Mixed Complementarity Research Articles

A compact MLCP-based projection recurrent neural network model to solve shortest path problem

ABSTRACT We develop a projection recurrent neural network (RNN) to obtain the solution of the shortest path problem (SPP). Our focus on the paper is to give a compact single-layer structure RNN model to solve the SPP. To present the RNN model, we utilise a mixed linear complementarity problem (MLCP). Moreover, the developed RNN is proved to be globally stable. Finally, some numerical simulations are stated to show the performance of the presented approach. We compare the results with some other methods.

A Unit-Consistent Error Measure for Multibody Systems With Unilateral Constraints

Modeling multibody systems subject to unilateral contacts and friction efficiently is challenging, and dynamic formulations based on the mixed linear complementarity problem (MLCP) are commonly used for this purpose. The accuracy of the MLCP solution method can be evaluated by determining the error introduced by it. In this paper, we find that commonly used MLCP error measures suffer from unit inconsistency leading to the error lacking any physical meaning. We propose a unit-consistent error measure, which computes energy error components for each constraint dependent on the inverse effective mass and compliance. It is shown by means of a simple example that the unit consistency issue does not occur using this proposed error measure. Simulation results confirm that the error decreases with convergence toward the solution. If a pivoting algorithm does not find a solution of the MLCP due to an iteration limit, e.g., in real-time simulations, choosing the result with the least error can reduce the risk of simulation instabilities and deviation from the reference trajectory.

Nonconvex Equilibrium Models for Gas Market Analysis: Failure of Standard Techniques and Alternative Modeling Approaches

This paper provides a first approach to assess gas market interaction on a network with nonconvex flow models. In the simplest possible setup that adequately reflects gas transport and market interaction, we elaborate on the relation of the solution of a simultaneous competitive gas market game, its corresponding mixed nonlinear complementarity problem (MNCP), and a first-best benchmark. We provide conditions under which the solution of the simultaneous game is also the solution of the corresponding MNCP. However, equilibria cannot be determined by the MNCP as the transmission system operator's (TSO’s) first-order conditions are insufficient, which goes back to nonconvexities of the gas flow model. This also implies that the welfare maximization problem may have multiple solutions that sometimes do not even coincide with any of the market equilibria. Our analysis shows that, even in the absence of strategic firms, market interaction fails to implement desirable outcomes from a welfare perspective due to the TSO’s incentive structure. We conclude that the technical environment calls for a market design that commits the TSO to a welfare objective through regulation and propose a design where the market solution corresponds to a welfare maximum and vice versa.

A complementarity approach for the computation of periodic oscillations in piecewise linear systems

Piecewise linear (PWL) systems can exhibit quite complex behaviours. In this paper, the complementarity framework is used for computing periodic steady-state trajectories belonging to linear time-invariant systems with PWL, possibly set-valued, feedback relations. The computation of the periodic solutions is formulated in terms of a mixed quadratic complementarity problem. Suitable anchor equations are used as problem constraints in order to determine the unknown period and to fix the phase of the steady-state oscillation. The accuracy of the complementarity problem solution is shown through numerical investigations of stable and unstable oscillations exhibited by practical PWL systems: a neural oscillator, a deadzone feedback system, a stick–slip system, a repressilator and a relay feedback system.

A rolling horizon approach for stochastic mixed complementarity problems with endogenous learning: Application to natural gas markets

In this paper we present a new approach for solving energy market equilibria that is an extension of the classical Nash-Cournot approach. Specifically, besides allowing the market participants to decide on their own decision variables such as production, flows or the like, we allow them to compete in terms of adjusting the data in the problem such as scenario probabilities and costs, consistent with a dynamic, more realistic approach to these markets. Such a problem in its original form is very hard to solve given the product of terms involving decision-dependent data and the variables themselves. Moreover, in its more general form, the players can affect not only each others׳ objective functions but also the constraint sets of opponents making such a formulation a more complicated instance of generalized Nash problems. This new approach involves solving a sequence of stochastic mixed complementarity (MCP) problems where only partial foresight is used, i.e., a rolling horizon. Each stochastic MCP or roll, involves a look-ahead for a fixed number of time periods with learning on the part of the players to approximate the extended Nash paradigm. Such partial foresight stochastic MCPs also offer a realism advantage over more traditional perfect foresight formulations. Additionally, the rolling-horizon approach offers a computational advantage over scenario-reduction methods as is demonstrated with numerical tests on a natural gas market stochastic MCP. Lastly, we introduce a new concept, the Value of the Rolling Horizon (VoRH) to measure the closeness of different rolling horizon schemes to a perfect foresight benchmark and provide some numerical tests on it using a stylized natural gas market.

A damped semismooth Newton iterative method for solving mixed linear complementarity problems

In this paper, we propose a damped semismooth Newton iterative method for mixed linear complementarity problems. Under appropriate conditions, we prove that the method exhibits monotone and superlinear convergence properties.

The Equilibrium Model of Electricity Market with Consideration of the Pumped Storage GenCo

Pumped storage is the largest-capacity form of grid energy storage available. A multi-period oligopolistic model for analyzing the bidding strategies of pumped storage GenCo (PSG) is proposed in this paper. In the pumping periods, the pumped storage unit (PSU) is simulated as a special load. While in generating periods, PSU is treated as a normal generator. In this model, all GenCos are assumed to exercise Cournot strategies to maximize their own profits. The resulting equilibrium formulation is established in terms of a mixed linear complementarity problem. The purpose of this paper is to provide an efficient simulation tool for the PSG to determine its bidding strategy in an oligopolistic environment. The proposed model can also be used to study various factors that may impact PSG’s profit. Results of a six-bus test system are analyzed to illustrate the characteristics of the proposed model.

