Nonlinear Complementarity Constraints Research Articles

A posteriori error estimates for a compositional two-phase flow with nonlinear complementarity constraints

In this work, we develop an a-posteriori-steered algorithm for a compositional two-phase flow with exchange of components between the phases in porous media. As a model problem, we choose the two-phase liquid-gas flow with appearance and disappearance of the gas phase formulated as a system of nonlinear evolutive partial differential equations with nonlinear complementarity constraints. The discretization of our model is based on the backward Euler scheme in time and the finite volume scheme in space. The resulting nonlinear system is solved via an inexact semismooth Newton method. The key ingredient for the a posteriori analysis are the discretization, linearization, and algebraic flux reconstructions allowing to devise estimators for each error component. These enable to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver whenever the corresponding error components do not affect significantly the overall error. Numerical experiments are performed using the Newton-min algorithm as well as the Newton-Fischer-Burmeister algorithm in combination with the GMRES iterative linear solver to show the efficiency of the proposed adaptive method.

Competition and regulation in a new integrated transit system across jurisdictional borders

SummaryIt is often argued lately that some regional transit systems should be integrated to improve the transit services across jurisdictional borders. One of the important issues concerning this integrated transit system is how to make proper use of the competition and regulation to improve its performance and to increase relevant benefits to the authorities, the operators and the passengers. This paper deals with this problem in a new integrated transit system across jurisdictional borders. An optimization model is used to investigate the competition of transit operators in oligopolistic competition transit market (OCTM). An optimization model with nonlinear complementarity constraints is proposed to guide the competition of transit operators using transit subsidy as the regulation means in system optimum transit system (SOTM). By analyzing and comparing the performances of the integrated transit system in two markets, numerical studies show that the competition mechanism should be made full use of and transit subsidy is an important regulation means that should be valued by the transit authorities in the integrated transit system. Copyright © 2016 John Wiley & Sons, Ltd.

Tikhonov regularization for mathematical programs with generalized complementarity constraints

In this paper, in order to solve mathematical programs with generalized nonlinear comple-mentarity constraints, we present a new regularization method of Tikhonov type, based on regularizing an optimization problem with equality constraints. The stability and convergence of the regularized solutions are considered. Numerical examples are given for illustration.

An Implicit Smooth Conjugate Projection Gradient Algorithm for Optimization with Nonlinear Complementarity Constraints

This paper discusses a special class of mathematical programs with equilibrium constraints. At first, by using a generalized complementarity function, the discussed problem is transformed into a family of general nonlinear optimization problems containing additional variable μ. Furthermore, combining the idea of penalty function, an auxiliary problem with inequality constraints is presented. And then, by providing explicit searching direction, we establish a new conjugate projection gradient method for optimization with nonlinear complementarity constraints. Under some suitable conditions, the proposed method is proved to possess global and superlinear convergence rate.

Gas phase appearance and disappearance as a problem with complementarity constraints

The modeling of migration of hydrogen produced by the corrosion of the nuclear waste packages in an underground storage including the dissolution of hydrogen involves a set of nonlinear partial differential equations with nonlinear complementarity constraints. This article shows how to apply a modern and efficient solution strategy, the Newton-min method, to this geoscience problem and investigates its applicability and efficiency. In particular, numerical experiments show that the Newton-min method is quadratically convergent for this problem.

Superlinear Convergence of a Smooth Approximation Method for Mathematical Programs with Nonlinear Complementarity Constraints

Superlinear Convergence of a Smooth Approximation Method for Mathematical Programs with Nonlinear Complementarity Constraints

An SQP algorithm for mathematical programs with nonlinear complementarity constraints

In this paper, we describe a successive approximation and smooth sequential quadratic programming (SQP) method for mathematical programs with nonlinear complementarity constraints (MPCC). We introduce a class of smooth programs to approximate the MPCC. Using an l1 penalty function, the line search assures global convergence, while the superlinear convergence rate is shown under the strictly complementary and second-order sufficient conditions. Moreover, we prove that the current iterated point is an exact stationary point of the mathematical programs with equilibrium constraints (MPEC) when the algorithm terminates finitely.

