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30
Interior methods for mathematical programs with complementarity constraints
 SIAM J. Optim
, 2004
"... This paper studies theoretical and practical properties of interiorpenalty methods for mathematical programs with complementarity constraints. A framework for implementing these methods is presented, and the need for adaptive penalty update strategies is motivated with examples. The algorithm is sh ..."
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Cited by 32 (10 self)
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This paper studies theoretical and practical properties of interiorpenalty methods for mathematical programs with complementarity constraints. A framework for implementing these methods is presented, and the need for adaptive penalty update strategies is motivated with examples. The algorithm is shown to be globally convergent to strongly stationary points, under standard assumptions. These results are then extended to an interiorrelaxation approach. Superlinear convergence to strongly stationary points is also established. Two strategies for updating the penalty parameter are proposed, and their efficiency and robustness are studied on an extensive collection of test problems.
Stochastic mathematical programs with equilibrium constraints, modeling and . . .
 SCHOOL OF INDUSTRIAL AND SYSTEM ENGINEERING, GEORGIA INSTITUTE OF TECHNOLOGY
, 2005
"... In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both – the lower level equilibrium solution and objective integrand. We sho ..."
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Cited by 28 (6 self)
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In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both – the lower level equilibrium solution and objective integrand. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized KarushKuhnTucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally we present some preliminary numerical test results.
The impact of plugin hybrid electric vehicles on distribution networks: a review and outlook, “Renewable and Sustainable Energy Reviews
, 2011
"... This paper analyzes the potential impacts of plugin hybrid electric vehicles (PHEVs) on locational marginal prices (LMPs). PHEVs are the next generation of hybrid electric vehicles with batteries that can be recharged by plugging into a standard electric power outlet. On the one hand, PHEVs produce ..."
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Cited by 20 (0 self)
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This paper analyzes the potential impacts of plugin hybrid electric vehicles (PHEVs) on locational marginal prices (LMPs). PHEVs are the next generation of hybrid electric vehicles with batteries that can be recharged by plugging into a standard electric power outlet. On the one hand, PHEVs produce less emission, have higher mileage, and reduce dependency on foreign supply of oil. On the other hand, economical and technical obstacles still exist, and the potential impacts on both transportation and electric power systems need to be studied. Earlier studies have examined the potential impacts on nonLMPbased electricity markets. This paper analyzes how wholesale electricity prices in LMPbased electricity markets would be affected by PHEV penetration. Simulation results from a PJM fivebus test example show that, if the electricity load increases by 20 % due to recharging PHEVs, the loadweighted mean and standard deviation of LMPs would increase by more than 70 % and 140%, respectively. The effects of battery stations that provide hotswap service are also studied. If the load increases by 20%, by taking advantage of the spatial price differences and shipping batteries between different locations, the battery stations could recharge the batteries at 63 % of what it would cost for PHEV drivers to recharge in home garages. At the same time, the mean of LMPs would only increase by about 22 % and the standard deviation would even decrease. 1
Elasticmode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties
 Math. Program
, 2005
"... Abstract. The elasticmode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first and secondorder necessary optimality conditions for the ..."
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Cited by 18 (2 self)
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Abstract. The elasticmode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first and secondorder necessary optimality conditions for the original problem are also first and secondorder points of the elasticmode formulation. Here we study global convergence properties of methods based on this formulation, which involve generating an (exact or inexact) first or secondorder point of the formulation, for nondecreasing values of the penalty parameter. Under certain regularity conditions on the active constraints, we establish finite or asymptotic convergence to points having a certain stationarity property (such as strong stationarity, Mstationarity, or Cstationarity). Numerical experience with these approaches is discussed. In particular, our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elasticmode formulation is solved inexactly. Key words. Nonlinear programming, equilibrium constraints, complementarity constraints, elasticmode formulation, strong stationarity, Cstationarity, Mstationarity. AMS subject classifications 49M30, 49M37, 65K05, 90C30, 90C33 1.
Modeling and computing twosettlement oligopolistic equilibrium in a congested electricity network
 Operations Research
, 2008
"... A model of twosettlement electricity markets is introduced, which accounts for flow congestion, demand uncertainty, system contingencies and market power. We formulate the subgame perfect Nash equilibrium for this model as an equilibrium problem with equilibrium constraints (EPEC), in which each fi ..."
