Rachid Ellaia

Global optimization of Hölder functions

We propose a branch-and-bound framework for the global optimization of unconstrained Hölder functions. The general framework is used to derive two algorithms. The first one is a generalization of Piyavskiis algorithm for univariate Lipschitz functions. The second algorithm, using a piecewise constant upper-bounding function, is designed for multivariate Hölder functions. A proof of convergence is provided for both algorithms. Computational experience is reported on several test functions from the literature.

A new hybrid method for solving global optimization problem

Abstract In this paper we present a new hybrid method, called the SASP method. The purpose of this method is the hybridization of the simulated annealing (SA) with the descent method, where we estimate the gradient using simultaneous perturbation. Firstly, the new hybrid method finds a local minimum using the descent method, then SA is executed in order to escape from the currently discovered local minimum to a better one, from which the descent method restarts a new local search, and so on until convergence. The new hybrid method can be widely applied to a class of global optimization problems for continuous functions with constraints. Experiments on 30 benchmark functions, including high dimensional functions, show that the new method is able to find near optimal solutions efficiently. In addition, its performance as a viable optimization method is demonstrated by comparing it with other existing algorithms. Numerical results improve the robustness and efficiency of the method presented.

A continuous approach to combinatorial optimization: application of water system pump operations

In this paper, we have suggested a penalty method to modify the combinatorial optimization problem with the linear constraints to a global optimization problem with linear constraints. It also deals with a topic of vital significance of pump operation optimization in a water system. In this connection we have done a lot of work to formulate a model based on a simplified flow volume balance to resolve the problem of optimal pump operation settings of switching “ON” and “OFF” with the reduced gradient method. This global solution approach incorporates some benefits for practical application to a real system as is shown in the case study.

A new multi-objective particle swarm optimization for reactive power dispatch

This paper proposes a new approach of computation using particle swarm in order to resolve multiobjective problems quickly and effectively. This approach is called accelerated multiobjective particle swarm which incorporates vector function as objective function and uses matrix computation to develop the Pareto front unlike the existing multi-objective algorithms which use an external archive. We also propose a novel method of initialization that contributes also in the acceleration of the algorithm. We apply this approach to resolve multi-objective reactive power dispatch problem. Simulations of the proposed algorithm are encouraging for very short CPU time.

Multicriteria shape design of a sheet contour in stamping

One of the hottest challenges in automotive industry is related to weight reduction in sheet metal forming processes, in order to produce a high quality metal part with minimal material cost. Stamping is the most widely used sheet metal forming process; but its implementation comes with several fabrication flaws such as springback and failure. A global and simple approach to circumvent these unwanted process drawbacks consists in optimizing the initial blank shape with innovative methods. The aim of this paper is to introduce an efficient methodology to deal with complex, computationally expensive multicriteria optimization problems. Our approach is based on the combination of methods to capture the Pareto Front, approximate criteria (to save computational costs) and global optimizers. To illustrate the efficiency, we consider the stamping of an industrial workpiece as test-case. Our approach is applied to the springback and failure criteria. To optimize these two criteria, a global optimization algorithm was chosen. It is the Simulated Annealing algorithm hybridized with the Simultaneous Perturbation Stochastic Approximation in order to gain in time and in precision. The multicriteria problems amounts to the capture of the Pareto Front associated to the two criteria. Normal Boundary Intersection and Normalized Normal Constraint Method are considered for generating a set of Pareto-optimal solutions with the characteristic of uniform distribution of front points. The computational results are compared to those obtained with the well-known Non-dominated Sorting Genetic Algorithm II. The results show that our proposed approach is efficient to deal with the multicriteria shape optimization of highly non-linear mechanical systems.

Multi-objective optimization by a new hybridized method: applications to random mechanical systems

In this article two linear problems with random Gaussian loading are transformed into multi-objective optimization problems. The first problem is the design of a pillar geometry with respect to a compressive random load process. The second problem is the design of a truss structure with respect to a vertical random load process for several frequency bands. A new algorithm, motivated by the Pincus representation formula hybridized with the Nelder–Mead algorithm, is proposed to solve the two multi-objective optimization problems. To generate the Pareto curve, the normal boundary intersection method is used to produce a series of constrained single-objective optimizations. The second problem, depending on the frequency band of excitation, can have as Pareto curve a single point, a standard Pareto curve, or a discontinuous Pareto curve, a fact that has been reported here for the first time in the literature, to the best of the authors’ knowledge.

Global optimization through a stochastic perturbation of the Polak–Ribière conjugate gradient method

Abstract We develop a new modified Polak–Ribiere conjugate gradient method by considering a random perturbation. Our approach is suitable for solving a large class of optimization problems on a rectangle of R n or unconstrained problems. Theoretical results ensure that the proposed method converges to a global minimizer. Numerical experiments are achieved on some typical test problems, particularly the engineering problem of Lennard-Jones clusters. A comparison with well known methods is carried out to show the performance of our algorithm.

A New Hybrid Genetic Algorithm and Particle Swarm Optimization

In this paper, we present a new hybrid algorithm which is a combination of a hybrid genetic algorithm and particle swarm optimization. We focus in this research on a hybrid method combining two heuristic optimization techniques, genetic algorithms (GA) and particle swarm optimization (PSO) for the global optimization. Denoted as GA-PSO, this hybrid technique incorporates concepts from GA and PSO and creates individuals in a new generation not only by crossover and mutation operations as found in GA but also by mechanisms of PSO. The performance of the two algorithms has been evaluated using several experiments.

Random perturbation of the projected variable metric method for nonsmooth nonconvex optimization problems with linear constraints

Random perturbation of the projected variable metric method for nonsmooth nonconvex optimization problems with linear constraints We present a random perturbation of the projected variable metric method for solving linearly constrained nonsmooth (i.e., nondifferentiable) nonconvex optimization problems, and we establish the convergence to a global minimum for a locally Lipschitz continuous objective function which may be nondifferentiable on a countable set of points. Numerical results show the effectiveness of the proposed approach.

A NEW METHODOLOGY FOR AN OPTIMAL SHAPE DESIGN

The aim of this paper is to study the implementation of an efficient and reliable methodology for shape optimization problems where the objective function and constraints are not known explicitly and are dependent on the Finite Element Analysis (FEA). It is based on the Simultaneous Perturbation Stochastic Approximation (SPSA) method for solving unconstrained continuous optimization problems. We also propose Penalty SPSA (PSPSA) for solving constrained optimization problems, the constraints are handled using exterior point penalty functions within an algorithm that combines SPSA and exact penalty transformations. This paper presents a new structural optimization methodology that combines shape optimization, geometric modeling, FEA and PSPSA method to successfully optimize structural optimization problems. Several tests have been performed on some well known benchmark functions to demonstrate the robustness and high performance of the suggested methodology. In addition, an illustrative two-dimensional structural problem has been solved in a very efficient way. The numerical results demonstrate the robustness and high performance of the suggested methodology for structural optimization problems.