The Solution Set Characterization and Error Bound for the Extended Mixed Linear Complementarity Problem

For the extended mixed linear complementarity problem (EML CP), we first present the characterization of the solution set for the EMLCP. Based on this, its global error bound is also established under milder conditions. The results obtained in this paper can be taken as an extension for the classical linear complementarity problems.

Electricity market equilibrium of thermal and wind generating plants in emission trading environment

PurposeThe purpose of this paper is to investigate the gaming equilibrium among fossil‐fueled generation companies (GenCos), wind generation companies, the grid company and customers participating in an emission trading (ET) market and the day‐ahead electricity market.Design/methodology/approachThe complementarity method is used in this work to obtain the Nash equilibrium. By combining the Karush‐Kuhn‐Tucker (KKT) conditions of each kind of market participants with market clearing and consistency conditions, a mixed linear complementarity problem could be established.FindingsSimulation results show that: the enforcement of ET could increase the share of generation outputs of wind generation units, and decrease the emissions from fossil‐fueled generation units; the bilateral contracts between GenCos and customers could limit the ability of exercising market power by GenCos; and when the emissions allowances allocated by the government shrink, the price of emissions allowance will increase and as the result the dispatching order of fossil‐fueled generation units will change, and the shares of generation outputs from wind generation units and combined‐cycle gas turbines increase. However, it should be mentioned that because the cost of wind generation is still very high, the increase of the share from wind generation units in the electricity market should mainly rely on cost reduction rather than the enforcement of ET.Originality/valueThe original contribution and the value of this study lie in developing a model framework to explore the gaming equilibrium that thermal and wind generating plants both play in the emissions trading environment and electricity market.

A NOVEL RECURRENT NEURAL NETWORK FOR SOLVING MLCPs AND ITS APPLICATION TO LINEAR AND QUADRATIC PROGRAMMING PROBLEMS

In this paper, we present a recurrent neural network for solving mixed linear complementarity problems (MLCPs) with positive semi-definite matrices. The proposed neural network is derived based on an NCP function and has a low complexity respect to the other existing models. In theoretical and numerical aspects, global convergence of the proposed neural network is proved. As an application, we show that the proposed neural network can be used to solve linear and convex quadratic programming problems. The validity and transient behavior of the proposed neural network are demonstrated by using five numerical examples.

A damped semismooth Newton method for mixed linear complementarity problems

The mixed linear complementarity problem is reformulated as a non-smooth equation by the use of the median operator. On the basis of slant differentiability, a damped semismooth Newton method for solving the non-smooth equation is presented. Both monotone and finite termination properties of the proposed method are proved.

Weighted-Path-Following Interior-Point Algorithm to Monotone Mixed Linear Complementarity Problem

In this paper we propose a weighted-path-following interior-point algorithm to monotone mixed linear complementarity problem. The algorithm is based on a new technique for finding a class of search directions and the strategy of the central path. At each iteration, we only use full-Newton step. Finally, the currently best known iteration bound for the algorithm with a small-update method, namely, is derived, which is as good as the bound for the linear optimization analogue.

Mixed quasi complementarity problems in topological vector spaces

In this paper, we introduce and consider a new class of complementarity problems, which is called the mixed quasi complementarity problems in a topological vector space. We show that the mixed quasi complementarity problems are equivalent to the mixed quasi variational inequalities. Using the KKM mapping theorem, we study the existence of a solution of the mixed quasi variational inequalities and mixed quasi complementarity problems. Several special cases are also discussed. Results obtained in this paper can be viewed as extension and generalization of the previously known results.

Electricity market near-equilibrium under locational marginal pricing and minimum profit conditions

This paper provides a tool to determine the near-equilibrium of an electric energy market. This market works under locational marginal pricing, i.e., generating units and demand loads are paid and pay, respectively, the locational marginal prices corresponding to the nodes they are connected to. The near-equilibrium is defined as the energy transaction levels for which generating companies maximize their respective profits and consumption companies maximize their respective utilities. An independent system operator clears the market maximizing the social welfare. Conditions that ensure minimum profit for generating units can be included. However, these conditions may render a generating unit uncompetitive and expel it from the market. Demands are taken to be non-constant and values are determined as part of the solution. The near-equilibrium is obtained through the solution of a mixed-integer quadratic problem equivalent to a mixed linear complementarity problem that includes the minimum profit conditions. It is important to note that the near-equilibrium concept presented in this paper does not solve a market equilibrium when indivisibilities such as start up costs or the like are present. Lastly, we validate the proposed model on a case study using data from the IEEE Reliability Test System.

Applied general equilibrium, agricultural policy, environmental policy, mixed complementarity, endogenous technology

In this paper an applied general equilibrium AGE model written in mixed-complementarity format is used to analyse the effects of an increase in milk quota in the Netherlands when nitrogen emissions in agriculture are restricted. The model combines the strengths of AGE models and mathematical programming models, which allows economy-wide policy analyses when technology switches are allowed. Results show that a welfare gain can be achieved by increasing milk quota while keeping nitrate N emissions in agriculture at the same level. Under such a policy change inactive N-extensive technologies in dairy farming become active and partly replace N-intensive technologies. Output in other agricultural industries decreases

A Noninterior Continuation Method for Quadratic and Linear Programming

The noninterior point path-following algorithm presented by the authors in 1990 is specialized to the mixed linear complementarity problem and its special cases (quadratic and linear programming). The new algorithm is related to, but has several advantages over, the interior point path-following algorithms.