sparse LCP, and propose algorithms based on the iterative reweighted method to approach a sparse solution of the LCP, and then show the convergence to the stationary point of ℓ p regularization problem. Numerical results on simulated data exhibit an excellent performance of the proposed algorithms on approaching a sparse solution of the LCP. Finally, we apply this method to the frictional and frictionless contact problems. The numerical experiments demonstrate that the contact problems can be efficiently solved by the proposed algorithm. • The ℓ p regularization problem with nonnegative constraint is proposed for sparse LCP. • The iterative reweighted algorithms are proposed with convergence guarantees. • Different merit functions for LCP are discussed in numerical experiments. • Numerical simulations demonstrate the superiority of proposed algorithms. • Application to contact problem exhibits the effectiveness of proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on numerical algorithms to value American options under regime switching. The prices of such options satisfy a set of complementary parabolic problems on an unbounded domain. Based on our previous experience, the pricing model could be truncated into a linear complementarity problem (LCP) over a bounded domain. In addition, we transform the resulting LCP into an equivalent variational problem (VP), and discretize the VP by an Euler‐finite element method. Since the variational matrix in the discretized system is P‐matrix, a primal‐dual active set (PDAS) algorithm is proposed to evaluate the option prices efficiently. As a specialty of PDAS, the optimal exercise boundaries in all regimes are obtained without further computation cost. Finally, numerical simulations are carried out to test the performance of our proposed algorithm and compare it to existing methods. [ABSTRACT FROM AUTHOR]

In this study, we explore the valuation challenge posed by American options subject to regime switching, utilizing a model defined by a complex system of parabolic variational inequalities within an infinite domain. The initial pricing model is transformed into a linear complementarity problem (LCP) in a bounded rectangular domain, achieved through the application of a priori estimations and the introduction of an appropriate artificial boundary condition. To discretize the LCP, we employ a finite difference method (FDM), and address the resulting discretized system using a primal-dual active set (PDAS) strategy. The PDAS approach is particularly advantageous for its ability to concurrently determine the option's price and the optimal exercise boundary. This paper conducts an extensive convergence analysis, evaluating both the truncation error associated with the FDM and the iteration error of the PDAS. Comprehensive numerical simulations are performed to validate the method's accuracy and efficiency, underscoring its significant potential for application in the field of financial mathematics. [ABSTRACT FROM AUTHOR]

It is well known that the standard complementarity problem can be equivalently reformulated as a projected fixed-point equation, and this reformulation plays an important role in the theoretical and algorithmic research of complementarity problems. The vertical linear complementarity problem proposed by Cottle and Dantzig is an important generalization of the standard linear complementarity problem, and recently it has been further generalized to the case of tensors, called the vertical tensor complementarity problem (VTCP). In this paper, we give a projected fixed-point reformulation of the VTCP, and then, we design a fixed-point iteration method for solving the VTCP with a vertical block implicit Z-tensor. When there are non-positive diagonal entries in the representation subtensors of the tensor involved in the VTCP, we can reduce the computational cost of our method by solving a lower dimensional VTCP. Under the assumption that the problem under consideration is feasible, we prove that the designed method converges monotonically to a solution of the problem. The numerical results show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The progressive hedging algorithm (PHA) is an effective solution method for solving monotone stochastic variational inequalities (SVIs). However, this validity is based on the assumption of global maximal monotonicity. In this paper, we propose a localized PHA for solving nonmonotone SVIs and show that its validity is based on the weaker assumption of locally elicitable maximal monotonicity. Furthermore, we prove that such assumption holds when the mapping involved in the SVI is locally elicitable monotone or locally monotone. The local convergence of the proposed algorithm is established, and it is shown that the localized PHA has the rate of linear convergence under some mild assumptions. Some numerical experiments, including a two-stage orange market problem and randomly generated two-stage piecewise stochastic linear complementarity problems, indicate that the proposed algorithm is efficient. Funding: This work was supported by the National Natural Science Foundation of China [Grant 12171271]. [ABSTRACT FROM AUTHOR]

This study introduces a new MATLAB subroutine designed to simplify a rigid-plastic model for analysing the dynamic response of steal beam under blast loading. The numerical framework integrates a Linear Complementarity (LC) model to evaluate the impact of loading on skeletal structures. The study aims to enhance the computational, algorithmic and numerical stability of this robust procedure while assessing the response of steel beams subjected to blast loading conditions. The computational results are then validated using experimental and computational examples. The first example seeks a theoretical solution for a steel beam exposed to a uniformly distributed triangular pulse load of varying magnitudes, presenting and contrasting both numerical and theoretical outcomes. In the second example, far-field blast experimental and computational results are compared with predictions from the LCP model. The findings illustrate a strong alignment between the model and experimental data documented in existing literature. [ABSTRACT FROM AUTHOR]

