Optimization deterministic local methods

We start with continuous variable optimization and consider in the next section the solution of NLP problems with differentiable objective and constraint functions. If only local solutions are required for the NLP problem, then very efficient large-scale methods can be considered. This is followed by methods that are not based on local optimality criteria we consider direct search optimization methods that do not require derivatives as well as deterministic global optimization methods. Following this, we consider the solution of mixed integer problems and outline the main characteristics of algorithms for their solution. Finally, we conclude with a discussion of optimization modeling software and its implementation on engineering models. 

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. 

This recent strategy developed by Das and Dennis [7] produces an even spre td of points on the Pareto front by transforming the original non-linear multi-objective optimization problem into a set of NLPs which are solved sequentially. As we have already mentioned, this method can be considered as deterministic and local since the original MATLAB implementation solves a set of single objective NLPs by means of the SQP algorithm. 

The global optimization method aBB deterministically locates the global minimum solution of (1) based on the refinement of converging lower and upper bounds. The lower bounds are obtained by the solution of (15), which is formulated as a convex programming problem. Upper bounds are based on the solution of (1) using local minimization techniques. 

The problem of finding all local minimum energy conformations can also be formulated as a single global optimization problem, which can be deterministically solved using the aBB algorithm [23]. This method stems from the idea that all stationary points (i.e., minima, maxima, and transition states) of the energy hypersurface satisfy the constraint V (0) = 0. This can be written as ... 

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