| Global optimization is a field that has enjoyed much development and research activity in recent years. As scientists and engineers attempt to tackle more complex problems, within a more constrained socio-economic setting, global optimization techniques are being invoked so as to ensure that every margin of benefit is captured. This Thesis presents several fundamental contributions in the area of global optimization from an algorithmic, theoretical, and application point of view. The key problems addressed are:;I. Tight convex underestimators for C2-continuous problems. We develop a novel technique for the tight underestimation of twice-differentiable univariate continuous functions of arbitrary structure. The method is based on a piecewise application of the well-known alphaBB formula, and marks the first time in the open literature that the convex envelope of a function without some special structure can be constructed. An extension to the case of multivariate functions, through proper projections into select two-dimensional spaces, is also presented. We further propose the introduction of the technique on orthonormal transformations of the original problem, leading to an improvement in tightness through the accumulation of additional cuts in the relaxation. Comprehensive computational studies indicate that the method produces convex underestimators of high quality in terms of both lower bound and tightness over the whole domain under consideration.;II. Convexity of products of univariate functions. Based on analysis, we derive sufficient conditions for convexity of a product of two univariate functions, and we then show that these suffice for the general case of an arbitrary number of factors. The conditions consider each factor-function on an individual basis, rendering the identification of global-convexity of factorable multivariate terms a trivial task. We finally extend the results to address terms raised into arbitrary exponents, elucidating the characteristics that have to be possessed by a functional mapping so that it is suitable to be employed in a variable transformation for the convexification of posynomials.;III. Piecewise relaxation schemes in the context of the pooling problem. We explore piecewise extensions of the long-familiar technique based on the bilinear envelopes for the relaxation of the pooling problem. In total, 15 different MILP relaxation schemes are derived and compared through a collection of benchmark problems, providing valuable insight on how to efficiently address their peculiar nonconvexities.;IV. Formulation and relaxation of an extended pooling problem. We introduce an explicit MINLP representation of a statutory model that is used for the certification of gasoline in the United States. We also introduce an extended pooling problem formulation, that allows oil refineries to take this disjunctive model into account when they attempt to schedule/optimize their pooling processes. The relaxation of the overall formulation is discussed, including opportunities for tight piecewise schemes.;V. Rational design of shape selective separations and catalysis. We develop an integrated computational framework that can screen large databases of zeolite structures and offer predictions for their selectivity between potential sorbate molecules of interest. The method is based on identifying the guest conformation that is globally optimal for penetration into the opening of a host structure. We conduct a study using a large set of molecules and zeolitic host structures, and compile the results in a database that can serve as a useful tool for designing shape-selectivity driven processes. We then present a number of enhancements that include taking into account the flexibility of the opening and the entropic contributions due to multiplicity of penetrating conformations. We conclude with a sensitivity analysis on the effective radii of the portal-defining atoms, assessing the effect of their uncertainty on the framework's predictions. |
