The optimization problem

In Chapters 6 and 7 we have encountered two types of optimization problem (i) that which arises when we vary the orbitals in a wavefunc-tion of 1-determinant form and (ii) that which results when we vary the linear expansion coefficients in a wavefimction of many-determinant or Cl form such as (7.2.3). Optimization of the wavefunction with respect to linear parameters is a simple matter, depending only on solution of a large set of linear equations. But the optimization of even a relatively simple wavefunction with respect to orbital variations raises more difficult problems, typical of non-linear variation methods, as we have seen in both chapters. 

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The regression problem is here formulated as the optimization problem... 

Let u be a vector valued stochastic variable with dimension D x 1 and with covariance matrix Ru of size D x D. The key idea is to linearly transform all observation vectors, u , to new variables, z = W Uy, and then solve the optimization problem (1) where we replace u, by z . We choose the transformation so that the covariance matrix of z is diagonal and (more importantly) none if its eigenvalues are too close to zero. (Loosely speaking, the eigenvalues close to zero are those that are responsible for the large variance of the OLS-solution). In order to liiid the desired transformation, a singular value decomposition of /f is performed yielding... 

In the style of the Darwinian Theory, the quality of a chromosome is called its fitness. The quality or fitness of a ehromosome is usually caleulated with the help of an objeetive function, which is a mathematical function indicating how good the solution, and thus the chromosome, is for the optimization problem. This computation of the fitness is done for each chromosome in each population,... 

Since the flowrates of the streams fed to the sinks are to be determined as part of the optimization problem, the path equations represented by model (P7.1) should be revised as follows to allow for variable flowrate of the... 

The problem [Eq. (15)] is a minimax optimization problem. For the case (as it is here) where the approximating function depends linearly on the coefficients, the optimization problem [Eq. (15)] has the form of the Chebyshev approximation problem and has a known solution (Murty, 1983). Indeed, it can be easily shown that with the introduction of the dummy variables z, z, z the minimax problem can be transformed to the following linear program (LP) ... 

It is worth mentioning that in practice one often encounters situations when the benefits of meeting a specific requirement brings about a disadvantage in another area. The optimization problem often arises when it is necessary to match simultaneously several requirements sensitivity - selectivity, operational temperature - sensitivity, etc. For... 

In this case, there are n design variables, with p equality constraints and q inequality constraints. The existence of such constraints can simplify the optimization problem by reducing the size of the problem to be searched or avoiding problematic regions of the objective function. In general though, the existence of the constraints complicates the problem relative to the problem with no constraints. 

Now consider the influence of the inequality constraints on the optimization problem. The effect of inequality constraints is to reduce the size of the solution space that must be searched. However, the way in which the constraints bound the feasible region is important. Figure 3.10 illustrates the concept of convex and nonconvex regions. 

Whilst Example 3.1 is an extremely simple example, it illustrates a number of important points. If the optimization problem is completely linear, the solution space is convex and a global optimum solution can be generated. The optimum always occurs at an extreme point, as is illustrated in Figure 3.12. The optimum cannot occur inside the feasible region, it must always be at the boundary. For linear functions, running up the gradient can always increase the objective function until a boundary wall is hit. 

The nature of the optimization problem can mm out to be linear or nonlinear depending on the mass transfer model chosen14. If a model based on a fixed outlet concentration is chosen, the model turns out to be a linear model (assuming linear cost models are adopted). If the outlet concentration is allowed to vary, as in Figure 26.35a and Figure 26.35b, then the optimization turns out to be a nonlinear optimization with all the problems of local optima associated with such problems. The optimization is in fact not so difficult in practice as regards the nonlinearity, because it is possible to provide a good initialization to the nonlinear model. If the outlet concentrations from each operation are initially assumed to go to their maximum outlet concentrations, then this can then be solved by a linear optimization. This usually... 

The advantage of this approach stems from the fact that the minimization over x can be carried out analytically. Consequently, the dimensionality of the optimization problem is reduced from P, the number of pipe sections to S, the number of paths. Murtagh (M9) reported computer storage reduction of more than 50% and computing time reduction of up to 80% using the dual instead of primal formulation. 

Thirdly, the inlet and outlet concentrations were specified such that one was fixed directly and the other determined by mass balance using flowrate and mass load. However, a number of variations are possible in the way that the process constraints on quantity (or flowrate) present themselves. For instance, it could happen that there is no direct specification of the water quantity (or flow) in a particular stream, as long as the contaminant load and the outlet concentration are observed. Furthermore, the vessel probably has minimum and maximum levels for effective operation. In that case the water quantity falls away as an equality constraints, to become an inequality constraints, thereby changing the nature of the optimization problem. 

