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In this paper, a general algorithm is presented for the integration of multisurface plasticity models. The algorithm combines the return mapping with the technique of the sequential linear complementarity problem (SLCP) to update stress. The set of active constraints is naturally identified by the classical Lemke’s algorithm. With the assumptions of isotropic linear elasticity, perfect plasticity, and the associated flow rule, all details are provided in matrix notations to facilitate computer implementation. The extension to hardening/softening multisurface plasticity models is also presented. The application of the algorithm is demonstrated via simulation of three types of geotechnical problems in 2D and 3D. Both linear and nonlinear multisurface plasticity models, e.g. Mohr-Coulomb and generalized Hoek-Brown yield criteria, are examined within the framework of the proposed algorithm. The numerical results are in good agreement with the analytic solutions. Moreover, the accuracy of the proposed stress integration procedure is investigated through 3D iso-error map. The convergence using the consistent tangent matrix at the global level is examined by a one-increment example consisting of one element.

For the large sparse linear complementarity problem, by reformulating it as implicit fixed-point equations, we establish a modulus-based iteration method with the aid of an inexact non-alternating ...

In this paper, we study the absolute value equation (AVE) $$Ax-b=|x|$$ . One effective approach to handle AVE is by using concave minimization methods. We propose a new method based on concave minimization methods. We establish its finite convergence under mild conditions. We also study some classes of AVEs which are polynomial time solvable.

In this paper, we consider numerical methods for the linear complementarity problem (LCP). By introducing a positive diagonal parameter matrix, the LCP is transformed into an equivalent fixed-point equation and the equivalence is proved. Based on such equation, the general fixed-point (GFP) method with two cases are proposed and analyzed when the system matrix is a $ P $-matrix. In addition, we provide several concrete sufficient conditions for the proposed method when the system matrix is a symmetric positive definite matrix or an $ H_{+} $-matrix. Meanwhile, we discuss the optimal case for the proposed method. The numerical experiments show that the GFP method is effective and practical.

In this paper, to economically and fast solve the linear complementarity problem, based on a new equivalent fixed-point form of the linear complementarity problem, we establish a class of new modulus-based matrix splitting methods, which is different from the previously published works. Some sufficient conditions to guarantee the convergence of this new iteration method are presented. Numerical examples are offered to show the efficacy of this new iteration method. Moreover, the comparisons on numerical results show the computational efficiency of this new iteration method advantages over the corresponding modulus method, the modified modulus method and the modulus-based Gauss-Seidel method.

In this paper, we propose a primal–dual interior point method for Linear Complementarity Problem (LCP) based on a new parameterized kernel function. The investigation according to it yields the bes...

As an extension of the complementarity problem (CP), the weighted complementarity problem (wCP) is a large class of equilibrium problems with wide applications in science, economics, and engineering. If the weight vector is zero, the wCP reduces to a CP. In this paper, we present a full-Newton step infeasible interior-point method (IIPM) for the special weighted linear complementarity problem over the nonnegative orthant. One iteration of the algorithm consists of one feasibility step followed by a few centering steps. All of them are full-Newton steps, and hence, no calculation of the step size is necessary. The iteration bound of the algorithm is as good as the best-known polynomial complexity of IIPMs for linear optimization.

Retinex theory shows that the color of a scene perceived by human visual system is only dependent on the intrinsic reflectance and unrelated to the illumination. For images recorded by digital cameras, the intensities are the product of the scene reflectance and illumination. Image intensities are different when taken under different illuminations. The aim of image retinex problem is to decompose the reflectance from the image intensities. In this paper, we propose a variational model for image retinex problem. The model can be reduced to a linear complementarity problem and a modulus-based multigrid method is applied to solve it. A pointwise coupled modulus-based Gauss-Seidel iteration scheme is used as a smoother of the multigrid method. The smoothing factor of the smoother is analyzed by using the local Fourier analysis. Both theoretical analyses and numerical results show the efficiency of the proposed method for image retinex problem.

Time-varying linear complementarity problem (TLCP) has received a great deal of attention due to its broad variety of scientific and engineering applications. Several efficient Zhang neural networks are introduced for solving TLCP in this paper. Theoretical analysis shows that the related error function of the model proposed in this paper eventually tends to zero. The state convergence time periods of those Zhang neural networks with three types of activation functions are proved to be finite and can be quantitatively estimated by using some given parameters. Further, it is shown that the proposed neural network is of noise-tolerance, which means the neural network is more appropriate for a wider application. Moreover, in order to implement neural network numerically, a related discrete-time version is also studied. Finally, numerical simulations confirm the analysis of the proposed models concretely.