A new subclass of nonsingular H-matrix named S DD+1 matrices is studied in this paper. The relationships between S DD+1 matrices and other subclasses of nonsingular H-matrices are analyzed by numerical examples. Moreover, the infinity norm bounds of the inverse for S DD+1 matrices are derived in two different methods. As applications, two error bounds of the linear complementarity problems (LCP) for S DD+1 matrices are given. Finally, the effectiveness of corresponding results is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we apply Markov-modulated models to value continuous-installment options of the European style with a partial differential equation approach. Under regime-switching models and the opportunity for continuing or stopping to pay installments, the valuation problem can be formulated as coupled partial differential equations (CPDE) with free boundary features, which in many ways is similar to the free boundary problem for vanilla American options due to the possibility of early exercise. In this paper, to value the continuous-installment options under the proposed model with a numerical approach, we first express the truncated CPDE as a linear complementarity problem (LCP), and then a finite element method is applied to solve the resulting variational inequality. We studied the existence and uniqueness of the solution and analyzed the stability of the proposed method under some appropriate assumptions, then we illustrated the error estimates on the appropriate spaces. We presented some numerical results to examine the rate of convergence and accuracy of the proposed method for the pricing problem under the regime-switching model. [ABSTRACT FROM AUTHOR]

This special issue of the Journal of Optimization Theory & Applications focuses on the work of Professors Cornelis Roos and Florian A. Potra in the field of optimization. The issue includes a biographical sketch of both professors, highlighting their achievements. The papers in the special issue cover various topics such as conic optimization, interior point methods, proximal methods, convex optimization, and stochastic optimal control. The authors of the papers have close connections to Professors Roos and Potra, either as students, visitors, or co-authors. The document is a compilation of research papers on interior-point methods (IPMs) and their applications in optimization and conic programming. The papers cover theoretical developments of IPMs, algorithms for linear complementarity problems (LCPs), weighted LCPs, and sufficient linear complementarity problems. There are also papers on new computational approaches for IPMs, applications of IPMs in different fields such as mechanics and support vector machines, and the theory and application of conic optimization. The document also includes papers on proximal methods for convex and non-convex problems, as well as papers on convexity and convex optimization. The volume concludes with acknowledgments to the authors, reviewers, and editor-in-chief. [Extracted from the article]

In this paper, we consider a kernel-based full-Newton step feasible interior-point method (IPM) for P ∗ (κ) -Weighted Linear Complementarity Problem (WLCP). The specific eligible kernel function is used to define an equivalent form of the central path, the proximity measure, and to obtain search directions. Full-Newton steps are adopted to avoid the line search at each iteration. It is shown that with appropriate choices of the parameters, and a certain condition on the starting point, the iterations always lie in the defined neighborhood of the central path. Assuming strict feasibility of P ∗ (κ) -WLCP, it is shown that the IPM converges to the ε -approximate solution of P ∗ (κ) -WLCP in a polynomial number of iterations. Few numerical results are provided to indicate the computational performance of the algorithm. [ABSTRACT FROM AUTHOR]

This paper investigates various aspects of sufficient matrices, one of the most relevant matrix classes introduced in connection with linear complementarity problems. We summarize the most important theoretical results and properties related to sufficient matrices. Based on these, we propose different construction rules that can be used to generate new matrices that belong to this class. A nonnegative number can be assigned to each sufficient matrix, which is called its handicap and works as a measure of sufficiency. The handicap plays a crucial role in proving convergence and complexity results for interior point algorithms for linear complementarity problems. For a particular sufficient matrix, called Csizmadia's matrix, we give the exact value of the handicap, which is exponential in the size of the matrix. Another important topic that we address is deciding whether a matrix is sufficient. Tseng proved in 2000 that this decision problem is co-NP hard. We investigate three different algorithms for determining the sufficiency of a given matrix: Väliaho's algorithm, a linear programming-based algorithm, and an algorithm that facilitates nonlinear programming reformulations of the definition of sufficiency. We tested the efficiency of these methods on our recently launched benchmark data set that consists of four different sets of matrices. In this paper, we give the description and most important properties of the benchmark set, which can be used in the future to compare the performance of different interior point algorithms for linear complementarity problems. [ABSTRACT FROM AUTHOR]

