This paper has as a major objective to present a unified overview and derivation of mixedinteger nonlinear programming (MINLP) techniques, Branch and Bound, Outer-Approximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are expressed in algebraic form. The solution of MINLP problems with convex functions is presented first, followed by a brief discussion on extensions for the nonconvex case. The solution of logic based representations, known as generalized disjunctive programs, is also described. Theoretical properties are presented, and numerical comparisons on a small process network problem.

: Two new methodologies for the global optimization of MINLP models, the Special structure Mixed Integer Nonlinear ffBB, SMIN--ffBB, and the General structure Mixed Integer Nonlinear ffBB, GMIN--ffBB, are presented. Their theoretical foundations provide guarantees that the global optimum solution of MINLPs involving twice--differentiable nonconvex functions in the continuous variables can be identified. The conditions imposed on the functionality of the binary variables differ for each method : linear and mixed bilinear terms can be treated with the SMIN--ffBB; mixed nonlinear terms whose continuous relaxation is twice--differentiable are handled by the GMIN--ffBB. While both algorithms use the concept of a branch & bound tree, they rely on fundamentally different bounding and branching strategies. In the GMIN--ffBB algorithm, lower (upper) bounds at each node result from the solution of convex (nonconvex) MINLPs derived from the original problem. The construction of convex lower bound...

Two novel deterministic global optimization algorithms for nonconvex mixed-integer problems (MINLPs) are proposed, using the advances of the ffBB algorithm for nonconvex NLPs Adjiman et al. (1998a). The Special Structure Mixed-Integer ffBB algorithm (SMIN-ffBB addresses problems with nonconvexities in the continuous variables and linear and mixed-bilinear participation of the binary variables. The General Structure Mixed-Integer ffBB algorithm (GMIN-ffBB), is applicable to a very general class of problems for which the continuous relaxation is twice continuously differentiable. Both algorithms are developed using the concepts of branch-and-bound, but they differ in their approach to each of the required steps. The SMIN-ffBB algorithm is based on the convex underestimation of the continuous functions while the GMIN-ffBB algorithm is centered around the convex relaxation of the entire problem. Both algorithms rely on optimization or interval based variable bound updates to enhance effici...

The global optimization of mixed integer non-linear programming (MINLP) problems is an active research area in many engineering fields. In this work, Differential Evolution (DE), a hybrid Evolutionary Computation method, is used for the optimization of nonconvex MINLP problems and a comparison is made among the algorithms based on hybrid of Simplex & Simulated Annealing (M-SIMPSA), Genetic Algorithms (GA), and DE. It is found that DE, an exceptionally simple evolutionary computation method, is significantly faster and yields the global optimum for a wide range of the key parameters. Results indicate that DE is more reliable, efficient and hence a better approach to the optimization of nonconvex nonlinear problems. DE found to be the best evolutionary computation method in all the problems studied.

A branch and bound algorithm for computing globally optimal solutions to nonconvex nonlinear programs in continuous variables is presented. The algorithm is directly suitable for a wide class of problems arising in chemical engineering design. It can solve problems defined using algebraic functions and twice differentiable transcendental functions, in which finite upper and lower bounds can be placed on each variable. The algorithm uses rectangular partitions of the variable domain and a new bounding program based on convex/concave envelopes and positive definite combinations of quadratic terms. The algorithm is deterministic and obtains convergence with final regions of finite size. The partitioning strategy uses a sensitivity analysis of the bounding program to predict the best variable to split and the split location. Two versions of the algorithm are considered, the first using a local NLP algorithm (MINOS) and the second using a sequence of lower bounding programs in the search fo...

The use of networks allows the representation of a variety of important engineering problems. The treatment of a particular class of network applications, the process synthesis problem, is exposed in this paper. Process Synthesis seeks to develop systematically process flowsheets that convert raw materials into desired products. In recent years, the optimization approach to process synthesis has shown promise in tackling this challenge. It requires the development of a network of interconnected units, the process superstructure, that represents the alternative process flowsheets. The mathematical modeling of the superstructure has a mixed set of binary and continuous variables and results in a mixed-integer optimization model. Due to the nonlinearity of chemical models, these problems are generally classified as Mixed-Integer Nonlinear Programming (MINLP) problems. A number of local optimization algorithms, developed for the solution of this class of problems, are presented in this pap...

The need for improvement in process operations, logistics and supply chain management has created a great demand for the development of optimization models for planning and scheduling. In this paper we first review the major classes of planning and scheduling models that arise in process operations, and establish the underlying mathematical structure of these problems. As will be shown, the nature of these models is greatly affected by the time representation (discrete or continuous), and is often dominated by discrete decisions. We then briefly review the major recent developments in mixed-integer linear and nonlinear programming, disjunctive programming and constraint programming, as well as general decomposition techniques for solving these problems. We present a general formulation for integrating planning and scheduling to illustrate the models and methods discussed in this paper.

Abstract: This paper presents a meta-heuristic optimization algorithm, Tabu Search (TS), and describes how it can be used to solve a wide variety of chemical engineering problems. Modifications to the original algorithm and constraint handling techniques are described and integrated to extend its applicability. All components of TS are described in detail. Initial values for each key parameter of TS are provided. In addition, guidelines for adjusting these parameters are provided to relieve a significant amount of time-consuming trial-and-error experiments that are typically required with stochastic optimization. Several small NLP and MINLP test cases and three small- to middle- scale chemical process synthesis problems demonstrate the feasibility and effectiveness of the techniques with recommended parameters. 1.

BSTRACT In industry there is still lot of potential to make an energy system more efficient and thereby reduce the waste heat available. On the other hand there is an option to export the waste heat to another industry or to society. When the use of a heat exchanger network is considered for these tasks the optimization framework developed in this work can be implemented to calculate the cost of optimal investments. This thesis presents a framework for generating flexible heat exchanger networks (HEN) over a specified range of variations in the flow rates and temperatures of the streams, so that the total annual costs (TAC) as a result of utility charges, exchanger areas and selection of matches are minimized. The proposed framework includes (i) an initialization stage to reduce the problem size, (ii) a multiperiod simultaneous MINLP model to synthesize a flexible HEN configuration, (iii) a multiperiod LP feasibility test model to check the operability and identify critical conditions which are to be included in the possible resolve stage of the MINLP model, and (iv) an NLP improvement model for further optimization

Significant advances have been made in the last two decades for the effective solution of mixed integer non-linear programming (MINLP) problems, mainly by exploiting the special structure of the problem that results under certain convexity assumptions. This paper discusses the computational experience with a novel decomposition algorithm which is based on the idea of closely approximating the feasible region defined by the set of constraints by a convex polytope using the simplicial approximation approach (Goyal and Ierapetritou, 2003b). A variety of problems are solved including structural flowsheet optimization, design of batch processes and trim-loss minimization to illustrate the applicability and efficiency of the proposed approach.