Mixed Nonlinear Complementarity Problem Research Articles

An interior penalty approach to a large-scale discretized obstacle problem with nonlinear constraints

We propose an interior penalty method to solve a nonlinear obstacle problem arising from the discretization of an infinite-dimensional optimization problem. An interior penalty equation is proposed to approximate the mixed nonlinear complementarity problem representing the Karush-Kuhn-Tucker conditions of the obstacle problem. We prove that the penalty equation is uniquely solvable and present a convergence analysis for the solution of the penalty equation when the problem is strictly convex. We also propose a Newton’s algorithm for solving the penalty equation. Numerical experiments are performed to demonstrate the convergence and usefulness of the method when it is used for the two non-trivial test problems.

Automatic schemes for including generator Q and transformer tap limits in the fast decoupled load flow method

This study proposes a modified formulation for the popular fast decoupled load flow method to facilitate automatic incorporation of reactive power flow related constraints, i.e. generator reactive power and in-phase transformer tap limits. This formulation has been developed by viewing each device constraint as a set of complementary conditions and adopting the complementarity framework to define one equation to represent this set. The original set of load flow equations is augmented with this set of new equations representing the device constraints and solved to get the adjusted solutions directly. This approach is a significant departure from all the earlier approaches. The earlier approaches generally adopt a two-stage scheme. In the first stage, an approximate unadjusted solution is obtained and in the next stage, adjustments are attempted by ad hoc schemes. The new formulation presented here permits handling both the reactive power related constraints simultaneously. However, it can also be used in situations where only one type of adjustment is needed. Two alternate algorithms in complementarity framework based on mixed complementarity problem and non-linear complementarity problem formulations are presented here. Simulation results are presented to provide evidence of the impressive performance of the new algorithms as compared with the existing ones.

A power penalty approach to a discretized obstacle problem with nonlinear constraints

A novel power penalty method is proposed to solve a nonlinear obstacle problem with nonlinear constraints arising from the discretization of an infinite-dimensional optimization problem. This approach is based on the formulation of a penalty equation approximating the mixed nonlinear complementarity problem arising from the Karush–Kuhn–Tucker conditions of the optimization problem. We show that the solution to the penalty equation converges to that of the complementarity problem with an exponential convergence rate depending on the parameters used in the penalty equation. Numerical experiments are performed to confirm the theoretical convergence rate established.

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.

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.

Solving a class of non-convex quadratic problems based on generalized KKT conditions and neurodynamic optimization technique

In this paper, based on a generalized Karush-Kuhn-Tucker (KKT) method a modified recurrent neural network model for a class of non-convex quadratic programming problems involving a so-called $Z$-matrix is proposed. The basic idea is to express the optimality condition as a mixed nonlinear complementarity problem. Then one may specify conditions for guaranteeing the global solutions of the original problem by using results from the S-lemma. This process is proved by building up a dynamic system from the optimality condition whose equilibrium point is exactly the solution of the mixed nonlinear complementarity problem. By the study of the resulting dynamic system it is shown that under given assumptions, steady states of the dynamic system are stable. Numerical simulations and comparisons with the other methods are presented to illustrate the efficiency of the practical technique that is proposed in this paper.

A penalty approach to a discretized double obstacle problem with derivative constraints

This work presents a penalty approach to a nonlinear optimization problem with linear box constraints arising from the discretization of an infinite-dimensional differential obstacle problem with bound constraints on derivatives. In this approach, we first propose a penalty equation approximating the mixed nonlinear complementarity problem representing the Karush---Kuhn---Tucker conditions of the optimization problem. We then show that the solution to the penalty equation converges to that of the complementarity problem with an exponential convergence rate depending on the parameters used in the equation. Numerical experiments, carried out on a non-trivial test problem to verify the theoretical finding, show that the computed rates of convergence match the theoretical ones well.

