Fischer-Burmeister Function Research Articles

Supply chain network equilibrium with outsourcing for fresh agricultural products under stochastic demands

Abstract Accepted by: M. Zied Babai This paper examines a supply chain network focusing on the distribution of fresh agricultural products, incorporating outsourcing and stochastic demands considerations. Initially, we establish a generalized Nash equilibrium model with stochastic demands among fresh produce firms. Subsequently, we transform this model into a mixed complementarity system using the Karush–Kuhn–Tucker conditions. Utilizing the Fischer–Burmeister function, we further convert the mixed complementarity system into a set of nonlinear equations, amenable to solution via GAMS software. We conduct numerical experiments and perform sensitivity analysis on key parameters. Our findings suggest that higher subsidy rates incentivize firms towards production outsourcing, particularly benefiting low-income farmers, thereby potentially increasing profits. Moreover, market fluctuations play a pivotal role in ensuring the stability and profitability of firms, with moderate fluctuations presenting opportunities for fresh enterprises to adapt, innovate and capitalize on evolving market conditions. These results offer valuable insights for effective management of fresh produce supply chains.

A varying-parameter complementary neural network for multi-robot tracking and formation via model predictive control

In this paper, a varying-parameter complementary neural network (VPCNN) is designed and combined with model predictive control (MPC) to solve the multi-robot tracking and formation problems via a leader–follower strategy. First, multi-robot tracking and formation problems are transformed into quadratic programming (QP) problems employing an MPC approach. Second, a nonlinear complementary function approach, i.e., the Fischer–Burmeister function, is used to map the Karush–Kuhn–Tucker conditions of the QP problem with double-ended inequality constraints to a system of nonlinear equations. Finally, the VPCNN is designed to solve the multi-robot tracking and formation problem. The effectiveness of the proposed method is demonstrated by numerical simulations, and the advantage of VPCNN in terms of solution speed is indicated by comparisons with a primal–dual neural network.

A fresh look at nonsmooth Levenberg–Marquardt methods with applications to bilevel optimization

In this paper, we revisit the classical problem of solving over-determined systems of nonsmooth equations numerically. We suggest a nonsmooth Levenberg–Marquardt method for its solution which, in contrast to the existing literature, does not require local Lipschitzness of the data functions. This is possible when using Newton-differentiability instead of semismoothness as the underlying tool of generalized differentiation. Conditions for local fast convergence of the method are given. Afterwards, in the context of over-determined mixed nonlinear complementarity systems, our findings are applied, and globalized solution methods, based on a residual induced by the maximum and the Fischer–Burmeister function, respectively, are constructed. The assumptions for local fast convergence are worked out and compared. Finally, these methods are applied for the numerical solution of bilevel optimization problems. We recall the derivation of a stationarity condition taking the shape of an over-determined mixed nonlinear complementarity system involving a penalty parameter, formulate assumptions for local fast convergence of our solution methods explicitly, and present results of numerical experiments. Particularly, we investigate whether the treatment of the appearing penalty parameter as an additional variable is beneficial or not.

Inexact sequential injective algorithm for weakly univalent vector equation and its application to regularized smoothing Newton algorithm for mixed second-order cone complementarity problems

<p style='text-indent:20px;'>It is known that the complementarity problems and the variational inequality problems are reformulated equivalently as a vector equation by using the natural residual or Fischer-Burmeister function. In this paper, we first propose an inexact sequential injective algorithm (ISIA) for a vector equation, and show the global convergence under weak univalence assumption. Roughly speaking, the ISIA generates the sequence of inexact solutions of approximate vector equations, which consist of the injectives converging to the original vector-valued function. Although the ISIA is simple and conceptual, it can be a prototype to many other algorithms such as a smoothing Newton algorithm, semismooth Newton algorithm, etc. Next, we apply the ISIA prototype to the regularized smoothing Newton algorithm (ReSNA) for mixed second-order cone complementarity problems (MSOCCPs). Exploiting the ISIA convergence scheme, we prove that the ReSNA is globally convergent under Cartesian <inline-formula><tex-math id="M1">\begin{document}$ P_0 $\end{document}</tex-math></inline-formula> assumption.</p>

Real-time pricing method for smart grid based on social welfare maximization model

