Optimization problems definitions

No single method or algorithm of optimization exists that can be apphed efficiently to all problems. The method chosen for any particular case will depend primarily on (I) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-HiU, New York, 1988). You do not have to follow the cited order exac tly, but vou should cover all of the steps eventually. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps I, 2, and 3 deal with the mathematical definition of the problem ideutificatiou of variables and specification of the objective function and statement of the constraints. If the process to be optimized is very complex, it may be necessaiy to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objec tive function (the economic model). 

Let s try to define in a practical sense the structural optimization problem. To do so, let s work our way through the complete definition by examining a sequence of useful lesser definitions. 

The resulting optimization problem is solved using ILOG CPLEX [4], which generates a schedule for all major process steps, as well as the main material requirements for the production (optimal recipe definition for each batch). The schedule obtained is furthermore passed on to a crane movement simulation module, which... 

If the subspace S contains a positive definite element S, neither the energy problem nor the spectral optimization problem has an optimal solution since there is no positive semidefinite matrix P satisfying both the conditions (P, = 0, (P, I)j = 1, the convex set Pq n 5 is empty, and the energy pro-... 

It is important to understand that all of these steps equally impact the success of the projea. Not understanding the system can lead to a problem definition that when solved does not help improve the system. An unreasonable hypothesis is by definition defining the wrong problem. Poorly designed experiments and ulty experimental technique leads to bad data. And improper data analysis can transform even good data into useless results. Careful attention to all steps in this process is therefore required in order to achieve optimal results. 

This chapter introduces the reader to elementary concepts of modeling, generic formulations for nonlinear and mixed integer optimization models, and provides some illustrative applications. Section 1.1 presents the definition and key elements of mathematical models and discusses the characteristics of optimization models. Section 1.2 outlines the mathematical structure of nonlinear and mixed integer optimization problems which represent the primary focus in this book. Section 1.3 illustrates applications of nonlinear and mixed integer optimization that arise in chemical process design of separation systems, batch process operations, and facility location/allocation problems of operations research. Finally, section 1.4 provides an outline of the three main parts of this book. 

This section presents first the formulation and basic definitions of constrained nonlinear optimization problems and introduces the Lagrange function and the Lagrange multipliers along with their interpretation. Subsequently, the Fritz John first-order necessary optimality conditions are discussed as well as the need for first-order constraint qualifications. Finally, the necessary, sufficient Karush-Kuhn-Dicker conditions are introduced along with the saddle point necessary and sufficient optimality conditions. 

This chapter focuses on heat exchanger network synthesis approaches based on optimization methods. Sections 8.1 and 8.2 provide the motivation and problem definition of the HEN synthesis problem. Section 8.3 discusses the targets of minimum utility cost and minimum number of matches. Section 8.4 presents synthesis approaches based on decomposition, while section 8.5 discusses simultaneous approaches. 

This chapter presents optimization-based approaches for the synthesis of heat exchanger networks. Sections 8.1 and 8.2 introduce the reader to the overall problem definition, key temperature approaches, and outline the different types of approaches proposed in the last three decades. For further reading, refer to the review paper of Gundersen and Naess (1988) and the suggested references. 

In the case of potential energy functions, unconstrained optimization problems can generally be formulated for large, nonlinear, and smooth functions. Obtaining first and second derivatives may be tedious but is definitely... 

While these optimization-based approaches have yielded very useful results for reactor networks, they have a number of limitations. First, proper problem definition for reactor networks is difficult, given the uncertainties in the process and the need to consider the interaction of other process subsystems. Second, all of the above-mentioned studies formulated nonconvex optimization problems for the optimal network structure and relied on local optimization tools to solve them. As a result, only locally optimal solutions could be guaranteed. Given the likelihood of extreme nonlinear behavior, such as bifurcations and multiple steady states, even locally optimal solutions can be quite poor. In addition, superstructure approaches are usually plagued by the question of completeness of the network, as well as the possibility that a better network may have been overlooked by a limited superstructure. This problem is exacerbated by reaction systems with many networks that have identical performance characteristics. (For instance, a single PFR can be approximated by a large train of CSTRs.) In most cases, the simpler network is clearly more desirable. 