Inspection of this equation indicates that the pressure drop will be quite sensitive to the pipe diameter employed. To minimize the pressure drop, large tubes are desired, but the larger the pipe, the lower the heat transfer area per unit volume. The three specified pipe sizes have been chosen to illustrate the optimization problem. We are now prepared to carry out the calculations for a single length of each of these pipe sizes. For hand calculations we have the necessary relations in equations 13.1.24, 13.1.29, and... 

As shown in Fig. 3-53, optimization problems that arise in chemical engineering can be classified in terms of continuous and discrete variables. For the former, nonlinear programming (NLP) problems form the most general case, and widely applied specializations include linear programming (LP) and quadratic programming (QP). An important distinction for NLP is whether the optimization problem is convex or nonconvex. The latter NLP problem may have multiple local optima, and an important question is whether a global solution is required for the NLP. Another important distinction is whether the problem is assumed to be differentiable or not. 

A broad class of optimization strategies does not require derivative information. These methods have the advantage of easy implementation and little prior knowledge of the optimization problem. In particular, such methods are well suited for quick and dirty optimization studies that explore the scope of optimization for new problems, prior to investing effort for more sophisticated modeling and solution strategies. Most of these methods are derived from heuristics that naturally spawn numerous variations. As a result, a very broad literature describes these methods. Here we discuss only a few important trends in this area. 

For optimization problems that are derived from (ordinary or partial) differential equation models, a number of advanced optimization strategies can be applied. Most of these problems are posed as NLPs, although recent work has also extended these models to MINLPs and global optimization formulations. For the optimization of profiles in time and space, indirect methods can be applied based on the optimality conditions of the infinite-dimensional problem using, for instance, the calculus of variations. However, these methods become difficult to apply if inequality constraints and discrete decisions are part of the optimization problem. Instead, current methods are based on NLP and MINLP formulations and can be divided into two classes ... 

Optimization pervades the fields of science, engineering, and business. In physics, many different optimal principles have been enunciated, describing natural phenomena in the fields of optics and classical mechanics. The field of statistics treats various principles termed maximum likelihood, minimum loss, and least squares, and business makes use of maximum profit, minimum cost, maximum use of resources, minimum effort, in its efforts to increase profits. A typical engineering problem can be posed as follows A process can be represented by some equations or perhaps solely by experimental data. You have a single performance criterion in mind such as minimum cost. The goal of optimization is to find the values of the variables in the process that yield the best value of the performance criterion. A trade-off usually exists between capital and operating costs. The described factors—process or model and the performance criterion—constitute the optimization problem. ... 

EXAMPLE 1.5 OPTIMAL SCHEDULING FORMULATION OF THE OPTIMATION PROBLEM... 

How many days per year (365 days) should each plant operate processing each kind of material Hints Does the table contain the variables to be optimized How do you use the information mathematically to formulate the optimization problem What other factors must you consider ... 

If the objective function and constraints in an optimization problem are nicely behaved, optimization presents no great difficulty. In particular, if the objective function and constraints are all linear, a powerful method known as linear programming can be used to solve the optimization problem (refer to Chapter 7). For this specific type of problem it is known that a unique solution exists if any solution exists. However, most optimization problems in their natural formulation are not linear. 

The objective function may exhibit many local optima, whereas the global optimum is sought. A solution to the optimization problem may be obtained that is less satisfactory than another solution elsewhere in the region. The better solution may be reached only by initiating the search for the optimum from a different starting point. 

For each of the following six problems, formulate the objective function, the equality constraints (if any), and the inequality constraints (if any). Specify and list the independent variables, the number of degrees of freedom, and the coefficients in the optimization problem. 

The required removal in each unit should be adjusted so that the final exit concentration x3 must be less than 0.05. Formulate (only) the optimization problem listing the objective fimction and constraints. 

Before carrying out the actual modeling, it is important to evaluate the economic justification for (and benefits of) the modeling effort and the capability of support staff for carrying out such a project. Primarily, determine that a successfully developed model will indeed help solve the optimization problem. 

Formulate the optimization problem using only the following notation (as needed) ... 

The optimization problem in this example comprises a linear objective function and linear constraints, hence linear programming is the best technique for solving it (refer to Chapter 7). 

Use the following notation to formulate the optimization problem, and solve it for the values of Ix and /2 as well as the values of Sy. Each plant has a maximum capacity of 500 units per day. 

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