MINLP Problems Research Articles

The heat exchanger network design as a paradigm of technology integration

The work presents evidence from experiments that attempt to combine optimisation technologies of different qualities and advantages. The evidence is illustrated in relevance to the design problem of heat exchanger networks (HEN). The HEN problem has been addressed separately by Pinch technology, mathematical programming, and stochastic optimisation. Pinch methods account for conceptual principles driven by an overall understanding of the problem. Mathematical programming applications have spanned across the entire spectrum of formulations and include LP, MILP, NLP, and MINLP problems. Stochastic optimisation is presented mostly in the form of simulated annealing and genetic algorithms. Considered in isolation, the different optimisation methods are both blessed and baffled in that they simultaneously enable and disable particular features. The paper presents scientific experiments as a paradigm for the development of an integrated methodology with a purpose to restrain limitations, compound merits and produce a syncretism of complimentary venues for synthesis and optimisation.

On the Optimum Choice of Decision Variables for Equation-Oriented Global Optimization

In design problems, where the set of variables is larger than the set of equations, the difference corresponds to the degrees of freedom available to the designer. The use of equation-oriented simulators is particularly useful for the global optimization of nonconvex problems, such as those that usually describe chemical processes. This paper shows that, by coupling a combinatorial optimizer with a tearing/partitioning algorithm, the simulation step of an optimization problem can be posed as a combinatorial optimization problem, with the objective of minimizing the cost of the simulation step. The functional form in which the variables appear in the equations can easily be taken into account as constraints to the optimization problem. The concept is described in detail for several examples found in the chemical engineering literature, showing that the proposed method may be a useful preprocessing tool for the global optimization of nonconvex NLP or MINLP problems, where SQP-based methods may not be adequate.

Mathematical programming approaches to the synthesis of chemical process systems

This paper presents a review of advances that have taken place in the mathematical programming approach to process design and synthesis. A review is first presented on the algorithms that are available for solving MINLP problems, and its most recent variant, Generalized Disjunctive Programming models. The formulation of superstructures, models and solution strategies is also discussed for the effective solution of the corresponding optimization problems. The rest of the paper is devoted to reviewing recent mathematical programming models for the synthesis of reactor networks, distillation sequences, heat exchanger networks, mass exchanger networks, utility plants, and total flowsheets. As will be seen from this review, the progress that has been achieved in this area over the last decade is very significant.

Studies in adaptive random search optimization for MINLP problems

Stochastic optimization approaches are achieving growing interest in application to difficult optimization problems common in chemical and process engineering. They are easy to use especially in cases where there are disjunctive functions and logical conditions. M-LJ optimization algorithm from adaptive random search (ARS) strategy has been developed for most advantageous optimization problems, i.e. MINLP ones. This method is an extension of LJ algorithm developed by Jaakola and Luus (1974) for problems with continuous variables. The main modification in M-LJ method is the change of distribution function for integer variables. The results of tests are presented and the comparison with the performance of other stochastic methods as e.g. M-SIMPSA algorithm from Cardoso et al. (1997). M-LJ has shown high robustness and can be seen easy to use optimization tool for certain MINLP problems. Also, the re-formulation of difficult optimization MINLP problem for batch processes from Kocis and Grossmann (1988) has been developed which allows for solving it as NLP task.

Convexification of different classes of non-convex MINLP problems

In the present paper, convexification strategies for certain kinds of discrete and integer non-convex optimization problems are introduced and discussed. We show how to solve problems with both posynomial and negative binomial terms in the constraints. The convexification technique may in some cases be generalized to include continuous variables. Posynomial functions are non-convex and for such functions no straightforward methods for finding the optimal solution exist. Such functions appear frequently in different kinds of chemical engineering problems. The different transformation techniques are illustrated in the form of short examples. The techniques are finally applied to a large, bilinear, trim loss problem regularly encountered at paper-converting mills.

Solving Irregular Econometric and Mathematical Optimization Problems with a Genetic Hybrid Algorithm

In the present paper we apply a new Genetic Hybrid Algorithm (GHA) to globally minimize a representative set of ill-conditioned econometric/mathematical functions. The genetic algorithm was specifically designed for nonconvex mixed integer nonlinear programming problems and it can be successfully applied to both global and constrained optimization. In previous studies, we have demonstrated the efficiency of the GHA in solving complicated NLP, INLP and MINLP problems. The present study is a continuation of this research, now focusing on a set of highly irregular optimization problems. In this paper we discuss the genetic hybrid algorithm, the nonlinear problems to be solved and present the results of the empirical tests.

