1,250 research outputs found
Strong Stationarity Conditions for Optimal Control of Hybrid Systems
We present necessary and sufficient optimality conditions for finite time
optimal control problems for a class of hybrid systems described by linear
complementarity models. Although these optimal control problems are difficult
in general due to the presence of complementarity constraints, we provide a set
of structural assumptions ensuring that the tangent cone of the constraints
possesses geometric regularity properties. These imply that the classical
Karush-Kuhn-Tucker conditions of nonlinear programming theory are both
necessary and sufficient for local optimality, which is not the case for
general mathematical programs with complementarity constraints. We also present
sufficient conditions for global optimality.
We proceed to show that the dynamics of every continuous piecewise affine
system can be written as the optimizer of a mathematical program which results
in a linear complementarity model satisfying our structural assumptions. Hence,
our stationarity results apply to a large class of hybrid systems with
piecewise affine dynamics. We present simulation results showing the
substantial benefits possible from using a nonlinear programming approach to
the optimal control problem with complementarity constraints instead of a more
traditional mixed-integer formulation.Comment: 30 pages, 4 figure
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.Comment: Accepted for Publication in Mathematics and Computers in Simulation.
Available online 6 August 2013, Mathematics and Computers in Simulation
(2013
Mathematical programs with complementarity constraints: convergence properties of a smoothing method
In this paper, optimization problems with complementarity constraints are considered. Characterizations for local minimizers of of Orders 1 and 2 are presented. We analyze a parametric smoothing approach for solving these programs in which is replaced by a perturbed problem depending on a (small) parameter . We are interested in the convergence behavior of the feasible set and the convergence of the solutions of for In particular, it is shown that, under generic assumptions, the solutions are unique and converge to a solution of with a rate . Moreover, the convergence for the Hausdorff distance , between the feasible sets of and is of order
Simulation-based solution of stochastic mathematical programs with complementarity constraints: Sample-path analysis
We consider a class of stochastic mathematical programs with complementarity constraints, in which both the objective and the constraints involve limit functions or expectations that need to be estimated or approximated. Such programs can be used for modeling \\average" or steady-state behavior of complex stochastic systems. Recently, simulation-based methods have been successfully used for solving challenging stochastic optimization problems and equilibrium models. Here we broaden the applicability of so-called the sample-path method to include the solution of certain stochastic mathematical programs with equilibrium constraints. The convergence analysis of sample-path methods rely heavily on stability conditions. We first review necessary sensitivity results, then describe the method, and provide sufficient conditions for its almost-sure convergence. Alongside we provide a complementary sensitivity result for the corresponding deterministic problems. In addition, we also provide a unifying discussion on alternative set of sufficient conditions, derive a complementary result regarding the analysis of stochastic variational inequalities, and prove the equivalence of two different regularity conditions.simulation;mathematical programs with equilibrium constraints;stability;regularity conditions;sample-path methods;stochastic mathematical programs with complementarity constraints
Solving MPCC problem with the hyperbolic penalty function
The main goal of this work is to solve mathematical program with complementarity constraints (MPCC) using
nonlinear programming techniques (NLP). An hyperbolic penalty function is used to solve MPCC problems by including the
complementarity constraints in the penalty term. This penalty function [1] is twice continuously differentiable and combines
features of both exterior and interior penalty methods. A set of AMPL problems from MacMPEC [2] are tested and a
comparative study is performed
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