383 research outputs found
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
Solving mathematical programs with complementarity constraints with nonlinear solvers
MPCC can be solved with specific MPCC codes or in its nonlinear
equivalent formulation (NLP) using NLP solvers. Two NLP solvers - NPSOL and
the line search filter SQP - are used to solve a collection of test problems in AMPL.
Both are based on SQP (Sequential Quadratic Programming) philosophy but the
second one uses a line search filter scheme.(undefined
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
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.stochastic processes;mathematics;stability;simulation;regulations;general equilibrium
Strongly stable C-stationary points for mathematical programs with complementarity constraints
In this paper we consider the class of mathematical programs with complementarity constraints (MPCC). Under an appropriate constraint qualification of Mangasarian–Fromovitz type we present a topological and an equivalent algebraic characterization of a strongly stable C-stationary point for MPCC. Strong stability refers to the local uniqueness, existence and continuous dependence of a solution for each sufficiently small perturbed problem where perturbations up to second order are allowed. This concept of strong stability was originally introduced by Kojima for standard nonlinear optimization; here, its generalization to MPCC demands a sophisticated technique which takes the disjunctive properties of the solution set of MPCC into account.publishedVersio
Simulation-based solution of stochastic mathematical programs with complementarity constraints: sample-path analyis
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
Numerical experiments with a modified regularization scheme for mathematical programs with complementarity constraints
On this paper we present a modified regularization scheme for Mathematical Programs with Complementarity Constraints. In the regularized formulations the complementarity condition is replaced by a constraint involving a positive parameter that can be decreased to zero. In our approach both the complementarity condition and the nonnegativity constraints are relaxed. An iterative algorithm is implemented in MATLAB language and a set of AMPL problems from MacMPEC database were tested
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
A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints
It is known in the literature that local minimizers of mathematical programs
with complementarity constraints (MPCCs) are so-called M-stationary points, if
a weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds.
In this paper we present a new elementary proof for this result. Our proof is
significantly simpler than existing proofs and does not rely on deeper
technical theory such as calculus rules for limiting normal cones. A crucial
ingredient is a proof of a (to the best of our knowledge previously open)
conjecture, which was formulated in a Diploma thesis by Schinabeck.Comment: 7 page
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