A Parametric Mixed-Integer Optimization Algorithm for Multiobjective Engineering Problems Involving Discrete Decisions

An iterative, decomposition-based algorithm is proposed in this paper, for the solution of convex parametric MINLP problems and, in extension, the identification of the noninferior solution set in multiobjective problems involving continuous and discrete decisions. The parametric optimal solution is constructed via an upper- and lower-bound procedure. Parametric upper bounds are identified with the solution of parametric NLP problems, while the parametric lower bound is updated in each iteration via the solution of a parametric MILP Master problem, which involves only the binary variables of the initial problem. Convergence properties and computational requirements are discussed in example problems from process synthesis under uncertainty and simultaneous product/process design.

An outer-approximation algorithm for the solution of multiparametric MINLP problems

Process synthesis problems involving uncertainty can be mathematically represented as multiparametric mixed integer nonlinear programming (mp-MINLP) models. In this paper, we present an outer-approximation algorithm for the solution of such mp-MINLPs, described by convex process models, linear in the vectors of binary variables and uncertain parameters. The algorithm follows decomposition principles, i.e., constructing a converging sequence of valid upper and lower bounds through the solution of parametric primal and master subproblems. The solution is characterized in different sub-domains of the uncertain parameter space by (i) linear parametric profiles, and (ii) the corresponding integer solutions.

Coupling genetic algorithm with a grid search method to solve mixed integer nonlinear programming problems

A new hybrid algorithm is being introduced for solving Mixed Integer Nonlinear Programming ( minlp) problems which arise from study of many real-life engineering problems such as the minimum cost development of oil fields and the optimization of a multiproduct batch plant. This new algorithm employs both the Genetic Algorithm and a modified grid search method interfacing in such a way that the resulting hybrid algorithm is capable of solving many minlp problems efficiently and accurately. Testings indicate that this algorithm is efficient and robust even for some ill-conditioned problems with nonconvex constraints.

Global optimization of MINLP problems in process synthesis and design

Two new methodologies for the global optimization of MINLP models, the Special structure Mixed Integer Nonlinear αBB, SMIN-αBB, and the General structure Mixed Integer Nonlinear αBB, GMIN-αBB, 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-αBB; mixed nonlinear terms whose continuous relaxation is twice-differentiable are handled by the GMIN-αBB. While both algorithms use the concept of a branch & bound tree, they rely on fundamentally different bounding and branching strategies. In the GMIN-αBB 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 bounding MINLPs, using the techniques recently developed for the generation of valid convex underestimators for twice-differentiable functions ( Adjiman et al., 1996; Adjiman and Floudas, 1996), is an essential task as it allows to solve the underestimating problems to global optimality using the GBD algorithm or the OA algorithm, provided that the binary variables participate separably and linearly. Moreover, the inherent structure of the MINLP problem can be fully exploited as branching is performed on the binary and the continuous variables. In the case of the SMIN-αBB algorithm, the lower and upper bounds are obtained by solving continuous relaxations of the original MINLP. Using the αBB algorithm, these nonconvex NLPs are solved as global optimization problems and hence valid lower bounds are generated. Since branching is performed exclusively on the binary variables, the maximum size of the branch-and-bound tree is smaller than that for the SMIN-αBB. The two proposed approaches are used to generate computational results on various nonconvex MINLP problems that arise in the areas of Process Synthesis and Design.

A hybrid global optimization method for design of batch chemical processes

Combining Simulated Annealing (SA) algorithm with Linear Programming (LP), SA/LP algorithm is proposed for solving nonconvex MINLP problems. An MINLP model for optimal design of multipurpose batch process with multiple routes is developed. It allows the existence of intermediate storage, parallel units in and out of phase as well as discrete equipment size. SA/LP algorithm is implemented to solve the proposed MINLP model. The results of illustrating examples are conducted and compared with those of some previous published paper. SA/LP procedure find the global optima and is shown to be far more efficient in computing time.

