Software global optimization

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. 

In addition to the Premium Excel Solver and Optquest, there are many other software systems for constrained global optimization see Pinter (1996b), Horst and Pardalos (1995), and Pinter (1999) for further information. Perhaps the most widely used of these is LGO (Pinter, 1999), (Pinter, 2000), which is intended for smooth problems with continuous variables. It is available as an interactive development environment with a graphical user interface under Microsoft Windows, or as a callable library, which can be invoked from an application written by the user in... 

Pinter, J. D. Continuous Global Optimization Software a Brief Review. Optima 52 1-8 (1996b). 

Depending on the form of the objective function, the final formulation obtained by replacing the nonlinear Eq. (17) by the set of linear inequalities corresponds to a MINLP (nonlinear objective), to a MIQP (quadratic objective) or to a MILP (linear objective). For the cases where the objective function is linear, solution to global optimal solution is guaranteed using currently available software. The same holds true for the more general case where the objective function is a convex function. 

M. Tawarmalani, V. Sahinidis, Global Optimization in Continuous and Mixed-Integer Nonlinear Programming Theory, Algorithms, Software, and Applications, Kluwer, Boston, 2001. 

Tawarmalani M. and Sahinidis N.V. 2002. Convexification and global optimization in continuous and mixed-integer nonlinear programming Theory, algorithms, software, and apphcations. In, Nonconvex Optimization and Its Apphcations Series, Vol. 65. Kluwer Academic Publishers, Dordrecht. 

Mockus, J., Eddy, W., Mockus, A., Mockus, L. and Reklaitis, G. Bayesian Heuristic Approach to Dsicrete and Global Optimization Algorithm, Ksualization, Software, and Applications. Kluwer Academic Publishers, Dordrecht, The Netholands, 1996. 

In this article we deal with the problem of unconstrained minimization, which is central to the development of optimization software. The problem is stated as follows given a sufficiently smooth function f IR —> IR, we look for a point X G El s,t /(x ) < f x) for all x G R near x. So we seek for a local minimum of the objective function. We do not discuss global optimization. 

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