Deterministic Global Optimization Theory, Methods and Applications

The vast majority of important applications in science, engineering and applied science are characterized by the existence of multiple minima and maxima, as well as first, second and higher order saddle points. The area of Deterministic Global Optimization introduces theoretical, algorithmic and computational ad­ vances that (i) address the computation and characterization of global minima and maxima, (ii) determine valid lower and upper bounds on the global minima and maxima, and (iii) address the enclosure of all solutions of nonlinear con­ strained systems of equations. Global optimization applications are widespread in all disciplines and they range from atomistic or molecular level to process and product level representations. The primary goal of this book is three fold : first, to introduce the reader to the basics of deterministic global optimization; second, to present important theoretical and algorithmic advances for several classes of mathematical prob­ lems that include biconvex and bilinear; problems, signomial problems, general twice differentiable nonlinear problems, mixed integer nonlinear problems, and the enclosure of all solutions of nonlinear constrained systems of equations; and third, to tie the theory and methods together with a variety of important applications.

1. Introduction.- 2. Basic Concepts of Global Optimization.- I Biconvex and Bilinear Problems.- 3. The GOP Primal — Relaxed Dual Decomposition Approach: Theory.- 4. The GOP Approach: Implementation and Computational Studies.- 5. The GOP Approach in Bilevel Linear and Quadratic Problems.- 6. The GOP Approach in Phase and Chemical Equilibrium Problems.- 7. The GOP Approach: Distributed Implementation.- II Signomial Problems.- 8. Generalized Geometric Programming: Theory.- 9. Generalized Geometric Programming: Computational Studies.- III Towards General Twice Differentiable NLPs.- 10. From Biconvex to General Twice Differentiable NLPs.- 11. The ?BB for Box Constrained Twice-Differentiable NLPs: Theory.- 12. The ?BB for Constrained Twice -Differentiable NLPs: Theory.- 13. Computational Studies of the ?BB Approach.- 14. Global Optimization in Microclusters.- 15. The ?BB Approach in Molecular Structure Prediction.- 16. The ?BB Approach in Protein Folding.- 17. The ?BB Approach in Peptide Docking.- 18. The ?BB Approach in Batch Design under Uncertainty.- 19. The ?BB Approach in Parameter Estimation.- IV Nonlinear and Mixed-Integer Optimization.- 20. Introduction to Nonlinear and Mixed-Integer Optimization.- 21. The SMIN-?BB Approach: Theory and Computations.- 22. The GMIN-?BB Approach: Theory and Computations.- V Nonlinear Constrained Systems of Equations.- 23. All Solutions of Nonlinear Constrained Systems of Equations.- 24. Locating All Homogeneous Azeotropes.- References.