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.

Solving a signalized traffic intersection problem with an hyperbolic penalty function

Mathematical Program with Complementarity Constraints (MPCC) finds many applications in fields such as engineering design, economic equilibrium and mathematical programming theory itself. A queueing system model resulting from a single signalized intersection regulated by pre-timed control in traffic network is considered. The model is formulated as an MPCC problem. A MATLAB implementation based on an hyperbolic penalty function is used to solve this practical problem, computing the total average waiting time of the vehicles in all queues and the green split allocation. The problem was codified in AMPL.

Numerical optimization experiments using the hyperbolic smoothing strategy to solve MPCC

In this work we solve Mathematical Programs with Complementarity Constraints using the hyperbolic smoothing strategy. Under this approach, the complementarity condition is relaxed through the use of the hyperbolic smoothing function, involving a positive parameter that can be decreased to zero. An iterative algorithm is implemented in MATLAB language and a set of AMPL problems from MacMPEC database were tested.

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.

A Java expression evaluator for nonlinear programming

Finding the optimal value for a problem is usual in many areas of knowledge where in many cases it is needed to solve Nonlinear Optimization Problems. For some of those problems it is not possible to determine the expression for its objective function and/or its constraints, they are the result of experimental procedures, might be non-smooth, among other reasons. To solve such problems it was implemented an API contained methods to solve both constrained and unconstrained problems. This API was developed to be used either locally on the computer where the application is being executed or remotely on a server. To obtain the maximum flexibility both from the programmers’ and users’ points of view, problems can be defined as a Java class (because this API was developed in Java) or as a simple text input that is sent to the API. For this last one to be possible it was also implemented on the API an expression evaluator. One of the drawbacks of this expression evaluator is that it is slower than the Java native code. In this paper it is presented a solution that combines both options: the problem can be expressed at run-time as a string of chars that are converted to Java code, compiled and loaded dynamically. To wide the target audience of the API, this new expression evaluator is also compatible with the AMPL format.

A filter algorithm: comparison with NLP solvers

The purpose of this work is to present an algorithm to solve nonlinear constrained optimization problems, using the filter method with the inexact restoration (IR) approach. In the IR approach two independent phases are performed in each iteration—the feasibility and the optimality phases. The first one directs the iterative process into the feasible region, i.e. finds one point with less constraints violation. The optimality phase starts from this point and its goal is to optimize the objective function into the satisfied constraints space. To evaluate the solution approximations in each iteration a scheme based on the filter method is used in both phases of the algorithm. This method replaces the merit functions that are based on penalty schemes, avoiding the related difficulties such as the penalty parameter estimation and the non-differentiability of some of them. The filter method is implemented in the context of the line search globalization technique. A set of more than two hundred AMPL test problems is solved. The algorithm developed is compared with LOQO and NPSOL software packages.

A filter inexact-restoration method for nonlinear programming

A new iterative algorithm based on the inexact-restoration (IR) approach combined with the filter strategy to solve nonlinear constrained optimization problems is presented. The high level algorithm is suggested by Gonzaga et al. (SIAM J. Optim. 14:646–669, 2003) but not yet implement—the internal algorithms are not proposed. The filter, a new concept introduced by Fletcher and Leyffer (Math. Program. Ser. A 91:239–269, 2002), replaces the merit function avoiding the penalty parameter estimation and the difficulties related to the nondifferentiability. In the IR approach two independent phases are performed in each iteration, the feasibility and the optimality phases. The line search filter is combined with the first one phase to generate a “more feasible” point, and then it is used in the optimality phase to reach an “optimal” point. Numerical experiences with a collection of AMPL problems and a performance comparison with IPOPT are provided.