The linear complementarity problem arising from a free boundary problem can be equivalently reformulated as a fixed-point equation. We present a modified modulus-based multigrid method to solve this fixed-point equation. This modified method is a full approximation scheme using the modulus-based splitting iteration method as the smoother and avoids the transformation between the auxiliary and the original functions which was necessary in the existing modulus-based multigrid method. We predict its asymptotic convergence factor by applying local Fourier analysis to the corresponding two-grid case. Numerical results show that the W-cycle possesses an h-independent convergence rate and a linear elapsed CPU time, and the convergence rate of the V-cycle can be improved by increasing the smoothing steps. Compared with the existing modulus-based multigrid method, the modified method is more straightforward and is a standard full approximation scheme, which makes it more convenient and efficient in practical applications.

In this paper we study sufficient matrices, which play an important role in theoretical analysis of interior-point methods for linear complementarity problems. We present new characterisations of these matrices which imply new necessary and sufficient conditions for sufficiency. We use these results to develop an algorithm with exponential iteration complexity which in each iteration solves a simple instance of linear programming problem and is capable to reveal whether given symmetric matrix is sufficient or not. This algorithm demonstrates 100 % accuracy on all tested instances of matrices.

For the extended linear complementarity problem (ELCP), we establish a global error bound estimation for ELCP under milder condition. Based on this, we propose a smoothing algorithm for solving the...

Retinex theory was proposed by Land in the 1970s. Its theoretical basis is that the color of an object is determined by the reflection ability of the object to the light of long wave, medium wave, and short wave, rather than by the absolute value of the reflected light intensity. There is color sense consistency in Retinex theory. The main purpose of Retinex problem is to recover the reflection properties of objects from the images obtained under a certain illumination intensity, so as to obtain the real colors of objects. There are many methods for solving the Retinex problem. In this paper, based on a variational minimization model with physical constraints of reflectance value, we first transform the Retinex problem into a linear complementarity problem, and then propose a class of two-step modulus-based matrix splitting iteration methods to solve this problem. We also prove the convergence of the two-step modulus-based matrix splitting iteration methods for solving the linear complementarity problem. The experimental results show that the convergence speeds of the proposed methods are much faster than the existing methods for solving the Retinex problem, and that the advantages of the new methods over the existing ones could be significant in the quality of the reflectance recovery.

We consider the linear complementarity problem with uncertain data modeled by intervals, representing the range of possible values. Many properties of the linear complementarity problem (such as solvability, uniqueness, convexity, finite number of solutions etc.) are reflected by the properties of the constraint matrix. In order that the problem has desired properties even in the uncertain environment, we have to be able to check them for all possible realizations of interval data. This leads us to the robust properties of interval matrices. In particular, we will discuss S-matrix, Z-matrix, copositivity, semimonotonicity, column sufficiency, principal nondegeneracy, $$R_0$$ -matrix and R-matrix. We characterize the robust properties and also suggest efficiently recognizable subclasses.

We consider the weighted linear complementarity problem (denoted by wLCP). Many numerical algorithms have been proposed for solving the monotone wLCP. In this paper, we propose a damped Gauss–Newto...

In this work, we model the relationship between prey and predators by studying the interactive behavior of this prey-predator model and using the change of prey. The objective is to maximize the profit function of each predator by seeking the strategy provided by each predator to maximize its profit. To do so, we maximize this utility function being constrained by balance equations between biomass and trophic, and we show that this last problem is completely equivalent to finding the generalized Nash equilibrium point. To calculate it, we use the conditions of Karush-Kuhn-Tucker and we show that it is indeed a linear complementarity problem.

In this paper, we present a finite difference scheme for a linear complementarity problem with a mixed boundary condition arising from pricing a Russian option with a finite time horizon. An implic...