Global optimisation of nonconvex MINLPs

The recent advances in mathematical programming approaches applied to process design and operation problems have produced a need for the ability to find the global optimum of a nonconvex problem containing discrete variables (a nonconvex MINLP). This paper presents a modified version of the reformulation/spatial branch-and-bound algorithm of Smith and Pantelides (1996) for the solution of such problems. The algorithm is implemented within the gPROMS modelling environment (Barton and Pantelides, 1994) and tested on several MINLP problems arising from process engineering applications.

Design and synthesis of batch plants — MINLP solution based on a stochastic method

Design and synthesis of batch plants must consider scheduling aspect simultaneously. In this paper, MINLP problems arising in very general flexible batch plants design and synthesis are solved by a stochastic MINLP optimiser. The formulations considered include linear/nonlinear models for capital cost processing units/storage vessels and processing task models with fixed, linear and nonlinear processing time in a general form. The MINLP optimiser, based on an Evolutionary Algorithm with Simulated Annealing does not require convexity and smoothness of the problem formulation, and a suitable original encoding procedure is reported. Several examples are given to demonstrate the method's efficiency and the quality of the solution.

Global Optimization of MINLP Problems in Process Synthesis and Design

Global Optimization of MINLP Problems in Process Synthesis and Design

A simulated annealing approach to the solution of minlp problems

An algorithm (M-SIMPSA) suitable for the optimization of mixed integer non-linear programming (MINLP) problems is presented. A recently proposed continuous non-linear solver (SIMPSA) is used to update the continuous parameters, and the Metropolis algorithm is used to update the complete solution vector of decision variables. The M-SIMPSA algorithm, which does not require feasible initial points or any problem decomposition, was tested with several functions published in the literature, and results were compared with those obtained with a robust adaptive random search method. For ill-conditioned problems, the proposed approach is shown to be more reliable and more efficient as regards the overcoming of difficulties associated with local optima and in the ability to reach feasibility. The results obtained reveal its adequacy for the optimization of MINLP problems encountered in chemical engineering practice.

An extended cutting plane method for solving convex MINLP problems

An extended cutting plane method for solving convex MINLP problems

Design and production planning for multiproduct dehydration plants

Drying operations constitute an important field of chemical engineering, which is still governed by empiricism. This paper deals with two important aspects related to the construction and operation of dehydration plants. The design problem involves the determination of process flowsheet structure, when a nominal production level is specified for all products processed in the plant. The production planning problem describes the production policy of the plant within a long-range planning horizon, under time-varying product demand and market prices of raw materials and saleable products. The policy adopted assigns production levels and duration of production runs for each product processed, in each one of the plant processors, at a certain time period, within the planning horizon. The objective in each approach is to optimize the total annual profit resulting from the construction of a new plant or the operation of an existing one. The process was described by deducing the mathematical model of conveyor-belt dryers. For the forementioned problems, appropriate formulations were developed and studied. The most general model presented involves numerous integer and continuous decision variables and a large number of space variables and constraints, resulting in cumbersome calculations and tremendous computational load. For the reduction of the computational effort, a shortcut modeling of total annual plant cost was proposed and evaluated from the operational data of a large number of possible flowsheet structures, through a simple analytical equation. The parameters of the proposed shortcut equation were estimated by nonlinear regression over an extensive number of computed points; each one of them was determined by solving an NLP optimization problem. The design and production problems were formulated as MINLP problems in which use of the shortcut cost equation reduced drastically the computational effort involved. Related problems (i.e. process modification and design under production planning criteria) were also taken into consideration. Characteristic examples were presented in order to demonstrate the effectiveness of each proposed approach.

