The Monotone Extended Second-Order Cone and Mixed Complementarity Problems

In this paper, we study a new generalization of the Lorentz cone $$\mathcal{L}^n_+$$ L + n , called the monotone extended second-order cone (MESOC). We investigate basic properties of MESOC including computation of its Lyapunov rank and proving its reducibility. Moreover, we show that in an ambient space, a cylinder is an isotonic projection set with respect to MESOC. We also examine a nonlinear complementarity problem on a cylinder, which is equivalent to a suitable mixed complementarity problem, and provide a computational example illustrating applicability of MESOC.

A Modified Nonsmooth Levenberg–Marquardt Algorithm for the General Mixed Complementarity Problem

As is well known, the mixed complementarity problem is equivalent to a nonsmooth equation by using a median function. By investigating the generalized Jacobi of a composite vector-valued maximum function, a nonsmooth Levenberg–Marquardt algorithm is proposed in this paper. In the present algorithm, we adopt a new LM parameter form and discuss the local convergence rate under the local error bound condition, which is weaker than nonsingularity. Finally, the numerical experiments and the application for the real-time pricing in smart grid illustrate the effectiveness of the algorithm.

Complementarity Modeling of a Ramsey-Type Equilibrium Problem with Heterogeneous Agents

We contribute to the field of Ramsey-type equilibrium models with heterogeneous agents. To this end, we state such a model in a time-continuous and time-discrete form, which in the latter case leads to a finite-dimensional mixed complementarity problem. We prove the existence of solutions of the latter problem using the theory of variational inequalities and present further properties of its solutions. Finally, we compute the growth dynamics in a calibrated model in which households differ with respect to their relative risk aversion, their discount factors, their initial wealth, and with respect to their interest rates on savings.

$$\Gamma $$-Robust electricity market equilibrium models with transmission and generation investments

We consider uncertain robust electricity market equilibrium problems including transmission and generation investments. Electricity market equilibrium modeling has a long tradition but is, in most of the cases, applied in a deterministic setting in which all data of the model are known. Whereas there exist some literature on stochastic equilibrium problems, the field of robust equilibrium models is still in its infancy. We contribute to this new field of research by considering $$\Gamma $$ Γ -robust electricity market equilibrium models on lossless DC networks with transmission and generation investments. We state the nominal market equilibrium problem as a mixed complementarity problem as well as its variational inequality and welfare optimization counterparts. For the latter, we then derive a $$\Gamma $$ Γ -robust formulation and show that it is indeed the counterpart of a market equilibrium problem with robustified player problems. Finally, we present two case studies to gain insights into the general effects of robustification on electricity market models. In particular, our case studies reveal that the transmission system operator tends to act more risk-neutral in the robust setting, whereas generating firms clearly behave more risk-averse.

The Impacts of Soy-Based Biodiesel on the Main Soy Producers in the International Market

This study analyzes the impacts of soy-based biodiesel consumption on the main soybean-producing countries. To achieve this goal, a partial equilibrium model formulated as a Mixed Complementarity Problem (PCM) is used, which allows the inclusion of trade barriers and the construction of alternative scenarios. Four scenarios are analyzed. The first simulates a demand shock for soybean meal and soybean oil, considering the observed growth in demand over the last ten years. The second evaluates the effects of an increase in the demand for meal, oil and biodiesel. The third considers the elimination of the demand for soy-based biodiesel. The last scenario assesses the effects of the elimination of tariffs and subsidies on the grain and food markets. In general terms, the results show that: in the second scenario, Brazil is the only exporting country that shows a greater increase in consumption than in production, reflecting the country’s smaller share in the world soybean trade; in the fourth scenario the gains are relatively smaller than the potential gains in the first scenario; in the third scenario, soybean prices are about 8% higher because of the use of soybeans for biodiesel production, demonstrating the relevance of mandatory policies in this market.

Numerical modeling and assessment of natural gas pipeline separation in China: the data from Henan Province

China’s natural gas market is focusing on price reform and aims to reconstruct vertically integrated industrial chains in the future. Based on the mixed complementarity problem model of gas markets with nodes in Henan Province, China, as an example, this paper applies numerical modeling to simulate the effects of social welfare and equilibrium prices on nodes in two scenarios: pipeline integration and pipeline separation. The findings reveal the following: (1) Pipeline separation yields greater overall social welfare than pipeline integration, with the welfare shifting from gas producers to consumption markets. (2) Pipeline separation lowers the equilibrium consumption prices by driving competition among gas supply sources. (3) Pipeline separation will increase the contribution of natural gas to primary energy.

Rigid Body Dynamic Simulation With Multiple Convex Contact Patches

We present a principled method for dynamic simulation of rigid bodies in intermittent contact with each other where the contact is assumed to be a non-convex contact patch that can be modeled as a union of convex patches. The prevalent assumption in simulating rigid bodies undergoing intermittent contact with each other is that the contact is a point contact. In recent work, we introduced an approach to simulate contacting rigid bodies with convex contact patches (line and surface contact). In this paper, for non-convex contact patches modeled as a union of convex patches, we formulate a discrete-time mixed complementarity problem where we solve the contact detection and integration of the equations of motion simultaneously. Thus, our method is a geometrically-implicit method and we prove that in our formulation, there is no artificial penetration between the contacting rigid bodies. We solve for the equivalent contact point (ECP) and contact impulse of each contact patch simultaneously along with the state, i.e., configuration and velocity of the objects. We provide empirical evidence to show that if the number of contact patches between two objects is less than or equal to three, the state evolution of the bodies is unique, although the contact impulses and ECP may not be unique. We also present simulation results showing that our method can seamlessly capture transition between different contact modes like non-convex patch to point (or line contact) and vice-versa during simulation.

The supply chain of Brazilian maize and soybeans: the effects of segregation on logistics and competitiveness

Despite the significant advances of Brazilian agriculture, transportation and storage costs still constitute the main barriers to the Brazilian agribusiness. The aim of this article is to analyze the effect of segregation of maize and soybeans in the Brazilian transport and storage logistics, especially genetically modified grains. In the context of the guidelines of the Cartagena Protocol on Biosafety (CPB) as well as of the competitiveness in the international market, we develop a spatial equilibrium model in the form of a mixed complementarity problem. The competitiveness of Brazilian maize and soybeans on the international market is compromised by the inefficient logistics and slow responses to the demands of the CPB. The contribution of the paper is to evaluate how regulatory issues of a segment, in this case biotechnology, may interfere with logistic infrastructure projects.

A Fast Algorithm for Contact Dynamics of Multibody Systems Using the Box Friction Model

One of the major challenges in dynamics of multibody systems is to handle redundant constraints appropriately. The box friction model is one of the existing approaches to formulate the contact and friction phenomenon as a mixed linear complementarity problem (MLCP). In this setting, the contact redundancy can be handled by relaxing the constraints, but such a technique might suffer from certain drawbacks, specially in the case of large number of redundant constraints. Most of the common pivoting algorithms used to solve the resulting mixed complementarity problem might not converge when the relaxation terms are chosen as small as they should be. To overcome the aforementioned shortcoming, we propose a novel approach which takes advantage of the sparse structure of the formulated MLCP. This novel approach reduces the sensitivity of the solution of the problem to the relaxation terms and decreases the number of required pivots to obtain the solution, leading to shorter computational times. Furthermore, as a result of the proposed approach, much smaller relaxation terms can be used while the solution algorithms converge.