The linear complementarity problem (LCP) is a general set membership problem that includes quadratic cone programming as a special case. In this work we consider a homogeneous embedding of the LCP, which encodes both the optimality conditions of the original LCP as well as certificates of infeasibility. The resulting problem can be expressed as a monotone inclusion problem involving the sum of two maximal monotone operators, to which we apply Douglas-Rachford splitting. The resulting algorithm has almost identical per-iteration cost as applying the splitting to the LCP directly. Specifically, each iteration requires solving a linear system and projecting onto a convex cone. The resulting algorithm has no explicit hyper-parameters and is able to return a primal-dual solution should one exist or a certificate of infeasibility otherwise. If a sequence of related problems are solved it can easily be warm-started and make use of factorization caching of the linear system. We demonstrate on a range of public and synthetic datasets that for feasible problems our approach tends to be somewhat faster than applying operator splitting directly to the LCP, and in cases of infeasibility our approach can be significantly faster than alternative approaches based on diverging iterates. The algorithm we describe has been implemented in C and is available open-source as SCS-3.0.

Solutions to the linear complementarity problem (LCP) are naturally sparse in many applications such as bimatrix games and portfolio section problems. Despite that it gives rise to the hardness, sp...

In this paper, we propose a generalized Newton method for solving a class of discrete-time linear complementarity systems consisting of a system of linear equations and a linear complementarity constraints with a Z-matrix. We obtain a complete characterization of the least element solution of a linear complementarity problem with a Z-matrix that a solution is the least element solution if and only if the principal submatrix corresponding to the nonzero components of the solution is an M-matrix. We present a Newton method for solving a linear complementarity problem with a Z-matrix. We propose a generalized Newton method for solving the discrete-time linear complementarity system where the linear complementarity problem constraint is solved by the proposed Newton method. Under suitable conditions, we show that the generalized Newton method converges globally and finds a solution in finitely many iterations. Preliminary numerical results show the efficiency of the proposed method.

In this paper, we present a generalized SOR-like iteration method to solve the non-Hermitian positive definite linear complementarity problem (LCP), which is obtained by reformulating equivalently the implicit fixed-point equation of the LCP as a two-by-two block nonlinear equation. The convergence properties of the generalized SOR-like iteration method are discussed under certain conditions. Numerical experiments show that the generalized SOR-like method is efficient, compared with the SOR-like method and the modulus-based SOR method.

In this paper, a full-Newton step Interior-Point Method for solving monotone Weighted Linear Complementarity Problem is designed and analyzed. This problem has been introduced recently as a generalization of the Linear Complementarity Problem with modified complementarity equation, where zero on the right-hand side is replaced with the nonnegative weight vector. With a zero weight vector, the problem reduces to a linear complementarity problem. The importance of Weighted Linear Complementarity Problem lies in the fact that it can be used for modelling a large class of problems from science, engineering and economics. Because the algorithm takes only full-Newton steps, the calculation of the step size is avoided. Under a suitable condition, the algorithm has a quadratic rate of convergence to the target point on the central path. The iteration bound for the algorithm coincides with the best iteration bound obtained for these types of problems.

In this paper, we propose a new full-Newton step interior-point method (IPM) for $$P_{*}(\kappa )$$ linear complementarity problem (LCP). The search direction is obtained by applying algebraic equivalent transformation on the centering equation of the central path which is introduced by Darvay et al. for linear optimization (Optim Lett 12(5):1099–1116, 2018). They point out that the search direction can also be obtained by using a positive-asymptotic kernel function. This kernel function has not been used in the complexity analysis of IPMs for $$P_{*}(\kappa )$$ -LCP before. Assuming a strictly feasible starting point is available, we show that the algorithm has the iteration complexity bound $$O((1+4\kappa )\sqrt{n}\log {\frac{n}{\epsilon }})$$ , which is the best known complexity result for such methods. Some numerical results have been provided.

In this paper, we introduce a class of SOR-like iteration methods for solving the systems of the weakly nonlinear equation, which is by reformulating equivalently the weakly nonlinear equation as a...

We present a feasible full step interior-point algorithm to solve the P∗(κ) horizontal linear complementarity problem defined on a Cartesian product of symmetric cones, which is not based on a usua...

