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- Multistart coupled with a derivative-free filter local search for locating multiple solutionsPublication . Fernandes, Florbela P.; Pereira, Ana I.; Costa, Maria F.P.; Fernandes, Edite M.G.P.A multistart technique coupled with a derivative-free lter local search algorithm for locating all the optimal solutions of a nonconvex constrained optimization is presented. To reach a fast convergence to the optimal solutions, the local search procedure is based on descent directions. The lter-set concept is introduced to handle the constraints of the problem. The generated direction vector is descent for the objective function if the sample point is feasible; otherwise, it is descent for the constraint violation. Numerical experiments with benchmark problems are reported and a comparison with other stochastic methods is included.
- Numerical experiments with nonconvex MINLP problemsPublication . Fernandes, Florbela P.; Costa, Maria F.P.; Fernandes, Edite M.G.P.We present a methodology to solve nonconvex Mixed-Integer Nonlinear Programming problems, that combines the Branch-and-Bound and simulated annealing type methods, which was implemented in MATLAB. A set of benchmark functions with simple bounds and different dimensions was used to analyze its practical behaviour. We exhibit computational results showing the good performance of the method.
- Multistart hooke and jeeves filter method for mixed variable optimizationPublication . Fernandes, Florbela P.; Costa, M. Fernanda P.; Fernandes, Edite M.G.P.; Rocha, Ana Maria A.C.In this study, we propose an extended version of the Hooke and Jeeves algorithm that uses a simple heuristic to handle integer and/or binary variables and a filter set methodology to handle constraints. This proposal is integrated into a multistart method as a local solver and it is repeatedly called in order to compute different optimal solutions. Then, the best of all stored optimal solutions is selected as the global optimum. The performance of the new method is tested on benchmark problems. Its effectiveness is emphasized by a comparison with other well-known stochastic solvers.
- A deterministic-stochastic method for nonconvex MINLP problemsPublication . Fernandes, Florbela P.; Costa, M. Fernanda P.; Fernandes, Edite M.G.P.A mixed-integer programming problem is one where some of the variables must have only integer values. Although some real practical problems can be solved with mixed-integer linear methods, there are problems occurring in the engineering area that are modelled as mixed-integer nonlinear programming (MINLP) problems. When they contain nonconvex functions then they are the most difficult of all since they combine all the difficulties arising from the two sub-classes: mixed-integer linear programming and nonconvex nonlinear programming (NLP). Efficient deterministic methods for solving MINLP are clever combinations of Branch-and-Bound (B&B) and Outer-Approximations classes. When solving nonconvex NLP relaxation problems that arise in the nodes of a tree in a B&B algorithm, using local search methods, only convergence to local optimal solutions is guaranteed. Pruning criteria cannot be used to avoid an exhaustive search in the solution space. To address this issue, we propose the use of a simulated annealing algorithm to guarantee convergence, at least with probability one, to a global optimum of the nonconvex NLP relaxation problem. We present some preliminary tests with our algorithm.
- Multilocal programming: a derivative-free filter multistart algorithmPublication . Fernandes, Florbela P.; Costa, M. Fernanda P.; Fernandes, Edite M.G.P.Multilocal programming aims to locate all the local solutions of an optimization problem. A stochastic method based on a multistart strategy and a derivative-free filter local search for solving general constrained optimization problems is presented. The filter methodology is integrated into a coordinate search paradigm in order to generate a set of trial approximations that might be acceptable if they improve the constraint violation or the objective function value relative to the current one. Preliminary numerical experiments with a benchmark set of problems show the effectiveness of the proposed method.
- A derivative-free filter driven multistart technique for global optimizationPublication . Fernandes, Florbela P.; Costa, M. Fernanda P.; Fernandes, Edite M.G.P.A stochastic global optimization method based on a multistart strategy and a derivative-free filter local search for general constrained optimization is presented and analyzed. In the local search procedure, approximate descent directions for the constraint violation or the objective function are used to progress towards the optimal solution. The algorithm is able to locate all the local minima, and consequently, the global minimum of a multi-modal objective function. The performance of the multistart method is analyzed with a set of benchmark problems and a comparison is made with other methods.
- Stopping rules effect on a derivative-free filter multistart algorithm for multilocal programmingPublication . Fernandes, Florbela P.; Costa, M. Fernanda P.; Fernandes, Edite M.G.P.Multilocal programming aims to identify all the local solutions of constrained optimization problems. The purpose of this paper is to analyze the effect of stopping rules on the performance of a particular multistart method, which relies on a derivative-free local search procedure to converge to a solution, when solving multilocal optimization problems. The method herein presented implements the approximate descent direction method combined with a filter methodology to handle the constraints by forcing the local search towards the feasible region. Two stopping rules are tested on five classical multimodal problems.
- Improving efficiency of a multistart with interrupted hooke-and-jeeves filter search for solving MINLP problemsPublication . Fernandes, Florbela P.; Costa, M. Fernanda P.; Rocha, Ana Maria A.C.; Fernandes, Edite M.G.P.This paper addresses the problem of solving mixed-integer nonlinear programming (MINLP) problems by a multistart strategy that invokes a derivative-free local search procedure based on a filter set methodology to handle nonlinear constraints. A new concept of componentwise normalized distance aiming to discard randomly generated points that are sufficiently close to other points already used to invoke the local search is analyzed. A variant of the Hooke-and-Jeeves filter algorithm for MINLP is proposed with the goal of interrupting the iterative process if the accepted iterate falls inside an ϵ-neighborhood of an already computed minimizer. Preliminary numerical results are included.
- Interrupted searches in the BBMCSFilter context for MINLP problemsPublication . Fernandes, Florbela P.; Costa, M. Fernanda P.; Fernandes, Edite M.G.P.The BBMCSFilter method was developed to solve mixed integer nonlinear programming problems. This kind of problems have integer and continuous variables and they appear very frequently in process engineering problems. The objective of this work is to analyze the performance of the method when the coordinate searches are interrupted in the context of the multistart strategy. From the numerical experiments, we observed a reduction on the number of function evaluations and on the CPU time.
- Overview on mixed integer nonlinear programming problemsPublication . Fernandes, Florbela P.; Costa, Maria F.P.; Fernandes, Edite M.G.P.Many optimization problems involve integer and continuous variables that can be modeled as mixed integer nonlinear programming (MINLP) problems. This has led to a wide range of applications, in particular in some engineering areas. Here, we provide a brief overview on MINLP, and present a simple idea for a future nonconvex MINLP solution technique.