In this paper, we present a full-Newton step interior-point method for solving monotone Weighted Linear Complementarity Problem. We use the technique of algebraic equivalent transformation (AET) of the nonlinear equation of the system which defines the central path. The AET is based on the square root function which plays an important role in computing the new search directions. The algorithm uses only full-Newton steps at each iteration, and hence, line searches are no longer needed. We prove that the algorithm has a quadratic rate of convergence to the target point on the central path. The obtained iteration bound coincides with the best known iteration bound for these types of problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new long-step interior point method for solving sufficient linear complementarity problems. The new algorithm combines two important approaches from the literature: the main ideas of the long-step interior point algorithm introduced by Ai and Zhang and the algebraic equivalent transformation technique proposed by Darvay. Similar to the method of Ai and Zhang, our algorithm also works in a wide neighborhood of the central path and has the best known iteration complexity of short-step variants. However, due to the properties of the applied transforming function in Darvay's technique, the wide neighborhood definition in the analysis depends on the value of the handicap. We implemented not only the theoretical algorithm but a greedy variant of the new method (working in a neighborhood independent of the handicap) in MATLAB and tested its efficiency on both sufficient and non-sufficient problem instances. In addition to presenting our numerical results, we also make some interesting observations regarding the analysis of Ai–Zhang type methods. [ABSTRACT FROM AUTHOR]

We propose a new predictor–corrector interior-point algorithm for solving Cartesian symmetric cone horizontal linear complementarity problems, which is not based on a usual barrier function. We generalize the predictor–corrector algorithm introduced in Darvay et al. (SIAM J Optim 30:2628–2658, 2020) to horizontal linear complementarity problems on a Cartesian product of symmetric cones. We apply the algebraically equivalent transformation technique proposed by Darvay (Adv Model Optim 5:51–92, 2003), and we use the difference of the identity and the square root function to determine the new search directions. In each iteration, the proposed algorithm performs one predictor and one corrector step. We prove that the predictor–corrector interior-point algorithm has the same complexity bound as the best known interior-point methods for solving these types of problems. Furthermore, we provide a condition related to the proximity and update parameters for which the introduced predictor-corrector algorithm is well defined. [ABSTRACT FROM AUTHOR]

In this work, we introduce two accurate and efficient finite difference methods based on radial basis functions (RBF–FD) for pricing European and American options under liquidity shocks. The problem is formulated as the semi-linear complementarity and PDE systems for American and European options, respectively. In the context of temporal semi-discretization, we provide two backward difference formulas of order one and two (BDF1 & BDF2). Furthermore, we discuss the stability and convergence properties of the proposed methods. For the American option, to solve the semi-linear system of complementarity problems (LCPs), we combine the RBF-FD approaches with an operator splitting (OS) method. To illustrate the efficiency and accuracy of the suggested methods, we provide numerical examples for both European and American call options and verify them with the existing work in the literature. In numerical discussion, we show the Greeks (Delta & Gamma) plots for the American options. [ABSTRACT FROM AUTHOR]

This paper provides the process of incremental constitutive integration for the unified hardening model combined with the transformation stress method. The dimension-extending technique takes the hardening function of the hardening/softening model as the same position as the stress components, so that the constitutive integration of the plasticity can be reduced to an initial value problem of differential–complementarity equations, which is solved using the Gauss–Seidel algorithm-based Projection–Correction for the mixed complementarity problem. The Gauss–Seidel based Projection–Correction algorithm does not require the calculation of the Jacobean matrix of the potential function, making it relatively easy to implement in programming. The unified hardening model is proposed based on the modified Cam–Clay model and the sub-loading surface model, and the elastic properties are pressure-dependent. Two processing methods, backward Euler integration and exact elastic property, are used for the variable elasticity properties. The constitutive integration of the increased dimensional unified hardening model is reduced to a special mixed complementarity problem and solved by the proposed algorithm, which does not need to calculate the Jacobean matrix of the potential function, and greatly simplifies the derivation process. Several numerical examples are given to verify the feasibility of the incremental constitutive integration in the unified hardening model, including the single integral point and the boundary value problems. The research results have expanded the scope of use of the Gauss–Seidel based Projection–Correction algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the tensor complementarity problem (TCP). From the perspective of non-coupled equality constraint minimization problem for the symmetric TCP, we propose a generalized alternating direction method of multipliers (G-ADMM) to solve the general TCP in which the tensor may not be symmetric. The global convergence and the O (1 k) convergence rate of the proposed method are proved under the assumption which is much weaker than the monotone assumption. Numerical results show that the method is efficient for solving the TCP. [ABSTRACT FROM AUTHOR]