Recently, Bai and Zhang [Numerical Linear Algebra with Applications, 2013, 20, 425Ƀ439] constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations and studied the convergence of them; Li et al. [Journal of Nanchang University (Natural Science), 2013, 37, 307Ƀ312] studied synchronous block multisplitting iteration methods; Zhang and Li [Computers and Mathematics with Application, 2014, 67, 1954Ƀ1959] analyzed and obtained the weaker convergence results for linear complementarity problems. In this paper, we generalize their algorithms and further study global relaxed modulus-based synchronous block multisplitting multi-parameters methods for linear complementarity problems. Furthermore, we give the weaker convergence results of our new method in this paper when the system matrix is a block $ H_{+}- $matrix. Therefore, new results provide a guarantee for the optimal relaxation parameters, please refer to [A. Hadjidimos, M. Lapidakis and M. Tzoumas, SIAM Journal on Matrix Analysis and Applications, 2012, 33, 97Ƀ110, (dx.doi.org/10.1137/100811222)], where optimal parameters are determined.

In this work we propose a novel numerical method for a finite-dimensional optimization problem arising from the discretization of an infinite-dimensional constrained optimization problem, called an obstacle problem, with interval coefficients. In this method, the two different ways of characterizing the optimal solutions, i.e., minimizing the mid-point and one end-point (the worst-case scenario) or the mid-point and the width of the objective interval, are formulated as a single constrained multi-objective minimization problem and the KKT conditions of the optimization problem defining the Pareto optimal solution to the multi-objective problem are of the form of a Linear Complementarity Problem (LCP) which is shown to have a unique solution. The LCP is the approximated by a non-linear equation using an interior penalty approach. We prove that the penalty equation is uniquely solvable and its solution converges to that of LCP as the penalty constant approaches to zero. Numerical results are presented to demonstrate the usefulness of the numerical method proposed.

In this work we develop partial differential equation (PDE) based computational models for pricing real options to contract the production or to transfer part/all of the ownership of a project when the underlying asset price of the project satisfies a geometric Brownian motion. The developed models are similar to the Black-Scholes equation for valuing conventional European put options or the partial differential linear complementarity problem (LCP) for pricing American put options. A finite volume method is used for the discretization of the PDE models and a penalty approach is applied to the discretized LCP. We show that the coefficient matrix of the discretized systems is a positive-definite \begin{document}$ M $\end{document} -matrix which guarantees that the solution from the penalty equation converges to that of the discretized LCP. Numerical experiments, performed to demonstrate the usefulness of our methods, show that our models and numerical methods are able to produce financially meaningful numerical results for the two non-trivial test problems.

Nonlinear optimal control problems are frequently transformed into nonlinear programming problems for which the solving process is generally time-consuming. In this paper, a linear direct transcription method is proposed for nonlinear optimal control problems. Taking advantage of the state-dependent coefficient parameterization method and the spectral discretization method, the nonlinear optimal control problem is successively linearized and turned into a sequence of efficiently solvable mixed linear complementarity problems. The proposed direct transcription method is linear in two ways: the nonlinear system is linearized using state-dependent coefficient parameterization; the resulting quadratic programming problem is converted into a mixed linear complementarity problem. Simulations are implemented, and numerical results verify the effectiveness and efficiency of the proposed method.

In 2013, Bai and Zhang (Numer Linear Algebra Appl, 20:425–439 2013) constructed modulus-based synchronous multisplitting iteration methods by an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations and studied the convergence of them. In this paper, we generalize Bai and Zhang’s method and study relaxed modulus-based synchronous multisplitting multi-parameter methods for linear complementarity problems. Furthermore, when the system matrix is an H+-matrix, we give convergence results for our new methods under weaker conditions than those in Bai and Zhang’s. Finally, numerical experiments are presented to illustrate the efficiency of the proposed methods.

This paper is concerned with the rank-deficient problem of least squares adjustment models with inequality constraints. First, the problem under study is transformed into a linear complementarity problem (LCP) with a P0-matrix (P0-LCP), which is a non-symmetric matrix whose principal minors are all nonnegative. Second, both Mangasarian’s symmetric successive over relaxation and a perturbed LCP with arbitrarily small perturbation are employed to solve the P0-LCP. As a result, a new iterative algorithm for rank-deficient adjustment models with inequality constraints is presented, and its convergence is proven. Finally, examples are given to demonstrate the efficiency of the proposed algorithm. It is shown that the proposed algorithm can not only assess the stability of points but also provide additional adjustment criteria to guarantee the uniqueness of solutions to the problem of rank-deficient free-network adjustment.