A penalty method for a finite-dimensional obstacle problem with derivative constraints

We propose a power penalty method for an obstacle problem arising from the discretization of an infinite-dimensional optimization problem involving differential operators in both its objective function and constraints. In this method we approximate the mixed nonlinear complementarity problem (NCP) arising from the KKT conditions of the discretized problem by a nonlinear penalty equation. We then show the solution to the penalty equation converges exponentially to that of the mixed NCP. Numerical results will be presented to demonstrate the theoretical convergence rates of the method.

A penalty method for a mixed nonlinear complementarity problem

We propose a power penalty method for a mixed nonlinear complementarity problem (MNCP) and show that the solution to the penalty equation converges to that of the MNCP exponentially as the penalty parameter approaches infinity, provided that the mapping involved in the MNCP is both continuous and ξ -monotone. Furthermore, a convergence theorem is established when the monotonicity assumption on the mapping is removed. To demonstrate the usefulness and the convergence rates of this method, we design a non-trivial test MNCP problem arising in shape-preserving bi-harmonic interpolation and apply our method to this test problem. The numerical results confirm our theoretical findings.

A mixed nonlinear complementarity technique for solving the dynamics of a dexterous manipulation system

The versatility of a robot to perform a task is limited principally by the flexibilityof its end-effector. In the last years, research has been focused on the development of a hand with several fingers since these devices are capable of manipulating and grasping objects of different forms. A dexterous manipulation system, composed of a robot hand with several fingers and an object that will be held or manipulated, could be modeled as a set of rigid bodies in contact. The dynamics of several rigid bodies in contact tries to predict the accelerations and forces at the contact points of the set of rigid bodies with Coulomb friction. The calculation of such forces allows us to determine if the contact is maintained or disappears and to plan a determined action. The equations that describe the problem form a system of differential algebraic equations. In this contribution the problem is reformulated as a mixed nonlinear complementarity problem (MNCP). Then, an optimization problem with box constraints associated to the MNCP is presented using an adequate merit function. Conditions about the equivalence between the problems are established. Finally, the optimization problem is solved using a robust and efficient algorithm. Encouraging numerical results are reported.

A Mixed Complementarity-Based Equilibrium Model of Natural Gas Markets

We present a new multiseasonal, multiyear, natural gas market equilibrium model based on the concept of a competitive equilibrium involving the market participants: producers, storage reservoir operators, peak gas operators, pipeline operators, marketers, and consumers. The first three classes are depicted as price-takers consistent with perfect competition. The pipeline operations are described with regulated tariffs, but also involve “congestion pricing” as a mechanism to allocate scarce pipeline capacity. The marketers are price-takers in all markets except in sales to consumers, in which they compete as Nash-Cournot players. Finally, consumers are described by demand curves for each of the four sectors: residential, commercial, industrial, and electric power. We show that the equilibrium model is an instance of a mixed nonlinear complementarity problem (NCP) and provide sufficient detail not generally seen in previous complementarity models of natural gas. The NCP formulation is derived from considering the Karush-Kuhn-Tucker optimality conditions of the optimization problems faced by these participants. Under mild conditions, we show that this NCP has a solution, and under additional reasonable conditions, we show that the market prices are unique. We also validate the model on a representative sample network with nine market participants and three seasons, using four scenarios.

Mixed Nonlinear Complementarity Problems via Nonlinear Optimization: Numerical Results on Multi-Rigid-Body Contact Problems with Friction

In this work we show that the mixed nonlinear complementarity problem may be formulated as an equivalent nonlinear bound-constrained optimization problem that preserves the smoothness of the original data. One may thus take advantage of existing codes for bound-constrained optimization. This approach is implemented and tested by means of an extensive set of numerical experiments, showing promising results. The mixed nonlinear complementarity problems considered in the tests arise from the discretization of a motion planning problem concerning a set of rigid 3D bodies in contact in the presence of friction. We solve the complementarity problem associated with a single time frame, thus calculating the contact forces and accelerations of the bodies involved.