<p style='text-indent:20px;'>With the applications of big data, the research of real-time pricing method for smart grid has become increasingly important. Based on the demand side management and the real-time pricing model, the social welfare maximization model of smart grid is considered. We transform it by Karush-Kuhn-Tucker condition, then the social welfare maximization model is transformed into a nonsmooth equation by Fischer-Burmeister function. Then, taking advantage of simple calculation and small storage, we propose a new smoothing conjugate gradient method to solve real-time pricing problem for smart grid based on the social welfare maximization. Under general conditions, the global convergence of the new proposed method is proved. Finally, the numerical simulation results show the effectiveness of the proposed method for solving the real-time pricing problems for smart grid based on the social welfare maximization.</p>

Two reduced non-smooth Newton's methods for discretised state constrained optimal control problem governed by advection-diffusion equation

We are interested in solving numerically an optimal control problem governed by advection-diffusion equation with state inequality constraints. This problem can be implemented to modelise the minimisation of air pollution. First the infinite dimensional problem is discretised by application of the upwind symmetric interior penalty galerkin (SIPG) method. Then, we write the optimality conditions for the discretised problem. We reformulate these conditions into an equivalent system of nonlinear and non-smooth equations by the use of the Fischer-Burmeister function. Then, two effective algorithms that combine the non-smooth Newton's method and an active (index) set technique or pseudo inverse method are applied to solve this system. At the end, numerical examples are presented to illustrate the performance of our methods.

A new difference of anisotropic and isotropic total variation regularization method for image restoration.

Total variation (TV) regularizer has diffusely emerged in image processing. In this paper, we propose a new nonconvex total variation regularization method based on the generalized Fischer-Burmeister function for image restoration. Since our model is nonconvex and nonsmooth, the specific difference of convex algorithms (DCA) are presented, in which the subproblem can be minimized by the alternating direction method of multipliers (ADMM). The algorithms have a low computational complexity in each iteration. Experiment results including image denoising and magnetic resonance imaging demonstrate that the proposed models produce more preferable results compared with state-of-the-art methods.

A nonconvex function activated noise-tolerant neurodynamic model aided with Fischer-Burmeister function for time-varying quadratic programming in the presence of noises

Time-varying quadratic programming problems (TVQPPs) with equality and inequality constraints often arise in the fields of scientific computation and engineering application. Zeroing neural network (ZNN), being a special kind of recurrent neural network, has shown powerful capabilities to compute a variety of zeroing finding problems with monotonically increasing odd activation function. However, the projection sets of nonconvex activation function are obviously excluded, which means a general conclusion remain unexplored. In addition, noises are always ubiquitous in actual applications. Nevertheless, most existing ZNN-based models usually assume that the solving process is free of noise before the calculation. In this paper, a general zeroing neural network model with nonconvex activated function (GZNNM-NAF) and a general noise-tolerant zeroing neural network model with nonconvex activated function (GNTZNNM-NAF), which are also viewed as ZNN-type models, are developed by the inspiration of the traditional ZNN model from a control-based perspective. The ZNN-type models break the limitation of the traditional ZNN models of activation function, which allows nonconvex sets for projection operations and combines nonlinear complementary function for dealing with inequality constraints arising in TVQPPs. Moreover, theoretical results indicate that the ZNN-type models globally converge to time-varying optimal solution of TVQPPs with equality and inequality constraints under the noise circumstance. According to the different cases of nonconvex activation function, robustness analyses are demonstrated in detail for TVQPPs. It may enlarge the scope of the proposed method, especially in the field of practical application. Finally, a numerical example and an application example to manipulator motion generation are analyzed to verify the superiority and robustness of the developed ZNN-type models for TVQPPs with different measurement noises.

An Active-Set Fischer–Burmeister Trust-Region Algorithm to Solve a Nonlinear Bilevel Optimization Problem

In this paper, the Fischer–Burmeister active-set trust-region (FBACTR) algorithm is introduced to solve the nonlinear bilevel programming problems. In FBACTR algorithm, a Karush–Kuhn–Tucker (KKT) condition is used with the Fischer–Burmeister function to transform a nonlinear bilevel programming (NBLP) problem into an equivalent smooth single objective nonlinear programming problem. To ensure global convergence for the FBACTR algorithm, an active-set strategy is used with a trust-region globalization strategy. The theory of global convergence for the FBACTR algorithm is presented. To clarify the effectiveness of the proposed FBACTR algorithm, applications of mathematical programs with equilibrium constraints are tested.