The constraints imposed by natural antibody affinity maturation often culminate in antibodies with suboptimal binding affinities, thereby limiting their therapeutic efficacy. As such, the augmentation of antibody binding affinity is pivotal for the advancement of efficacious antibody-based therapies. Classical experimental paradigms for antibody engineering are financially and temporally prohibitive due to the extensive combinatorial space of sequence variations in the complementarity-determining regions (CDRs). The advent of computational techniques presents a more expeditious and economical avenue for the systematic design and optimization of antibodies. In this investigation, we assess the performance of AlphaFold2 coupled with the binder hallucination technique for the computational refinement of antibody sequences to elevate the binding affinity of pre-existing antigen-antibody complexes. These methodologies exhibit the capability to predict protein tertiary structures with remarkable fidelity, even in the absence of empirically derived data. Our results intimate that the proposed approach is adept at designing antibodies with improved affinities for antigen-antibody complexes unrepresented in AlphaFold2's training dataset, underscoring its potential as a robust and scalable strategy for antibody engineering. [ABSTRACT FROM AUTHOR]

Aim: Patterns of benthic biodiversity at the macroecological scale remain poorly characterised throughout the Chilean latitudinal gradient, in part due to the lack of integrated databases, uneven sampling effort, and the use of species richness alone to quantify biodiversity. Different diversity measures, encompassing taxonomic and functional components, may give us extra information on biodiversity relevant to conservation planning and management. Thus, evaluating the spatial complementarity of these measures is essential. Location: Coast and continental shelf of Chile. Methods: The latitudinal gradient of Chile was divided into five ecoregions according to the Marine Ecosystems of the World classification. Using a 55 × 55 km equal area grid, we estimated the incidence coverage‐based estimator (ICE), taxonomic distinctness (Δ+) and three measures of functional diversity: functional richness (FRic), functional evenness (FEve) and functional divergence (FDiv). For each measure, we described spatial patterns, identified hotspots, evaluated hotspot congruence and evaluated complementarity between measures. Results: Diversity patterns varied between ecoregions and over the latitudinal gradient. ICE and Δ+ peaked in the Chiloense and Channels and Fjords ecoregions. Δ+ and FRic present a similar pattern at mid‐latitudes. FEve showed a contrary pattern, principally with FRic. Areas with high numbers of hotspots differed spatially according to each metric, and three latitudinal bands were observed. ICE, Δ+ and FRic were positively correlated, but the hotspot overlap at the grid cell level was more limited. Main Conclusions: The complementarity between taxonomic and functional diversity measures is limited when we observe the overlap between grid cells representing hotspots. However, some regions are consistently identified as highly diverse, with the Magellanic Province (Chiloense and Channels and Fjords ecoregions) being the most important for the richness, taxonomic and functional diversity of benthos. Confirmation of the importance of this region can help prioritise conservation efforts. [ABSTRACT FROM AUTHOR]

We aim to develop efficient and robust algorithms for nonsmooth nonlinear systems arising from complementarity problems. The semismooth Newton algorithm is popular due to its reliability and efficiency. However, it struggles with issues with imbalanced nonlinearities of the problems, leading to degraded convergence rates or failure despite help from the globalization techniques like linesearch or trust region. We introduce a right nonlinearly preconditioned semismooth Newton algorithm to address this difficulty. The critical success ingredient is that before each global Newton update, a nonlinear preconditioning step implicitly removes the so-called 'bad components' causing trouble via nonlinear subspace correction, inspired by Gaussian elimination but adapted nonlinearly to balance system nonlinearities. Additionally, our method integrates with a domain decomposition framework, enhancing parallelism. Numerical results on two classes of problems demonstrate significantly improved convergence over standard semismooth Newton methods. [ABSTRACT FROM AUTHOR]

We first propose a new error bound for the linear complementarity problems when the involved matrices are generalized Nekrasov matrices, which generalizes the recent result obtained by Li et al. (Numer. Algorithms 74:997–1009, 2017). Then we present two new error bounds for the linear complementarity problems when the involved matrices are Nekrasov matrices. Numerical examples are given to illustrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