Multi-period transportation network design under demand uncertainty

The ability to make optimal transportation network investments decision is central to the strategic management of transportation systems. The presence of uncertainty in transportation systems presents new challenges in making optimal network investment decisions. In this paper, we develop a multi time period network design problem considering both demand uncertainty and demand elasticity. Such an approach affords the planner the flexibility to delay, change, or even abandon the future network investment. We measure the flexibility of investing over multiple time periods as compared to a single-stage network design decision. Initially, we provide a taxonomy and define many dimensions of transportation network flexibility. This is followed with the development of a flexible network design formulation (FNDP), in which the investment is staged over multiple time periods. The demand is assumed to be separable and the demand elasticity is captured using a negative exponential distribution. We develop the FNDP formulation as bilevel stochastic mathematical programming with complementarity constraints (STOCH-MPEC) in which the bi-level formulation is converted to a single level using non-linear complementarity constraints conditions for user equilibrium (UE) problem. The formulation is implemented on two test networks and the results show the benefits of FNDP over single-stage NDP—measured in terms of increase in present expected system consumer surplus (PESCS)—are in the range of 10–30%. The results clearly demonstrate that under demand uncertainty there are potential benefits of introducing flexibility in investment decisions. Finally, we conduct a sensitivity analysis of FNDP with different budget values and it is observed that certain paradoxical sharp corners are observed at certain budget values.

Electricity market equilibrium analysis based on nonlinear interior point algorithm with complementarity constraints

This paper presents an interior point algorithm based on a.c. network model for determining the Nash supply function equilibrium (SFE) of bid-based electricity markets. The SFE problem is considered as a bi-level game. At the first level, the problem begins with the formulation of an optimal power flow (OPF) interior point-based algorithm to handle the independent system operator (ISO) problem for maximising social welfare. This algorithm is based on the OPF with a.c. network transmission model taking into account all the operating aspects such as the generation capacity limits, bus voltage limits, transmission line constraints, network losses and especially the effect of the reactive power. The resulting Karush-Kuhn-Tucker (KKT) conditions of the problem at the first level are then reformulated using nonlinear complementarity constraints and incorporated as equality constraints in the second-level formulation for maximising the individual profit for each strategic generating firm. By employing a special nonlinear complementarity function, the complementarity constraints are then transformed into nonlinear algebraic expressions, thus the KKT conditions of the resulting combined problem can be derived. The final problem is then solved iteratively based on the solution techniques of the interior point algorithm. Numerical examples of a three-bus system, the IEEE 14-bus system and the IEEE 30-bus system, show that the algorithm can successfully determine the electricity market equilibrium with the a.c. network model.

A Superlinearly Convergent Implicit Smooth SQP Algorithm for Mathematical Programs with Nonlinear Complementarity Constraints

This paper discusses a special class of mathematical programs with nonlinear complementarity constraints, its goal is to present a globally and superlinearly convergent algorithm for the discussed problems. We first reformulate the complementarity constraints as a standard nonlinear equality and inequality constraints by making use of a class of generalized smoothing complementarity functions, then present a new SQP algorithm for the discussed problems. At each iteration, with the help of a pivoting operation, a master search direction is yielded by solving a quadratic program, and a correction search direction for avoiding the Maratos effect is generated by an explicit formula. Under suitable assumptions, without the strict complementarity on the upper-level inequality constraints, the proposed algorithm converges globally to a B-stationary point of the problems, and its convergence rate is superlinear.

Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints

Mathematical programs with nonlinear complementarity constraints are reformulated using better posed but nonsmooth constraints. We introduce a class of functions, parameterized by a real scalar, to approximate these nonsmooth problems by smooth nonlinear programs. This smoothing procedure has the extra benefits that it often improves the prospect of feasibility and stability of the constraints of the associated nonlinear programs and their quadratic approximations. We present two globally convergent algorithms based on sequential quadratic programming (SQP) as applied in exact penalty methods for nonlinear programs. Global convergence of the implicit smooth SQP method depends on existence of a lower-level nondegenerate (strictly complementary) limit point of the iteration sequence. Global convergence of the explicit smooth SQP method depends on a weaker property, i.e., existence of a limit point at which a generalized constraint qualification holds. We also discuss some practical matters relating to computer implementations.

Solving two-Level variational inequality

An approach to solving a mathematical program with variational inequality or nonlinear complementarity constraints is presented. It consists in a variational re-formulation of the optimization criterion and looking for a solution of thus obtained variational inequality among the points satisfying the initial variational constraints.

On Optimization Problems with Variational Inequality Constraints

This paper is devoted to a class of optimization problems that contain variational inequality or nonlinear complementarity constraints. Problems of this kind arise, for example, in game theory, bilevel programming, and the design of networks subject to equilibrium conditions. For such problems first-order necessary optimality conditions are derived and a numerical approach is proposed, based on nondifferentiable optimization techniques. This approach is illustrated by simple examples.