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A new Pareto front approximation method is proposed for multiobjective optimization problems (MOPs) with bound constraints. The method employs a hybrid optimization approach using two derivative-free direct search techniques, and intends to solve black box simulation-based MOPs where the analytical form of the objectives is not known and/or the evaluation of the objective function(s) is very expensive. A new adaptive weighting scheme is proposed to convert a multiobjective optimization problem to a single objective optimization problem. Another contribution of this paper is the generalization of the star discrepancy-based performance measure for problems with more than two objectives. The method is evaluated using five test problems from the literature, and a realistic engineering problem. Results show that the method achieves an arbitrarily close approximation to the Pareto front with a good collection of well-distributed nondominated points for all six test problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a three-term conjugate gradient method which can produce sufficient descent condition, that is, [image omitted] . This property is independent of any line search used. When an exact line search is used, this method reduces to the standard Hestenes-Stiefel conjugate gradient method. We also introduce two variants of the proposed method which still preserve the sufficient descent property, and prove that these two methods converge globally with standard Wolfe line search even if the minimization function is nonconvex. We also report some numerical experiment to show the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the distributed and boundary optimal control of the Stokes equations in the presence of pointwise control constraints. The analysis of the problems involve existence results, as well as necessary conditions and optimality systems. A primal–dual active set strategy is studied for the numerical solution of the problems. For the distributed case, global and local convergence results for the infinite dimensional method are presented. For the boundary control case, due to the lack of regularity of the pointwise constraint multiplier, the method is applied to a discretized version of the original problem. Finally, some numerical experiments, which illustrate the performance of the method for both cases, are commented. [ABSTRACT FROM AUTHOR]

One of the most popular algorithms for solving minimax approximation problems is the Barrodale and Phillips simplex-based method [I. Barrodale and C. Phillips (1974). An improved algorithm for discrete Chebyshev linear approximation. In: Proceedings of the Fourth Manitoba Conference on Numerical Mathematics , pp. 177-190, University of Manitoba, Winnipeg, Canada.]. Our new research has found that the modified pivoting strategy in the first stage of the Barrodale and Phillips algorithm could result in an invalid starting point and system failure in later stages [D. Lei (2002). Robust and efficient algorithms for ℓ 1 and ℓ ∞ approximations. PhD Thesis, School of Computing and Engineering, University of Huddersfield, Huddersfield, UK.]. In contrast, a primal method that we propose in this paper is numerically more stable, and the approximating function does not suffer the effects of wild oscillation. However, on average, the primal method requires more searching steps to find an optimal solution. To improve the efficiency of the primal method, we have considered using the least squares solution to provide a better starting point. Experimental results show that if the least squares estimator is available, then it can speed up the convergence rate significantly in most cases. The number of steps required by using the least squares solution for the primal method also compares favourably with that of the dual method. [ABSTRACT FROM AUTHOR]