Local search

Progress in these areas will require a number of new supporting tools that can effectively handle and solve a variety of mathematical models involving thousands and millions of variables. These supporting tools in turn will require that chemical engineers become acquainted with new advances in numerical analysis, mathematical programming, and local search techniques. 

Local search make local changes to the solution and see whether there is an improvement of costs. Local changes might be changes of resource assignments, split or combine quants, add or remove links. 

So far, only techniques, starting from some initial point and searching locally for an optimum, have been discussed. However, most optimization problems of interest will have the complication of multiple local optima. Stochastic search procedures (cf Section 4.4.4.1) attempt to overcome this problem. Deterministic approaches have to rely on rigorous sampling techniques for the initial configuration and repeated application of the local search method to reliably provide solutions that are reasonably close to globally optimal solutions. 

Considering convergence properties of the evolutionary algorithm by comparing the number of different structures occurred in the process (about 655,000) to the size of the stock of structures (1070), it was suggested that local search was carried out in the network architecture and parameter spaces. 

Swaney and Grossmann (1985b) note that the assumption of vertex solutions can be relaxed somewhat by carrying out a local search from the solution vertex if a descent direction exists and give one example where this approach identified a nonvertex worst point. 

Exploiting the heuristic that many local minimizers of constraints lie at vertices of the parameter space through vertex searches and local searches from promising vertices. 

The local search procedures are more likely to suffer problems due to discontinuities either from changes in the identified local minima with respect to o (discontinuity in value or first derivative) or from changes in the active set of constraints at a minimum with respect to o (discontinuity in first derivative). This is likely to cause failure to converge to a local maximum with respect to v in at least some cases. 

In the early iterations of the design, local vertex searches as used by Grossmann and Swaney provide an efficient way to generate new maximizers and push the design toward the required robustness. Local searches from a subset of the vertices may be used to try to identify nonvertex maximizers. /oclmax is set to limit the number of local searches used to supplement the local vertex search. defines the maximum depth of the local vertex search. See Appendix A for further details. 

If the local search methods are unsuccessful on any iteration, global search... 

Carry out local searches into the interior of V from up to nloc vertices identified during the local vertex search as having ascent directions into V which did not lead to an increased value of c ax on the corresponding vertex. 

If ivert = 1 test all the vertices not already examined Carry out a local search from the largest new maximizer with an ascent direction into V... 

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