An efficient numerical method is propoesd for a parabolic linear complementarity problem (LCP) arising in the valuation of American options on zero-coupon bonds under the Cox–Ingersoll–Ross (CIR) model. With variable substitutions, we first transform the original pricing problem into a degenerated linear complementarity problem on a bounded domain, and present a corresponding variational inequality (VI). We then give the full discretization scheme of VI constructed by finite element and finite difference methods in spatial and temporal directions, respectively. Within the framework of VI, the stability and the rate of convergence are obtained. Moreover, for the resulted discretised variational inequality, we present a primal-dual active set (PDAS) method to solve it. Numerical results are carried out to test the usefulness of the proposed method compared with existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the resolution of the simplicial cone constrained convex quadratic optimization (abbreviated SCQO). It is known that the optimality conditions of SCQO is only a standard linear complementarity problem (LCP). Under a suitable condition, the solution of LCP is equivalent to find the solution of an absolute value equations AVE. For its numerical solution, we propose an efficient two-steps fixed point iterative method for solving the AVE. Moreover, we show that this method converges globally linear to the unique solution of the AVE and which is in turn an optimal solution of SCQO. Some numerical results are reported to demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, for vertical nonlinear complementarity problems, a two-step modulus-based matrix splitting iteration method is established by applying the two-step splitting technique to the modulus-based matrix splitting iteration method. The convergence theorems of the proposed method are given when the number of system matrices is larger than 2. Numerical results show that the convergence rate of the proposed method can be accelerated compared to the existing modulus-based matrix splitting iteration method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some constants with the tensors of special structures and present their some useful properties. Furthermore, some perturbation bounds of the tensor complementarity problem are obtained on the base of these constants. [ABSTRACT FROM AUTHOR]

Recently, Gowda and Sossa [Math. Program., 2019, 177: 149–171] studied the existence of solutions to weakly homogeneous variational inequalities. In particular, their main result, based on a degree-theoretic condition and a constraint on the corresponding cone complementarity problem, covers a majority of existence results on the subcategory problems of weakly homogeneous variational inequalities. In this paper, what we achieve is a new copositivity-type existence result for the weakly homogeneous variational inequality. The conditions we used are easier to check than the degree-theoretic condition and our result crosses each other with the main result established by Gowda and Sossa and the main result given by Ma, Zheng and Huang [SIAM J. Optim., 2020, 30(1): 132–148], respectively. Besides, we show the distinctiveness of our existence result by comparing it with the well-known coercivity result obtained for variational inequalities and a norm-coercivity result obtained for complementarity problems, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we give some existence theorems of solutions to Γ -robust counterparts of gap function formulations of uncertain linear complementarity problems, in which Γ plays a role in adjusting the robustness of the model against the level of conservatism of solutions. If the Γ -robust uncertainty set is nonconvex, it is hard to prove the existence of solutions to the corresponding robust counterpart. Using techniques of asymptotic functions, we establish existence theorems of solutions to the corresponding robust counterpart. For the case of nonconvex Γ -robust ellipsoidal uncertainty sets, these existence results are not proved in the paper [Krebs et al., Int. Trans. Oper. Res. 29 (2022), pp. 417–441]; for the case of convex Γ -robust ellipsoidal uncertainty sets, our existence theorems are obtained under the conditions which are much weaker than those in Krebs' paper. Finally, a case study for the uncertain traffic equilibrium problem is considered to illustrate the effects of nonconvex uncertainty sets on the level of conservatism of robust solutions. [ABSTRACT FROM AUTHOR]

In the present paper, the strict feasibility of the polynomial complementarity problem (PCP) is investigated. To this end, as a generalization of the concept of S-tensor, a concept of S-tensor tuple is introduced. Some properties of S-tensor tuples are investigated. In particular, several conditions are proposed to judge whether a tensor tuple is an S-tensor tuple or not. Then, based on the S-tensor tuple, the strict feasibility of PCP is investigated. [ABSTRACT FROM AUTHOR]

Tuft disentanglement involves the principal actions of opening and carding, which are essential processing in textile technology. In this article, tuft disentanglement is modeled as fiber withdrawal considering the fiber interactions within the tuft. Based on the reasonable tuft model, the kinematic constraints were constructed in the disentanglement considering the carding essence and fiber interactions. Combined with the dynamic equation, the state of the fibers can be solved as a linear complementarity problem due to the complementarity in the fiber interactions. The withdrawal forces verified the simulations, which capture the qualitative features of experiments. Also, the effect of the withdrawal parameters on disentanglement was studied. The simulation results showed that the withdrawal forces increased with increasing withdrawal velocities and decreasing gauge lengths. The simulation results are consistent with the experiments. The proposed computational framework can be extended for predicting the mechanical behaviors of other random fibrous materials involving fiber interactions. [ABSTRACT FROM AUTHOR]

This paper provides an overview of the necessary and sufficient conditions for guaranteeing the unique solvability of absolute value equations. In addition to discussing the basic form of these equations, we also address several generalizations, including generalized absolute value equations and matrix absolute value equations. Our survey encompasses known results as well as novel characterizations proposed in this study. [ABSTRACT FROM AUTHOR]