Christodoulos A. Floudas

Handbook of Test Problems in Local and Global Optimization

Preface. 1. Introduction. 2. Quadratic Programming Problems. 3. Quadratically Constrained Problems. 4. Univariate Polynomial Problems. 5. Bilinear Problems. 6. Biconvex and (D.C.) Problems. 7. Generalized Geometric Programming. 8. Twice Continuously Differentiable NLPs. 9. Bilevel Programming Problems. 10. Complementarity Problems. 11. Semidefinite Programming Problems. 12. Mixed-Integer Nonlinear Problems. 13. Combinatorial Optimization Problems. 14. Nonlinear Systems of Equations. 15. Dynamic Optimization Problems.

Continuous-time versus discrete-time approaches for scheduling of chemical processes: a review

An overview of developments in the scheduling of multiproduct/multipurpose batch and continuous processes is presented. Existing approaches are classified based on the time representation and important characteristics of chemical processes that pose challenges to the scheduling problem are discussed. In contrast to the discrete-time approaches, various continuous-time models have been proposed in the literature and their strengths and limitations are examined. Computational studies and applications are presented. The important issues of incorporating scheduling at the design stage and scheduling under uncertainty are also reviewed.

A global optimization method, αBB, for general twice-differentiable constrained NLPs — I. Theoretical advances

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 parameters for general twice-differentiable functions. A crucial phase in the design of such procedures is the use of interval arithmetic on the Hessian matrix or the characteristic polynomial of the function being investigated. Thanks to this step, the proposed schemes share the common property of computational tractability and preserve the global optimality guarantees of the algorithm. However, their accuracy and computational requirements differ so that no method can be shown to perform consistently better than others for all problems. Their use is illustrated on an unconstrained and a constrained example. The second part of this paper (Adjiman et al., 1998) is devoted to the discussion of issues related to the implementation of the αBB algorithm and to extensive computational studies illustrating its potential applications.

αBB: A global optimization method for general constrained nonconvex problems

A branch and bound global optimization method,αBB, for general continuous optimization problems involving nonconvexities in the objective function and/or constraints is presented. The nonconvexities are categorized as being either of special structure or generic. A convex relaxation of the original nonconvex problem is obtained by (i) replacing all nonconvex terms of special structure (i.e. bilinear, fractional, signomial) with customized tight convex lower bounding functions and (ii) by utilizing the α parameter as defined in [17] to underestimate nonconvex terms of generic structure. The proposed branch and bound type algorithm attains finiteε-convergence to the global minimum through the successive subdivision of the original region and the subsequent solution of a series of nonlinear convex minimization problems. The global optimization method,αBB, is implemented in C and tested on a variety of example problems.

Proteome-wide post-translational modification statistics: frequency analysis and curation of the swiss-prot database

Post-translational modifications (PTMs) broadly contribute to the recent explosion of proteomic data and possess a complexity surpassing that of protein design. PTMs are the chemical modification of a protein after its translation, and have wide effects broadening its range of functionality. Based on previous estimates, it is widely believed that more than half of proteins are glycoproteins. Whereas mutations can only occur once per position, different forms of post-translational modifications may occur in tandem. With the number and abundances of modifications constantly being discovered, there is no method to readily assess their relative levels. Here we report the relative abundances of each PTM found experimentally and putatively, from high-quality, manually curated, proteome-wide data, and show that at best, less than one-fifth of proteins are glycosylated. We make available to the academic community a continuously updated resource (http://selene.princeton.edu/PTMCuration) containing the statistics so scientists can assess “how many” of each PTM exists.

A review of recent advances in global optimization

This paper presents an overview of the research progress in deterministic global optimization during the last decade (1998–2008). It covers the areas of twice continuously differentiable nonlinear optimization, mixed-integer nonlinear optimization, optimization with differential-algebraic models, semi-infinite programming, optimization with grey box/nonfactorable models, and bilevel nonlinear optimization.