Oligopolistic Equilibrium Analysis for Electricity Markets: A Nonlinear Complementarity Approach

Oligopolistic equilibrium models are widely used in electricity market analysis, among which the supply function equilibrium model has been chosen as the basis of many power market models. A nonlinear complementarity approach has been proposed in this paper to calculate the Nash supply function equilibrium for a bid-based-pool generation market with a dc transmission model. A mixed nonlinear complementarity problem (NCP) is presented by combining the Karush-Kuhn-Tucker conditions of all strategic generating firms. Using a special nonlinear complementarity function, the mixed NCP is reformulated as a set of nonlinear algebraic equations and thus can be solved by an inexact Levenberg-Marquardt algorithm. Numerical examples are presented to verify the effectiveness of the proposed method. The results show that the generating firms could exercise their market power by over-production under congestion, or by capacity withholding in case of power shortage. The approach developed in this paper provides an efficient way to the solution of large-scale, complicated equilibrium models for electricity markets.

Inexact SQP Interior Point Methods and Large Scale Optimal Control Problems

Optimal control problems with partial differential equations lead to large scale nonlinear optimization problems with constraints. An efficient solver which takes into account the structure and also the size of the problem is an inexact sequential quadratic programming method where the quadratic problems are solved iteratively. Based on a reformulation as a mixed nonlinear complementarity problem we give a measure of when to terminate the iterative quadratic program solver. For the latter we use an interior point algorithm. Under standard assumptions, local linear, superlinear, and quadratic convergence can be proved. The numerical application is an optimal control problem from nonlinear heat conduction.

A Hybrid Smoothing Method for Mixed Nonlinear ComplementarityProblems

In this paper, we describe a new, integral-based smoothing method for solving the mixed nonlinear complementarity problem (MNCP). This approach is based on recasting MNCP as finding the zero of a nonsmooth system and then generating iterates via two types of smooth approximations to this system. Under weak regularity conditions, we establish that the sequence of iterates converges to a solution if the limit point of this sequence is regular. In addition, we show that the rate is Q-linear, Q-superlinear, or Q-quadratic depending on the level of inexactness in the subproblem calculations and we make use of the inexact Newton theory of Dembo, Eisenstat, and Steihaug. Lastly, we demonstrate the viability of the proposed method by presenting the results of numerical tests on a variety of complementarity problems.

QPCOMP: A quadratic programming based solver for mixed complementarity problems

QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel [14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudo-monotone mixed nonlinear complementarity problem. QPCOMP also extends the NE/SQP method for the nonlinear complementarity problem to the more general mixed nonlinear complementarity problem. Computational results are provided, which demonstrate the effectiveness of the algorithm.

Properties of an Interior-Point Mapping for Mixed Complementarity Problems

Using a unified theory of local homeomorphic maps, we establish some basic properties of a fundamental mapping associated with the family of path-following interior-point methods for solving a mixed nonlinear complementarity problem.

Stability Analysis of Variational Inequalities and Nonlinear Complementarity Problems, via the Mixed Linear Complementarity Problem and Degree Theory

This paper is concerned with the mixed linear complementarity problem and the role it and its variants play in the stability analysis of the nonlinear complementarity problem and the Karush-Kuhn-Tucker system of a variational inequality problem. Under a nonsingular assumption, the mixed linear complementarity problem can be converted to the standard problem; in this case, the rich theory of the latter can be directly applied to the former. In this work, we employ degree theory to derive some sufficient conditions for the existence of a solution to the mixed linear complementarity problem in the absence of the nonsingularity property. Next, we extend this existence theory to the mixed nonlinear complementarity problem and establish a main stability result under a certain degree-theoretic assumption concerning the linearized problem. We then specialize this stability result and its consequences to the parametric variational inequality problem under the assumption of a unique set of multipliers. Finally, we consider the latter problem with the uniqueness assumption of the multipliers replaced by a convexity assumption and obtain stability results under some weak second-order conditions. In addition to the new existence results for the mixed linear complementarity problem, the main contributions of this paper in the stability category are the following: a resolution to a conjecture concerning the local solvability of a parametric variational inequality, the use of the generalized linear complementarity problem as a tool to broaden the second-order conditions, the characterization of the solution stability of the linear complementarity problem and the affine variational inequality problem in terms of the solution isolatedness under some weak hypotheses, and various stability theorems under some weak second-order conditions.