Acceleration of reduced Hessian methods for large-scale nonlinear programming

Process optimization problems are frequently characterized by large models, with many variables and constraints but relatively few degrees of freedom. Thus, reduced Hessian decomposition methods applied to Successive Quadratic Programming (SQP) exploit the low dimensionality of the subspace of the decision variables, and have been very successful for a wide variety of process application. However, further development is needed for improving the efficient large-scale use of these tools. In this study we develop an improved SQP algorithm decomposition with coordinate bases that includes an inexpensive second order correction term. The resulting algorithm is 1-step Q-superlinearly convergent. More importantly, though, the resulting algorithm is largely independent of the specific decomposition steps. Thus, the inexpensive factorization of the coordinate decomposition, which lends itself very well to tailoring, can be applied in a reliable and efficient manner. With this efficient and easy-to-implement NLP strategy, we continue to improve the efficiency of the optimization algorithm by exploiting the mathematical structure of existing process engineering models. Here we consider the tailoring of a reduced Hessian method for the block tridiagonal structure of the model equations for distillation columns. This approach is applied to the Naphthali-Sandholm algorithm implemented within the UNIDIST and programs. Our reduced Hessian SQP strategy is incorporated within the package with only minor changes In the program's interface and data structures. Through this integration, reductions of 20% to 80% in the total CPU time are obtained compared to general reduced space optimization; an order of magnitude reduction Is obtained when compared to conventional sequential strategies. Consequently, this approach shows considerable potential for efficient and reliable large-scale process optimization, particularly when complex Newton-based process models are already available. For large systems, various decomposition strategies have been proposed which reduce the dimensionality of the problem and thus allow for an efficient algorithm. A decomposition strategy using coordinate basis matrices was used by Locke et aL (1983); while the decomposition itself is efficient, it frequently leads to inconsistent convergence results. To remedy this, Vasantharajan and Biegler (1988) investigate the use of orthogonal basis representations. Here, although the computational effort per iteration is higher, especially as the number of degrees of freedom increases, the resulting SQP method is more robust. However, orthogonal projections are not always easy to adapt to the sparsity structure of the process model. In section 2 we briefly describe the concept of decomposition for SQP with details on the use of coordinate and orthogonal bases. While these reduced Hessian methods show much promise, they may still be inefficient when faced with large, complex process engineering problems. The performance of SQP for large-scale models can be improved considerably through a closer examination of the mathematical structure inherent to individual classes of chemical systems; for example, the tridiagonal structure of the Jacobian matrix for distillation problems. To tailor the SQP algorithm to take advantage of this underlying structure we adapt the reduced space SQP strategy so that the appropriate equation solver is used directly as part of the optimization procedure, in order to eliminate the dependent variables. In fact, any model specific structures and procedures used to generate the Newton step can be exploited directly here. As will be discussed later, coordinate basis representations are best suited for this task and thus we first discuss how to guarantee consistent convergence results with this decomposition strategy. Section 3 covers the details of our improved coordinate basis method and outlines three approaches for the calculation of a necessary second order correction term. We also present a series of test problem results using uncorrected orthogonal and coordinate bases as well as our improved algorithm. Finally, using the improved coordinate bases algorithm, we exploit the structure of specific classes of process engineering problems in section 4. In particular, we consider the optimization of distillation columns and interface directly to the UNIDIST and NRDIST packages (Andersen et at, 1991). Here, the model equations form a blocktridiagonal matrix, and are solved using the Naphthali-Sandholm distillation algorithm. After making only minor changes within the interfaces and model solution algorithms, we are able to realize up to an order of magnitude increase in efficiency over the conventional sequential procedure. The final section of the paper summarizes this work and briefly outlines future work. 2. Decomposition Strategies for SQP In this section we discuss the background for successive quadratic programming and various decomposition strategies. The nonlinear programming problem (NLP) to be solved is formulated as.