Kernel functions play an important role in the design and complexity analysis of interior point algorithms for solving convex optimization problems. They determine both search directions and the proximity measure between the iterate and the central path. In this paper, we introduce a primal-dual interior point algorithm for solving $$P_*(\kappa ) $$-horizontal linear complementarity problems based on a new kernel function that has a trigonometric function in its barrier term. By using some simple analysis tools, we present some properties of the new kernel function. Our analysis shows that the algorithm meets the best known complexity bound i.e., $$O\left( (1+2\kappa )\sqrt{n}\log n\log \frac{n}{\varepsilon }\right) $$ for large-update methods. Finally, we present some numerical results illustrating the performance of the algorithm.

In this paper we introduce three numerical methods to evaluate the prices of European, American, and barrier options under a regime-switching jump-diffusion model (RSJD model) where the volatility and other parameters are considered as variable coefficients. The prices of the European option, which is one of the financial derivatives, are given by a partial integro-differential equation (PIDE) problem and those of the American option are evaluated by solving a linear complementarity problem (LCP). The proposed methods are constructed to avoid the use of any fixed point iteration techniques at each state of the economy and time step. We analyze the stability of the proposed methods with respect to the discrete l2-norm in the time and spatial variables. A variety of numerical experiments are carried out to show the second-order convergence of the three numerical methods under the regime-switching jump-diffusion model with variable coefficients.

A new procedure for dealing with elastoplasticity during the time integration of multi degree of freedom (MDOF) structural problems in dynamics is developed and numerically implemented. It is shown here how the solution of MDOF elastic–plastic dynamics structural problems formulated in generalized variables, when piecewise linear yield functions are usually available, can be obtained utilizing any time integration technique—here the Newmark scheme is adopted—in conjunction with a linear complementarity problem (LCP) approach for dealing with plasticity. The same method could be extended to general 3D continuum dynamics problems with general yield conditions exploiting techniques developed in the past for the quasi-static case. The paper summarizes first the quasi-static problem in terms of a finite step LCP formulation; the elastic–plastic dynamic integration scheme is then formulated as an extension of the quasi-static LCP formulation. Two simple examples illustrate the effectiveness of the approach.

Tensors (hypermatrices) are multidimensional analogs of matrices. The tensor complementarity problem is a class of nonlinear complementarity problems with the involved function being defined by a tensor, which is also a direct and natural extension of the linear complementarity problem. In the last few years, the tensor complementarity problem has attracted a lot of attention, and has been studied extensively, from theory to solution methods and applications. This work, with its three parts, aims at contributing to review the state-of-the-art of studies for the tensor complementarity problem and related models. In this part, we describe the theoretical developments for the tensor complementarity problem and related models, including the nonemptiness and compactness of the solution set, global uniqueness and solvability, error bound theory, stability and continuity analysis, and so on. The developments of solution methods and applications for the tensor complementarity problem are given in the second part and the third part, respectively. Some further issues are proposed in all the parts.

Following the central-path, as a guide line to optimal solution of mathematical problems, is one of the main difficulty of interior-point methods in practice. These iterative methods, follow the central path step by step to get close enough to the optimal solution of underlying problem. Based on estimating the central path by an ellipse, we propose an infeasible interior-point method for linear complementarity problem. In each iteration, the algorithm follows the ellipsoidal approximation of the central-path to find an $$\varepsilon $$ -approximate solution of the problem. We prove that under certain conditions the proposed algorithm is well-defined and the generated points by the algorithm converge to an $$\varepsilon $$ -approximate solution of the linear complementarity problem.

In this paper, we define a backward error for the linear complementarity problem (LCP), and then present an expression of it which can be employed to examine the stability of algorithms solving the LCP. Some numerical examples are given to show the efficiency of the proposed theoretical results.

In this paper, a preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problems associated with an M-matrix is proposed. The convergence analysis of the presented method is given. In particular, we provide a comparison theorem between preconditioned two-step modulus-based Gauss–Seidel (PTMGS) iteration method and two-step modulus-based Gauss–Seidel (TMGS) iteration method, which shows that PTMGS method improves the convergence rate of original TMGS method for linear complementarity problem. Numerical tested examples are used to illustrate the theoretical analysis.

