Optimization indirect methods

The second class of multivariable optimization techniques in principle requires the use of partial derivatives, although finite difference formulas can be substituted for derivatives such techniques are called indirect methods and include the following classes ... 

For optimization problems that are derived from (ordinary or partial) differential equation models, a number of advanced optimization strategies can be applied. Most of these problems are posed as NLPs, although recent work has also extended these models to MINLPs and global optimization formulations. For the optimization of profiles in time and space, indirect methods can be applied based on the optimality conditions of the infinite-dimensional problem using, for instance, the calculus of variations. However, these methods become difficult to apply if inequality constraints and discrete decisions are part of the optimization problem. Instead, current methods are based on NLP and MINLP formulations and can be divided into two classes ... 

The first one is based on a classical variation method. This approach is also known as an indirect method as it focuses on obtaining the solution of the necessary conditions rather than solving the optimization directly. Solution of these conditions often results in a two-point boundary value problem (TPBVP), which is accepted that it is difficult to solve [15], Although several numerical techniques have been developed to address the solution of TPBVP, e.g. control vector iteration (CVI) and single/multiple shooting method, these methods are generally based on an iterative integration of the state and adjoint equations and are usually inefficient [16], Another difficulty relies on the fact that it requires an analytical differentiation to derive the necessary conditions. 

The method of choice is dependent upon the analyte, the assay performance required to meet the intended application, the timeline, and cost-effectiveness. The assay requirements include sensitivity, selectivity, linearity, accuracy, precision, and method robustness. Assay sensitivity in general is in the order of IA > LC-MS/MS > HPLC, while selectivity is IA LC-MS/MS > HPLC. However, IA is an indirect method which measures the binding action instead of relying directly on the physico-chemical properties of the analyte. The IA response versus concentration curve follows a curvilinear relationship, and the results are inherently less precise than for the other two methods with linear concentration-response relationships. The method development time for IA is usually longer than that for LC/MS-MS, mainly because of the time required for the production and characterization of unique antibody reagents. Combinatorial tests to optimize multiple factors in several steps of some IA formats are more complicated, and also result in a longer method refinement time. The nature of IAs versus that of LC-MS/MS methods are compared in Table 6.1. However, once established, IA methods are sensitive, consistent, and very cost-effective for the analysis of large volumes of samples. The more expensive FTMS or TOF-MS methods can be used to complement IA on selectivity confirmation. 

The coefficients of variation of linear density for. short staple yarns specified in ASTM D 2645 95 are 5% for carded yarns and 4% for combed yarns. ISO 10290 does not differentiate between carded or combed yarns and has the CV% for linear density as 4% for both, but this is clearly the result of optimism. A method for the indirect measure-... 

Good historical review on the Characterization and Optimization of Polyethylene Blends is a part of the PhD thesis (Cran 2004). Zhao and Choi (2006) reviewed miscibility of PE blends in three parts (1) miscibility inferred from indirect methods, (2) interaction parameters, and (3) molecular simulation. Unfortunately, only selected binary blends of HOPE, LDPE, Z-N-LLDPE, and m-LLDPE, from ca. 1990 to 2000, are discussed. The authors postulate that the branch content, BC, of the two components determines miscibility, while MW, MWD, and branch length are secondary. However, the presence of a few long-chain branches could induce immiscibility in the blends. 

---