Guest Editorial: Situational awareness of integrated energy systems

Guest Editorial: Situational awareness of integrated energy systems

An active-set with barrier method and trust-region mechanism to solve a nonlinear Bilevel programming problem

<abstract><p>Nonlinear Bilevel programming (NBLP) problem is a hard problem and very difficult to be resolved by using the classical method. In this paper, Karush-Kuhn-Tucker (KKT) condition is used with Fischer-Burmeister function to convert NBLP problem to an equivalent smooth single objective nonlinear programming (SONP) problem. An active-set strategy is used with Barrier method and trust-region technique to solve the smooth SONP problem effectively and guarantee a convergence to optimal solution from any starting point. A global convergence theory for the active-set barrier trust-region (ACBTR) algorithm is studied under five standard assumptions. An applications to mathematical programs are introduced to clarify the effectiveness of ACBTR algorithm. The results show that ACBTR algorithm is stable and capable of generating approximal optimal solution to the NBLP problem.</p></abstract>

Power flow analysis of integrated energy microgrid considering non‐smooth characteristics

With the development of electric energy substitution technologies such as electric auxiliary heating, the relationship between electric energy and thermal energy is getting closer, and it is of great significance to in-depth study of integrated energy micro-grids containing electricity and heat. However, the integrated energy microgrid power flow (PF) model has non-smooth characteristics such as distributed generations (DGs) limits, converter stations limits and the unknown direction of pipeline mass flow, which can easily lead to the failure of PF convergence. This paper establishes an integrated energy microgrid model including three-phase unbalanced AC/DC hybrid microgrid and heating system. Firstly, the correctness of the converter station model is verified by electromagnetic transient simulation, and then the non-smooth constraint models considering DG and converter station's reactive power limits and pipeline mass flow direction unknown are established based on complementary constraints, and the non-smooth constraints are smoothed by Fischer-Burmeister (FB) function. Finally, the integrated energy microgrid is used for PF analysis, and the results verify the rationality of the model and the effectiveness of the proposed non-smooth characteristics treatment method.

Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets

This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness.

A VSC-based Model for Power Flow Assessment of Multi-terminal VSC-HVDC Transmission Systems

This paper puts forward a new practical voltage source converter (VSC) based AC-DC converter model suitable for conducting power flow assessment of multi-terminal VSC-based high-voltage direct current (VSC-MTDC) systems. The model uses an advanced method to handle the operational limits and control modes of VSCs into the power flow formulation. The new model is incorporated into a unified framework encompassing AC and DC power grids and is solved by using the Newton-Raphson method to enable quadratically convergent iterative solutions. The use of complementarity constraints, together with the Fischer-Burmeister function, is proposed to enable the seamless incorporation of operational control modes of VSC and automatic enforcement of any converter's operational limits that become violated during the iterative solution process. Thus, a dedicated process for checking limits is no longer required. Furthermore, all existing relationships between the VSC control laws and their operational limitsare considered directly during the solution of the power flow problem. The applicability of the new model is demonstrated with numerical examples using various multi-terminal AC-DC transmission networks, one of which is a utility-sized power system.

Efficient Power Flow Algorithm for AC/MTDC Considering Complementary Constraints of VSC's Reactive Power and AC Node Voltage

For a hybrid ac/dc power system incorporating voltage source converter-based multi-terminal direct current (VSC-MTDC) grids, it is essential to solve the power flow problems efficiently for reliable operation. This paper presents an efficient ac/dc power flow algorithm considering the complementary constraints of VSC's reactive power limit and ac node voltage. It can be distinguished from the existing works in terms of both modeling and algorithm aspects. First, the VSC model is formulated, in which the interactions between the reactive power limit and the ac node voltage of VSC are viewed as complementary relations. The complementary relations are expressed using a smooth Fischer-Burmeister function. Then, power flow models of the ac grid and dc grid with a constant Jacobian matrix are formulated based on the VSC model, respectively. To efficiently solve the power flow of an ac/dc grid, the structure characteristic of the Jacobian matrix of ac/dc power flow computation is analyzed. It shows that the Jacobian matrix can be approximated as a block-upper triangular one. Based on this characteristic, a decoupled ac/dc power flow computation algorithm is provided. Test results on two hybrid ac/dc systems under different scenarios validate the correctness, convergence, and efficiency of the proposed power flow algorithm.