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Cited by 18 (2 self)
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A model of twosettlement electricity markets is introduced, which accounts for flow congestion, demand uncertainty, system contingencies and market power. We formulate the subgame perfect Nash equilibrium for this model as an equilibrium problem with equilibrium constraints (EPEC), in which each firm solves a mathematical program with equilibrium constraints (MPEC). The model assumes linear demand functions, quadratic generation cost functions and a lossless DC network, resulting in equilibrium constraints as a parametric linear complementarity problem (LCP). We introduce an iterative procedure for solving this EPEC through repeated application of an MPEC algorithm. This MPEC algorithm is based on solving quadratic programming subproblems and on parametric LCP pivoting. Numerical examples demonstrate the effectiveness of the MPEC and EPEC algorithms and the tractability of the model for realistic size power systems. 1
On the Global Solution of Linear Programs with Linear Complementarity Constraints
, 2007
"... This paper presents a parameterfree integerprogramming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPEC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three ..."
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Cited by 15 (4 self)
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This paper presents a parameterfree integerprogramming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPEC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three outcomes—infeasibility, unboundedness, or solvability—of an LPEC. An extreme point/ray generation scheme in the spirit of Benders decomposition is developed, from which valid inequalities in the form of satisfiability constraints are obtained. The feasibility problem of these inequalities and the carefully guided linear programming relaxations of the LPEC are the workhorse of the algorithm, which also employs a specialized procedure for the sparsification of the satifiability cuts. We establish the finite termination of the algorithm and report computational results using the algorithm for solving randomly generated LPECs of reasonable sizes. The results establish that the algorithm can handle infeasible, unbounded, and solvable LPECs effectively.
Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints
, 2009
"... We consider a reformulation of mathematical programs with complementarity constraints, where by introducing an artificial variable the constraints are converted into equalities which are once but not twice differentiable. We show that the Lagrange optimality system of such a reformulation is semismo ..."
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Cited by 6 (4 self)
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We consider a reformulation of mathematical programs with complementarity constraints, where by introducing an artificial variable the constraints are converted into equalities which are once but not twice differentiable. We show that the Lagrange optimality system of such a reformulation is semismooth and BDregular at the solution under reasonable assumptions. Thus, fast local convergence can be obtained by applying the semismooth Newton method. Moreover, it turns out that the squared residual of the Lagrange system is continuously differentiable (even though the system itself is not), which opens the way for a natural globalization of the local algorithm.
Mathematical programs with vanishing constraints: Optimality conditions, sensitivity, and a relaxation method
 J. Optim. Theory Appl
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An Extended Mathematical Programming Framework
, 2009
"... Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to ..."
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Cited by 4 (1 self)
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Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to facilitate higher level structure identification. The structures, which often involve constraints on the solution sets of other models or complementarity relationships, can be exploited by modern large scale mathematical programming algorithms for efficient solution. A specific implementation of this framework is outlined that communicates structure from the GAMS modeling system to appropriate solvers in a computationally beneficial manner. Example applications are taken from chemical engineering.
Multiplier convergence in trustregion methods with application to convergence of decomposition methods for MPECs
 Math. Program
"... Abstract. We study piecewise decomposition methods for mathematical programs with equilibrium constraints (MPECs) for which all constraint functions are linear. At each iteration of a decomposition method, one step of a nonlinear programming scheme is applied to one piece of the MPEC to obtain the n ..."
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Cited by 3 (1 self)
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Abstract. We study piecewise decomposition methods for mathematical programs with equilibrium constraints (MPECs) for which all constraint functions are linear. At each iteration of a decomposition method, one step of a nonlinear programming scheme is applied to one piece of the MPEC to obtain the next iterate. Our goal is to understand global convergence to Bstationary points of these methods when the embedded nonlinear programming solver is a trustregion scheme, and the selection of pieces is determined using multipliers generated by solving the trustregion subproblem. To this end we study global convergence of a linear trustregion scheme for linearlyconstrained NLPs that we call a trustsearch method. The trustsearch has two features that are critical to global convergence of decomposition methods for MPECs: a robustness property with respect to switching pieces, and a multiplier convergence result that appears to be quite new for trustregion methods. These combine to clarify and strengthen global convergence of decomposition methods without resorting either to additional conditions such as eventual inactivity of the trustregion constraint, or more complex methods that require a separate subproblem for multiplier estimation.