A. Ismael F. Vaz

Robot trajectory planning with semi-infinite programming

Abstract In this paper we describe how robot trajectory planning can be formulated as a semi-infinite programming (SIP) problem. The formulation as a SIP problem allowed us to treat the problem with one of the three main classes of methods for solving SIP, the discretization class. Two of the robotics trajectory planning problems formulated were coded in the SIPAMPL environment which is publicly available. A B-Spline library was also created to allow the codification of the robotics trajectory problem.

SIPAMPL: Semi-infinite programming with AMPL

SIPAMPL is an environment for coding semi-infinite programming (SIP) problems. This environment includes a database containing a set of SIP problems that have been collected from the literature and a set of routines. It allows users to code their own SIP problems in AMPL, to use any problem already in the database, and to develop and test any SIP solver. The SIPAMPL routines support the interface between a potential SIP solver and test problems coded in AMPL. SIPAMPL also provides a tool that allows the selection of problems from the database with specified characteristics. As a concept demonstration, we show how MATLAB can use SIPAMPL to solve the problems in the database. The Linux and Microsoft Windows versions together with the database of coded problems are freely available via the web.

A Method for Constrained Multiobjective Optimization Based on SQP Techniques

We propose a method for constrained and unconstrained nonlinear multiobjective optimization problems that is based on an SQP-type approach. The proposed algorithm maintains a list of nondominated points that is improved both for spread along the Pareto front and optimality by solving single-objective constrained optimization problems. These single-objective problems are derived as SQP problems based on the given nondominated points. Under appropriate differentiability assumptions we discuss convergence to local optimal Pareto points. We provide numerical results for a set of unconstrained and constrained multiobjective optimization problems in the form of performance and data profiles, where several performance metrics are used. The numerical results confirm the superiority of the proposed algorithm against a state-of-the-art multiobjective solver and a classical scalarization approach, both in the quality of the approximated Pareto front and in the computational effort necessary to compute the approximation.

Modeling nearby FGK Population I stars: A new form of estimating stellar parameters using an optimization approach

Context. Modeling a single star with theoretical stellar evolutionary tracks is a nontrivial problem because of a large number of unknowns compared to the number of observations. A current way of estimating stellar age and mass consists of using interpolations in grids of stellar models and/or isochrones, assuming ad hoc values for the mixing length parameter and the metal-to-helium enrichment, which is normally scaled to the solar values. Aims. We present a new method to model the FGK main-sequence of Population I stars. This method is capable of simultaneously estimating a set of stellar parameters, namely the mass, the age, the helium and metal abundances, the mixing length parameter, and the overshooting. Methods. The proposed method is based on the application of a global optimization algorithm (PSwarm) to solve an optimization problem that in turn consists of finding the values of the stellar parameters that lead to the best possible fit of the given observations. The evaluation of the fitting objective function requires the use of a stellar evolution simulation code, however. Results. The methodology is tested using the Sun and five FGK fictitious stars, and is then adapted to 115 stars with a detailed spectroscopic analysis; half of these are planet-hosting stars. Conclusions. We present and discuss the stellar parameters estimated for each star in the context of previous works.

An MPCC approach on a Stackelberg game in an electric power market: changing the leadership

An electric power market is formulated as a Stackelberg game where two firms, A and B, produce energy. Two distinct situations, according to the firm who plays the leader role, are analysed. In the first one, the firm A is the leader and the firm B is the follower, and in the second situation the players reverse their roles. In order to select the optimal strategy, the leader uses as knowledge his own perception of the market and anticipates the reactions of the other followers. The main goal of this paper is to understand the behaviour of the various agents that compose the electric power network, such as transmissions capacity, quantities of power generated and demanded, when the leadership changes. The problem is formulated as a mathematical program with complementarity constraints (MPCC) and reformulated into a nonlinear program (NLP), allowing the use of robust NLP solvers. Computational results using Lancelot, Loqo and Snopt solvers are performed. The numerical experiments show that the firm profit is conditioned by the available information.

Tools for Robotic Trajectory Planning Using Cubic Splines and Semi-Infinite Programming

In this paper we describe how robot trajectory planning, using cubic splines to generate the trajectory, can be formulated as standard semi-infinite programming (SIP) problems and efficiently solved by a discretization method. These formulated problems were coded in the publicly available SIPAMPL environment and to allow the codification of these problems a cubic splines dynamic library for AMPL was developed. The discretization method used to solve the formulated problems is implemented in the NSIPS solver and numerical results with four particular problems are shown.

A quasi-Newton interior point method for semi-infinite programming

We propose an interior point method for solving nonlinear semi-infinite programming (SIP) problems. The method is based on a transcription of the SIP problem into a nonlinear finite problem. We apply the interior point strategy to the finite problem and obtain an algorithm for nonlinear SIP. The evaluation of the constraint functions and derivatives of the finite problem requires the numerical evaluation of integrals that we compute by an adaptative trapezoid formula. Numerical results with a collection of several problems from the SIPAMPL [A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes. SIPAMPL v2.0: Semi-infinite programming with AMPL. Technical Report ALG/EF/4-2002, Universidade do Minho, Braga, Portugal, December 2002. [http://www.norg.uminho.pt/aivaz/].] database are also shown. †E-mail: emgpf@dps.uminho.pt ‡E-mail: p.gomes@ic.ac.uk

Build Orientation Optimization Problem in Additive Manufacturing

Additive manufacturing (AM) is an emerging type of production technology to create three-dimensional objects layer-by-layer directly from a 3D CAD model. AM is being extensively used by engineers and designers.

A reduction method for nonlinear semi-infinite programming based on an exact penalty technique

Semi-infinite programming (SIP) problems arise in several areas of engineering, such as robot trajectory planning, production planning, design of digital filters and air pollution control. Despite its large applicability there is not much software available to solve such problems. The only available SIP solvers are the fseminf MATLAB function, FSQP and NSIPS, but none of these solvers provide an algorithm belonging to the class of reduction type methods. In this article we propose a reduction type method, based on a penalty technique to solve SIP problems. We report numerical results with 117 test problems from the SIPAMPL database, using a MATLAB implementation of the proposed algorithm, which we coined as SiRedAl.

A Scheduling Application to a Molding Injection Machine: A Challenge Addressed on the 109th European Study Group with Industry

This paper addresses a scheduling optimization problem applied to a molding injection machine. This optimization problem was posed as a challenge to the mathematical community present at the 109th European Study Group with Industry (ESGI), held in Portugal in 2015. We propose a mathematical model for the scheduling optimization problem, which was coded in a widely known modeling language. The model is validated through a set of numerical results obtained with a state-of-the-art solver.