Objective functions - such as yield, productivity or total cost - provide an evaluation of efficiency and quality of preparative chromatographic separations. Since values of the objective functions can be changed by altering operating and design parameters, they are dependent variables for any optimization problem. The definitions of each objective function are identical for every chromatographic process (e.g. batch or SMB) but different parameters must be applied for its calculation. 

Generally, two or more objective functions are defined for gene expression profiling and gene network analysis. Usually, these objectives are conflicting in nature. Use of traditional single objective optimization techniques to solve these multi-objective optimization problems suffer from many drawbacks. Single objective problems either use penalty function approach or use some of the objectives as constraints. Both of these approaches have user-defined biases. Thus, multi-objective optimization techniques are definitely needed to model and solve these and similar other problems. 

A search of parameters of the model Xopt, which provide the optimal, in a definite sense, approximation to the set of simultaneously unachieved properties, leads to the multicriteria optimization problem. It is formulated as follows To find... 

Enforcing stoichiometric, capacity, and thermodynamic constraints simultaneously leads to the definition of a solution space that contains all feasible steady-state flux vectors. Within this set, one can find a particular steady-state metabolic flux vector that optimizes the network behavior toward achieving one or more goals (e g., maximize or minimize the production of certain metabolites). Mathematically speaking, an objective function has to be defined that needs to be minimized or maximized subject to the imposed constraints. Such optimization problems are typically solved via linear programming techniques. 

The calculational procedures are presented first for conventional distillation columns and then for complex distillation columns. The conventional distillation column is completely determined by fixing the following variables (1) the complete definition of the feed (total flow rate, composition, and thermal condition), (2) the column pressure (or the pressure at one point in the column, say in the accumulator), (3) the type of condenser, (4) ku the number of plates above and including the feed plate, (5) /c2, the total number of plates, and (6) two other specifications which are usually taken to be the reflux ratio and the distillate rate LJD, D or two product specifications such as bjdh bh/d,, XDh xBh > Td, 7, or combinations of these. The subscript / is used to denote the light key and the subscript h is used to denote the heavy key. In all of the optimization problems considered herein, the variables listed in items (1), (2), and (3) are always fixed. For convenience these variables are referred to collectively as the usual specifications. The remaining four variables, ku /c2, and two other specifications such as LJD and D are called additional specifications. ... 

Expressions of the same form as those shown above are obtained for the heavy key by replacing the subscript / by the subscript h. For definiteness in the formulation of the optimization problem, the two product specifications are taken to be bt/dt and bh/dh (or dt and dh). 

They depend on the operating and design parameters of a plant. Hereby, they are potential variables for any optimization problem and the basis for the formulation of objective functions. The definitions of objective functions are often identical for diverse chromatographic processes (e.g., batch or SMB), but different operating and design parameters must be applied for their calculation. 

In this work, the PSO algorithm was properly modified in order to satisfy the requirements of leading with discrete type variables and other strategies were also included to solve MINLP-based models. In addition, as criteria for obtaining the synthesis of the reuse water network, the minimization of the total cost was applied. At first, the WAP problem definitions and its mathematical formulation are presented. Then, the proposed modified PSO is shown, and finally applied in two literature case studies, mono and multicomponent problems, considering the minimization of annual total cost as optimization criteria. 

GAMS has a powerful feature which allows sets to be declared. Sets allow for subscripted variables used in variable and constant declarations, as well as equation definitions. As an example of the utility of declaring sets, an optimization problem might contain the five constraints ... 

As far as multidimensional optimization problems are concerned, the matrix B can be a bad approximation of the Hessian (provided it is positive definite) yet still be able to guarantee a reduction in the merit function. Conversely, the matrix B involved in the solution of nonlinear systems should be a good estimate of the Jacobian. 

By applying the presented Nataf model the multivariate distribution function is obtained by solving the optimization problem with four parameters for each random variable independently. The successful application of the model requires a positive definite covari-ance matrix Czz and continuous and strictly increasing distribution functions Fxtixi). In our smdy Equation 21 is solved iteratively to obtain Py for each pair of marginal distributions from the known correlation coefficient pij. 

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