Preconditioned Nonlinear Conjugate Gradient methods based on a modified secant equation

This paper includes a twofold result for the Nonlinear Conjugate Gradient (NCG) method, in large scale unconstrained optimization. First we consider a theoretical analysis, where preconditioning is embedded in a strong convergence framework of an NCG method from the literature. Mild conditions to be satisfied by the preconditioners are defined, in order to preserve NCG convergence. As a second task, we also detail the use of novel matrix-free preconditioners for NCG. Our proposals are based on quasi-Newton updates, and either satisfy the secant equation or a secant-like condition at some of the previous iterates. We show that, in some sense, the preconditioners we propose also approximate the inverse of the Hessian matrix. In particular, the structures of our preconditioners depend on low-rank updates used, along with different choices of specific parameters. The low-rank updates are obtained as by-product of NCG iterations. The results of an extended numerical experience using large scale CUTEst problems is reported, showing that our preconditioners can considerably improve the performance of NCG methods

Biogeography-Based Optimization with Orthogonal Crossover

Biogeography-based optimization (BBO) is a new biogeography inspired, population-based algorithm, which mainly uses migration operator to share information among solutions. Similar to crossover operator in genetic algorithm, migration operator is a probabilistic operator and only generates the vertex of a hyperrectangle defined by the emigration and immigration vectors. Therefore, the exploration ability of BBO may be limited. Orthogonal crossover operator with quantization technique (QOX) is based on orthogonal design and can generate representative solution in solution space. In this paper, a BBO variant is presented through embedding the QOX operator in BBO algorithm. Additionally, a modified migration equation is used to improve the population diversity. Several experiments are conducted on 23 benchmark functions. Experimental results show that the proposed algorithm is capable of locating the optimal or closed-to-optimal solution. Comparisons with other variants of BBO algorithms and state-of-the-art orthogonal-based evolutionary algorithms demonstrate that our proposed algorithm possesses faster global convergence rate, high-precision solution, and stronger robustness. Finally, the analysis result of the performance of QOX indicates that QOX plays a key role in the proposed algorithm