Efficient algorithms and models for mechanical and structural design optimization

In the present work, different algorithms and penalty methods for design optimization of mechanical elements and structures are applied. Seven robust optimization techniques and seven penalty methods are thoroughly investigated and implemented in MATLAB codes. In addition, different optimization models are compared using two benchmark problems, namely, the minimal cost design of a welded beam structure and the optimal buckling design of a functionally graded material column. A performance measure factor is defined to determine the best technique among the implemented optimization algorithms. The results are arranged and nested to make it easy for the reader to figure out each technique characteristics, and hence choose the suitable one for a specific design problem and/or application. Comprehensive computer experimentations were performed, and the best optimization techniques and models have been thoroughly demonstrated. The attained optimal solutions show that, in general, the hybrid algorithms worked better than the stand-alone ones and the sequential quadratic programming (SQP) with global search indicates a superior performance than other techniques. Finally, based on the present study, the adaptive and dynamic penalties need further investigation to become more consistent with the implemented optimization algorithms.

Ensemble Linear Subspace Analysis of High-Dimensional Data

Regression models provide prediction frameworks for multivariate mutual information analysis that uses information concepts when choosing covariates (also called features) that are important for analysis and prediction. We consider a high dimensional regression framework where the number of covariates (p) exceed the sample size (n). Recent work in high dimensional regression analysis has embraced an ensemble subspace approach that consists of selecting random subsets of covariates with fewer than p covariates, doing statistical analysis on each subset, and then merging the results from the subsets. We examine conditions under which penalty methods such as Lasso perform better when used in the ensemble approach by computing mean squared prediction errors for simulations and a real data example. Linear models with both random and fixed designs are considered. We examine two versions of penalty methods: one where the tuning parameter is selected by cross-validation; and one where the final predictor is a trimmed average of individual predictors corresponding to the members of a set of fixed tuning parameters. We find that the ensemble approach improves on penalty methods for several important real data and model scenarios. The improvement occurs when covariates are strongly associated with the response, when the complexity of the model is high. In such cases, the trimmed average version of ensemble Lasso is often the best predictor.

Coupled versus decoupled penalization of control complementarity constraints

This paper deals with the numerical solution of optimal control problems with control complementarity constraints. For that purpose, we suggest the use of several penalty methods which differ with respect to the handling of the complementarity constraint which is either penalized as a whole with the aid of NCP-functions or decoupled in such a way that non-negativity constraints as well as the equilibrium condition are penalized individually. We first present general global and local convergence results which cover several different penalty schemes before two decoupled methods which are based on a classical $\ell_1$- and $\ell_2$-penalty term, respectively, are investigated in more detail. Afterwards, the numerical implementation of these penalty methods is discussed. Based on some examples, where the optimal boundary control of a parabolic partial differential equation is considered, some quantitative properties of the resulting algorithms are compared.