A novel nonsmooth approach for flexible multibody systems with contact and friction in 3D space

For computational multibody system dynamics, contact and friction problems are very complex and important problems. Therefore, this paper proposes a novel nonsmooth method for flexible multibody systems with contact and friction in 3D space. Considering the nonsmooth effect of contact and friction on the state variable of the multibody systems, the proposed method is divided into two parts: (i) the change of state variables under the action of smooth force and (ii) the change of state variables caused by contact and friction. Furthermore, for the contact part, the Newton’s impact law and the classical Coulomb friction model are employed to deal with normal impact contact impulse and tangential friction contact impulse, respectively. In addition, Fischer–Burmeister function and Karush–Kuhn–Tucker conditions are also used to solve the normal impact contact impulse and tangential friction contact impulse. What’s more, since the symplectic discrete format performs well robustness to numerical results, the discrete format of the proposed method is based on the symplectic discreteness, and it is expected to obtain the robust numerical results. Finally, several numerical examples are tested by the proposed nonsmooth method, and the numerical simulation results verify the effectiveness of the proposed approach.

Expected residual minimization method for monotone stochastic tensor complementarity problem

In this paper, we first introduce a new class of structured tensors, named strictly positive semidefinite tensors, and show that a strictly positive semidefinite tensor is not an $$R_0$$ tensor. We focus on the stochastic tensor complementarity problem (STCP), where the expectation of the involved tensor is a strictly positive semidefinite tensor. We denote such an STCP as a monotone STCP. Based on three popular NCP functions, the min function, the Fischer–Burmeister (FB) function and the penalized FB function, as well as the special structure of the monotone STCP, we introduce three new NCP functions and establish the expected residual minimization (ERM) formulation of the monotone STCP. We show that the solution set of the ERM problem is nonempty and bounded if the solution set of the expected value (EV) formulation for such an STCP is nonempty and bounded. Moreover, an approximate regularized model is proposed to weaken the conditions for nonemptiness and boundedness of the solution set of the ERM problem in practice. Numerical results indicate that the performance of the ERM method is better than that of the EV method.

An ordinary differential equation approach for nonlinear programming and nonlinear complementary problem

We consider an ordinary differential equation (ODE) approach for solving non- linear programming (NLP) and nonlinear complementary problem (NCP). The Karush- Kuhn Tucker (KKT) optimality conditions can be converted to NCP. Based on the Fischer-Burmeister (FB) function and the Natural-Residual (NR) function are obtained the new NCP-functions. A special technique is employed to reformulate of the NCP as the system of nonlinear algebraic equations (NAEs) later on reformulated once more by force. of an original time-like function into an ODE. Afterwards, a group preserving scheme (GPS) is a package to reformulate an ODE into the new numerical equation in a way the ODEs system is designed into a nonlinear dynamical system (NDS) and is continued to a discovery the new numerical equation through activating the Lorentz group SO0(n, 1) and its Lie algebra so(n, 1). Lastly, the fictitious time integration method (FTIM) is utilized into this new numerical equation to determine an approximation solution at the numerical experiments area.

Computational shape optimisation for a gradient-enhanced continuum damage model

An isotropic gradient-enhanced damage model is applied to shape optimisation in order to establish a computational optimal design framework in view of optimal damage distributions. The model is derived from a free Helmholtz energy density enriched by the damage gradient contribution. The Karush–Kuhn–Tucker conditions are solved on a global finite element level by means of a Fischer–Burmeister function. This approach eliminates the necessity of introducing a local variable, leaving only the global set of equations to be iteratively solved. The necessary steps for the numerical implementation in the sense of the finite element method are established. The underlying theory as well as the algorithmic treatment of shape optimisation are derived in the context of a variational framework. Based on a particular finite deformation constitutive model, representative numerical examples are discussed with a focus on and application to damage optimised designs.

Semi-smooth Newton methods for nonlinear complementarity formulation of compositional two-phase flow in porous media

Simulating compositional multiphase flow in porous media is a challenging task, especially when phase transition is taken into account. The main problem with phase transition stems from the inconsistency of the primary variables such as phase pressure and phase saturation, i.e. they become ill-defined when a phase appears or disappears. Recently, a new approach for handling phase transition has been developed, whereby the system is formulated as a nonlinear complementarity problem (NCP). Unlike the widely used primary variable switching (PVS) method which requires a drastic reduction of the time step size when a phase appears or disappears, this approach is more robust and allows for larger time steps. One way to solve an NCP system is to reformulate the inequality constraints as a non-smooth equation using a complementary function (C-function). Because of the non-smoothness of the constraint equations, a semi-smooth Newton method needs to be developed. In this work, we consider two methods for solving NCP systems used to model multiphase flow: (1) a semi-smooth Newton method for two C-functions: the minimum and the Fischer-Burmeister functions, and (2) a new inexact Newton method based on the Jacobian smoothing method for a smooth version of the Fischer-Burmeister function. We show that the new method is robust and efficient for standard benchmark problems as well as for realistic examples with highly heterogeneous media such as the SPE10 benchmark.