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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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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.
Mathematical programs with complementarity constraints: Convergence properties of a smoothing method
 Mathematics of Operations Research
, 2007
"... Abstract. In the present paper, optimization problems P with complementarity constraints are considered. Characterizations for local minimizers x of P of order one and two are presented. We analyze a parametric smoothing approach for solving these programs in which P is replaced by a perturbed probl ..."
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Abstract. In the present paper, optimization problems P with complementarity constraints are considered. Characterizations for local minimizers x of P of order one and two are presented. We analyze a parametric smoothing approach for solving these programs in which P is replaced by a perturbed problem P fi depending on a (small) parameter fi. We are interested in the convergence behavior of the feasible set F fi and the convergence of the solutions x fi of P fi for fi → 0. In particular, it is shown, that under generic assumptions the solutions x fi are unique and converge to a solution x of P with a rate O. fi/. Moreover, the convergence for the Hausdorff distance d.F fi;F / between the feasible sets of P fi and P is of order O. fi/.
EXTENDING MODELING SYSTEMS: STRUCTURE AND SOLUTION
"... Keywords Some extensions of a modeling system are described that facilitate higher level structure identification within a model. These structures, which often involve complementarity relationships, can be exploited by modern large scale mathematical programming algorithms. A specific implementation ..."
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Keywords Some extensions of a modeling system are described that facilitate higher level structure identification within a model. These structures, which often involve complementarity relationships, can be exploited by modern large scale mathematical programming algorithms. A specific implementation of an extended mathematical programming tool is outlined that communicates structure in a computationally beneficial manner from the GAMS modeling system to an appropriate solver. Modeling, complementarity, nonlinear programming, variational inequalities
MPEC Problem Formulations in Chemical Engineering Applications
, 2007
"... With the development and widespread use of largescale nonlinear programming (NLP) tools for process optimization, there has been an associated application of NLP formulations with complementarity constraints in order to represent discrete decisions. Also known as Mathematical Programs with Equil ..."
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With the development and widespread use of largescale nonlinear programming (NLP) tools for process optimization, there has been an associated application of NLP formulations with complementarity constraints in order to represent discrete decisions. Also known as Mathematical Programs with Equilibrium Constraints (MPECs), these formulations can be used to model certain classes of discrete events and can be more efficient than a mixed integer formulation. However, MPEC formulations and solution strategies are not yet fully developed in process engineering. In this study, we discuss MPEC properties, including concepts of stationarity and linear independence that are essential for welldefined NLP formulations. Nonlinear programming based solution strategies for MPECs are then reviewed and examples of complementarity drawn from chemical engineering applications are presented to illustrate the effectiveness of these formulations. 1
TITLE: Dynamic Optimization Formulations for Plant Operation Under Partial Shutdown Conditions
"... iii Our research focuses on the development of systematic strategies and formulations for the optimal operation of plants under partial shutdown conditions. A partial shutdown is a type of circumscribed plant unit shutdown that permits the rest of the plant to continue operating to some degree. The ..."
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iii Our research focuses on the development of systematic strategies and formulations for the optimal operation of plants under partial shutdown conditions. A partial shutdown is a type of circumscribed plant unit shutdown that permits the rest of the plant to continue operating to some degree. The goal of a partial shutdown strategy is to be able to manipulate the available degreesoffreedom in a plant—during and after a shutdown—such that production is restored in a cost optimal fashion while meeting all safety and operational constraints. This can be accomplished through adjustments of production rates, recycles and buffer levels. In order to compute these adjustments and to arrive at an openloop advisory policy, we solve a differentialalgebraicequation (DAE) based dynamic optimization problem containing a model of the plant. Within this broad framework, we take a novel multitiered dynamic optimization approach that allows us to prioritize multiple competing objectives and specify the tradeoffs from one tier to the next. The control trajectories from the solution of the dynamic optimization problem can then be used to inform the formulation of an inventory management policy for
Cournot Equilibrium in Twosettlement Electricity Markets: Formulation and Computation
, 2006
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THREE ESSAYS ON DECISION MAKING UNDER UNCERTAINTY IN ELECTRIC POWER SYSTEMS
, 2007
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ON THE CALCULATION OF NASH EQUILIBRIUM POINTS WITH THE AID OF THE SMOOTHING APPROACH
"... Let us consider a game with n players where the set of possible strategies depend on the decision of the other player. In this case, if the players behave rationally the solution is a point of generalized Nash equilibrium (GNE). These points can be obtained as solutions of a special class of bilevel ..."
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Let us consider a game with n players where the set of possible strategies depend on the decision of the other player. In this case, if the players behave rationally the solution is a point of generalized Nash equilibrium (GNE). These points can be obtained as solutions of a special class of bilevel programs. In this work, the bilevel problem is substituted by a simpler model which can be solved by the so called smoothing approach for mathematical programs with complementarity constraints. We discuss if the hypothesis for the convergence of this method are generically fulfilled or not.
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, 2008
"... A new class of smoothing methods for mathematical programs with equilibrium constraints ..."
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A new class of smoothing methods for mathematical programs with equilibrium constraints