Feature extraction algorithms for constrained global optimization II. Batch process scheduling application

The feature extraction algorithms developed in part I of this series are applied to solve nonconvex mixed integer nonlinear programming problems which arise in the optimal scheduling of multipurpose chemical plants. A general formulation of the multipurpose plant scheduling problem is developed which considers the allocation of plant equipment and secondary, limited resources to recipe operations so as to satisfy given production requirements while minimizing cost. Results obtained with a test example involving 135 binary and 922 continuous variables show that the successive refinement strategy is effective in identifying dominant regions of the solution space. Furthermore it is shown that multiple moment based characterization methods are superior to the interval analysis method reported in the literature. Trials using a second, larger nonlinear test problem involving 356 binary and 2402 continuous variables demonstrate that the focused successive refinement strategy is more efficient than a constant resolution strategy which employs genetic algorithm constructions. Although the conventional genetic algorithm can be significantly improved by introducing a heuristic mutation strategy which increases the likelihood of constant feasibility, the successive refinement strategy remains dominant. These studies demonstrate that the feature extraction strategy employing successive refinements and relatively low order moment based region characterization methods, offers an effective approach to solving an important class of large scale MINLP problems with multiple local optima.

An LP/NLP based branch and bound algorithm for convex MINLP optimization problems

This paper is aimed at improving the solution efficiency of convex MINLP problems in which the bottleneck lies in the combinatorial search for the 0–1 variables. An LP/NLP based branch and bound algorithm is proposed in which the explicit solution of an MILP master problem is avoided at each major iteration. Instead, the master problem is defined dynamically during the tree search to reduce the number of nodes that need to be enumerated. A branch and bound search is conduced to predict lower bounds by solving LP subproblems until feasible integer solutions are found. At these nodes nonlinear programming subproblems are solved, providing upper bounds and new linear approximations which are used to tighten the linear representation of the open nodes in the search tree. To reduce the size of the LP subproblems, new types of linear approximations are proposed which exploit linear substructures in the MINLP problem. Preliminary numerical results on several test problems are reported which show that the expense of solving the MI need to be enumerated, while in most cases the number of NLP subproblems to be solved remains the same.

On the Generalized Benders Decomposition

Generalized Benders Decomposition is a procedure to solve certain types of NLP and MINLP problems. The use of this procedure has been recently suggested as a tool for solving process design problems. This paper analyzes the solution of nonconvex problems through different implementations of the Generalized Benders Decomposition. It is demonstrated that in certain cases only local minima may be found, whereas in other cases not even convergence to local optima can be achieved. A criterion to identify whether the converged value is a candidate for being a local minimum is provided. It is also shown that in the presence of a dual gap, a particular implementation of the Generalized Benders Decomposition may provide upper and lower bounds on the global optimum. The conceptual steps undertaken in establishing the various implementations of GBD are summarized as follows: • -Convexity of X and of F ( x, y) and G ( x, y) in x, implies that (1) is equivalent to (4). When these conditions do not apply a gap between v( y) and its dual may exist (Remark 2.1). • -Problem (4) is replaced by a sequence of relaxed master problems (Remark 2.3). • -Convexity of X and of F ( x, y) and G ( x, y) in x (as well as satisfaction of certain other conditions) guarantees ϵ-convergence of the GBD iterations (Geoffrion, 1972, Theorem 2.5). It is implicitly understood, that for these results to hold, both the primal and the relaxed master problem must be solved globally. • -Global solution of the primal stems readily from convexity of X and of F ( x, y) and G ( x, y) in x. Global solution of the relaxed master may come either through the use of special algorithms or by the establishment of convexity of L *( y, u) and L * ( y, λ) in y. One case for which convexity of these functions can be established is when F ( x, y) and G ( x, y) are jointly convex in x and y (Geoffrion, 1972, Section 4.2). • -Prior to the solution of the relaxed master problem, the evaluation of L*( y, u*) and L * ( y, λ*) is first required through (5) and (6). • -One case for which L*( y, u*) and L * ( y, λ*) can be obtained in explicit form is when the global minimum (over x) of (5) and (6) can be obtained independently of y [ Property ( P)]. In this case one can construct the Benders iterations by selecting x to be the solution of the primal and u (or λ) the corresponding Lagrange multiplier. • -When the Benders iterations are built based on the primal solution, i.e. assuming that the solution of the minimization problem (5) or (6) is equal to the solution of the primal x*, [irrespective of the satisfaction of Property ( P)], the NP-GBD algorithm is obtained. When x* is indeed the solution of the minimization problem, it is said that Property ( P′) holds. • -When the Benders iterations are built through the introduction of additional constraints as in (11), and through global solutions of (5) and (6) (problems R and S), then the PL-GBD is obtained.