Linear complementarity problem (LCP) presents many nice properties when the associated matrix belongs to some special matrix classes, especially H-matrices. In this paper, we put forward a new subclass of H-matrices, called S-QN matrices, which is the proper generalization of the QN matrices. We have proved that for a given S-QN matrix A, there exists a diagonal scaling matrix W such that AW is a QN matrix. Then, we present two kinds of error bounds for LCP of S-QN matrices. The Error Bound I generalizes the error bound for LCP of QN matrices. The Error Bound II overcomes the limitation that the Error Bound I cannot be used. Numerical examples illustrate that the Error Bound I is better than other previous bounds for H-matrices in some cases. Moreover, in some special cases, the Error Bound II can improve considerably the Error Bound I.

A new operator splitting method is proposed for American options under time-fractional Black–Scholes models. The fractional linear complementarity problem is split into two easy sub-problems, with the leading coefficients separated from the convolution sum and matched through a general correction step. The method is implementation friendly in the sense that one can easily modify a fractional European solver to obtain the proposed method, since the correction step is decoupled and is trivial to solve. The method is validated through numerical experiments and demonstrated to be superior to the traditional approach. The paper also provides numerical studies including the effect of fractional orders and the comparison of fractional models.

In the setting of a Euclidean Jordan algebra V with symmetric cone $$V_+$$ , corresponding to a linear transformation M, a ‘weight vector’ $$w\in V_+$$ , and a $$q\in V$$ , we consider the weighted linear complementarity problem wLCP(M, w, q) and (when w is in the interior of $$V_+$$ ) the interior point system IPS(M, w, q). When M is copositive on $$V_+$$ and q satisfies an interiority condition, we show that both the problems have solutions. A simple consequence, stated in the setting of $$\mathbb {R}^{n}$$ is that when M is a copositive plus matrix and q is strictly feasible for the linear complementarity problem LCP(M, q), the corresponding interior point system has a solution. This is analogous to a well-known result of Kojima et al. on $$\mathbf{P}_*$$ -matrices and may lead to interior point methods for solving copositive LCPs.

We propose a wide neighborhood interior-point algorithm with arc-search for P ⁎ ( κ ) linear complementarity problem (LCP). Along the ellipsoidal approximation of the central path, the algorithm searches optimizers in every iteration. Assuming a strictly starting point is available, we show that the algorithm has O ( ( 1 + 2 κ ) ( 1 + 18 κ ) 2 n log ⁡ ( x 0 ) T s 0 e ) iteration complexity. It matches the currently best known iteration bound for P ⁎ ( κ ) LCP. Some preliminary numerical results show that the proposed algorithm is efficient and reliable.

In this paper, we present a general two-sweep modulus-based iteration method to solve a class of linear complementarity problems. Convergence analysis shows that the general two-sweep modulus-based matrix splitting iteration method will converge to the exact solution of linear complementarity problem under appropriate conditions. Numerical experiments further show that the proposed methods are superior to the existing methods in actual implementation.

An efficient numerical method for solving a symmetric positive definite second-order cone linear complementarity problem (SOCLCP) is proposed. The method is shown to be more efficient than recently developed iterative methods for small-to-medium sized and dense SOCLCP. Therefore it can serve as an excellent core computational engine in solutions of large scale symmetric positive definite SOCLCP solved by subspace projection methods, solutions of general SOCLCP and the quadratic programming over a Cartesian product of multiple second-order cones, in which small-to-medium sized SOCLCPs have to be solved repeatedly, efficiently, and robustly.

Stochastic $$R_0$$ matrix, which is a generalization of $$R_0$$ matrix, is instrumental in the study of the stochastic linear complementarity problem (SLCP). In this paper, we focus on the expected residual minimization (ERM) of the SLCP via the Fischer–Burmeister (FB) function, which enjoys better properties of the continuity and differentiability than the min-function-based ERM when the involved matrix is a stochastic $$R_0$$ matrix. First, we prove that the solution set of the FB-function-based ERM is nonempty and bounded if and only if the involved matrix is a stochastic $$R_0$$ matrix. Second, we show that its objective function is continuously differentiable in $${\mathbb {R}}^n$$ , which makes it possible for us to design the gradient-type algorithm. Finally, we implement the numerical experiments generated randomly via sample average approximation, and the numerical results indicate that the optimal solutions of the FB-function-based ERM are better than that of the min-function-based ERM in terms of preserving the nonnegativity of the involved linear function, especially when the involved matrix is a constant matrix.