In Part I (Floudas and Visweswaran, 1990), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the rigorous solution of the problem through a series of primal and relaxed dual problems until the upper and lower bounds from these problems converged to an ffl-global optimum. In this paper, theoretical results are presented for several classes of mathematical programming problems that include : (i) the general quadratic programming problem, (ii) quadratic programming problems with quadratic constraints, (iii) pooling and blending problems, and (iv) unconstrained and constrained optimization problems with polynomial terms in the objective function and/or constraints. For each class, a few examples are presented illustrating the approach. Keywords : Global Optimization, Quadratic Programming, Quadratic Constraints, Polynomial functions, Pooling and Blending Problems. Author to whom...

In this paper, the deterministic global optimization algorithm, αBB, (α-based Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twice-differentiable NLPs. The key idea is the construction of a converging sequence of upper and lower bounds on the global minimum through the convex relaxation of the original problem. This relaxation is obtained by (i) replacing all nonconvex terms of special structure (i.e., bilinear, trilinear, fractional, fractional trilinear, univariate concave) with customized tight convex lower bounding functions and (ii) by utilizing some α parameters as defined by Maranas and Floudas (1994b) to generate valid convex underestimators for nonconvex terms of generic structure. In most cases, the calculation of appropriate values for the α parameters is a challenging task. A number of approaches are proposed, which rigorously generate a set of α par...

. Quadratic optimization comprises one of the most important areas of nonlinear programming. Numerous problems in real world applications, including problems in planning and scheduling, economies of scale, and engineering design, and control are naturally expressed as quadratic problems. Moreover, the quadratic problem is known to be NP-hard, which makes this one of the most interesting and challenging class of optimization problems. In this chapter, we review various properties of the quadratic problem, and discuss different techniques for solving various classes of quadratic problems. Some of the more successful algorithms for solving the special cases of bound constrained and large scale quadratic problems are considered. Examples of various applications of quadratic programming are presented. A summary of the available computational results for the algorithms to solve the various classes of problems is presented. Key words: Quadratic optimization, bilinear programming, concave pro...

In Floudas and Visweswaran (1990, 1992), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the solution of the problem through a series of primal and relaxed dual problems that provide valid upper and lower bounds respectively on the global solution. The algorithm was proved to have finite convergence to an ffl-global optimum. In this paper, new theoretical properties are presented that help to enhance the computational performance of the GOP algorithm applied to problems of special structure. The effect of the new properties is illustrated through application of the GOP algorithm to a difficult indefinite quadratic problem, a multiperiod tankage quality problem that occurs frequently in the modeling of refinery processes, and a set of pooling/blending problems from the literature. In addition, extensive computational experience is reported for randomly generated concave and in...

Part I of this paper (Adjiman et al., 1997) described the theoretical foundations of a global optimization algorithm, the ffBB algorithm, which can be used to solve problems belonging to the broad class of twice-differentiable NPLs. For any such problem, the ability to automatically generate progressively tighter convex lower bounding problems at each iteration guarantees the convergence of the branchand -bound ffBB algorithm to within ffl of the global optimum solution. Several methods were presented for the construction of convex valid underestimators for general nonconvex functions. In this second part, the performance of the proposed algorithm and its alternative underestimators is studied through their application to a variety of problems. An implementation of the ffBB is described and a number of rules for branching variable selection and variable bound updates are shown to enhance convergence rates. A user-friendly parser facilitates problem input and provides flexibility in the...

Recently, Floudas and Visweswaran (1990, 1993) proposed a global optimization algorithm (GOP) for the solution of a large class of nonconvex problems through a series of primal and relaxed dual subproblems that provide upper and lower bounds on the global solution. Visweswaran and Floudas (1995a) proposed a reformulation of the algorithm in the framework of a branch and bound approach that allows for an easier implementation. They also proposed an implicit enumeration of all the nodes in the resulting branch and bound tree using a mixed integer linear (MILP) formulation, and a linear branching scheme that reduces the number of subproblems from exponential to linear. In this paper, a complete implementation of the new versions of the GOP algorithm, as well as detailed computational results of applying the algorithm to various classes of nonconvex optimization problems is presented. The problems considered including pooling and blending problems, problems with separation and heat exchang...

A global optimization algorithm, αBB, for twice-differentiable NLPs is presented. It operates within a branch-and-bound framework and requires the construction of a convex lower bounding problem. A technique to generate such a valid convex underestimator for arbitrary twice-differentiable functions is described. The αBB has been applied to a variety of problems and a summary of the results obtained is provided.

We address in this paper the mid-term planning of chemical complexes with integration of stochastic inventory management under supply and demand uncertainty. By using the guaranteed service approach to model the time delays in the chemical flows inside the chemical process network, we capture the stochastic nature of the supply and demand variations, and develop an equivalent deterministic optimization model to minimize the total cost including production cost, feedstock purchase cost, cycle inventory and safety stock costs. The model simultaneously determines the optimal purchases of the feedstocks, production levels of the processes, sales of final products and safety stock levels of all the chemicals, as well as the internal demand of the production processes. The model also captures “risk-pooling ” effects to allow centralization of inventory management for chemicals that are consumed/produced by multiple processes. We formulate the model as a mixed-integer nonlinear program (MINLP) with a nonconvex objective function and nonconvex constraints. To solve the global optimization problem with modest computational times, we exploit some model