A global optimization method, αBB, for general twice-differentiable constrained NLPs—II. Implementation and computational results

Abstract Part I of this paper ( Adjiman et al., 1998a ) described the theoretical foundations of a global optimization algorithm, the α BB algorithm, which can be used to solve problems belonging to the broad class of twicedifferentiable NPLs. For any such problem, the ability to automatically generate progressively tighter convex lower bounding problems at each iteration guarantees the convergence of the branch-and-bound α BB algorithm to within e of the global optimum solution. Several methods were presented for the construction of valid convex 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 α BB 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 selection of an underestimating strategy. In addition, the package features both automatic differentiation and interval arithmetic capabilities. Making use of all the available options, the α BB algorithm successfully identifies the global optimum solution of small literature problems, of small and medium size chemical engineering problems in the areas of reactors network design, heat exchanger network design, reactor–separator network design, of generalized geometric programming problems for design and control, and of batch process design problems with uncertainty.

ACTIVE CONSTRAINT STRATEGY FOR FLEXIBILITY ANALYSIS IN CHEMICAL PROCESSES

Abstract It is shown in this paper that by exploiting properties of limiting constraints for flexibility in a design, problems for flexibility analysis can be formulated as mixed-integer optimization problems. Formulations are derived when control variables are present or not, and when equalities are eliminated or handled explicitly. These formulations do not rely on the assumption that critical parameter values are vertices, nor do they require exhaustive vertex searches. The case of linear constraints reduces to standard MILP problems, while for the nonlinear case a novel active constraint strategy is proposed and its theoretical properties are analyzed. Examples are presented for both rigorous and screening calculations.

A new robust optimization approach for scheduling under uncertainty: I. Bounded uncertainty

The problem of scheduling under bounded uncertainty is addressed. We propose a novel robust optimization methodology, which when applied to mixed-integer linear programming (MILP) problems produces “robust” solutions which are in a sense immune against bounded uncertainty. Both the coefficients in the objective function, the left-hand-side parameters and the right-hand-side parameters of the inequalitie s are considered. Robust optimization techniques are developed for two types of uncertain data: bounded uncertainty and bounded and symmetric uncertainty. By introducing a small number of auxiliary variables and constraints, a deterministic robust counterpart problem is formulated to determine the optimal solution given the (relative) magnitude of uncertain data, feasibility tolerance, and “reliability level” when a probabilis tic measurement is applied. The robust optimization approach is then applied to the scheduling under uncertainty problem. Based on a novel and effective continuous-time short-term scheduling model proposed by Floudas and coworkers [Ind. Eng. Chem. Res. 37 (1998a) 4341; Ind. Eng. Chem. Res. 37 (1998b) 4360; Ind. Eng. Chem. Res. 38 (1999) 3446; Comp. Chem. Engng. 25 (2001) 665; Ind. Eng. Chem. Res. 41 (2002) 3884; Ind. Eng. Chem. Res. (2003)], three of the most common sources of bounded uncertainty in scheduling problems are addressed, namely processing times of tasks, market demands for products, and prices of products and raw materials. Computational results on several small examples and an industrial case study are presented to demonstrate the effectiveness of the proposed approach.

A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—I. Theory

Abstract A large number of nonlinear optimization problems involve bilinear, quadratic and/or polynomial functions in their objective function and/or constraints. In this paper, a theoretical approach is proposed for global optimization in constrained nonconvex NLP problems. The original nonconvex problem is decomposed into primal and relaxed dual subproblems by introducing new transformation variables if necessary and partitioning of the resulting variable set. The decomposition is designed to provide valid upper and lower bounds on the global optimum through the solutions of the primal and relaxed dual subproblems, respectively. New theoretical results are presented that enable the rigorous solution of the relaxed dual problem. The approach is used in the development of a Global OPtimization algorithm (GOP). The algorithm is proved to attain finite e-convergence and e-global optimality. An example problem is used to illustrate the GOP algorithm both computationally and geometrically. In an accompanying paper (Visweswaran and Floudas, Computers & Chemical Engineering 14 , 1419, 1990), application of the theory and the GOP algorithm to various classes of optimization problems, as well as computational results of the approach are provided.