Objective function, definition

This approach enables the incorporation of input, output, and final-time constraints and is flexible with respect to the crystallizer configuration, the objective function definition, and the choice of manipulated variables. Examples of the use of nonlinear programming to solve this problem are given subsequently. 

F2602sub.m objective function definition used by F2602.m for optimization... 

In this step, the optimization problem is formulated as MILP or MINLP problem depending on the objective function definition and constraints using appropriate software, in this case GAMS. The ouQ)ut is the optimal biorefmery configuration. The generic models and stracture of the optimization problem (MIP/MINLP) organized and used in this study are presented and explained in the following text. 

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). 

The traditional approach to optimize a process is schematically shown in Figure 2 its principle elements are the development of a model, model validation, definition of an objective function and an optimizing algorithn. The "model" can be (a) theoretical, (b) empirical or (c) a combination of the two. 

There is one additional reason why the IT empirical risk is a better objective function to use. With the empirical risk given by Eq. (6), which is by definition a pointwise measure, it is clear how to define in practice the... 

As matrix Q, is positive definite, the above equation gives the minimum of the objective function. 

A free interpretation, loosely speaking, is a minimal one - one in which we make as few decisions as possible in fulfilling the definition of an interpretation of P. In particular, we establish no relations whatsoever among members of the domain and establish no connections between objects, functions, and the values of functions on those objects, except those required by formal identity. Thus f(x,g(y,z)) must be equal to itself and must be the result of applying f to x and g(y,z) - but we assume that it is distinct from, say, g(f(x,x),y)) and that there is no relationship whatsoever between g(y,z) and f(x,x) or g(x,x). ... 

We can conclude that the continuous-time STN and RTN models based on the definition of global time points are quite general. They are capable of easily accommodating a variety of objective functions such as profit maximization or makespan minimization. However, events taking place during the time horizon such as multiple due dates and raw material receptions are more complex to implement given that the exact position of the time points is unknown. 

Steps 1, 2, and 3 deal with the mathematical definition of the problem, that is, identification of variables, specification of the objective function, and statement of the constraints. We devote considerable attention to problem formulation in the remainder of this chapter, as well as in Chapters 2 and 3. If the process to be optimized is very complex, it may be necessary to reformulate the problem so that it can be solved with reasonable effort. 

Steepest descent can terminate at any type of stationary point, that is, at any point where the elements of the gradient of /(x) are zero. Thus you must ascertain if the presumed minimum is indeed a local minimum (i.e., a solution) or a saddle point. If it is a saddle point, it is necessary to employ a nongradient method to move away from the point, after which the minimization may continue as before. The stationary point may be tested by examining the Hessian matrix of the objective function as described in Chapter 4. If the Hessian matrix is not positive-definite, the stationary point is a saddle point. Perturbation from the stationary point followed by optimization should lead to a local minimum x. ... 

Is it necessary that the Hessian matrix of the objective function always be positive-definite in an unconstrained minimization problem ... 

Show how to make the Hessian matrix of the following objective function positive-definite at x = [1 l]r by using Marquardt s method ... 

If no active constraints occur (so x is an unconstrained stationary point), then (8.32a) must hold for all vectors y, and the multipliers A and u are zero, so V L = V /. Hence (8.32a) and (8.32b) reduce to the condition discussed in Section 4.5 that if the Hessian matrix of the objective function, evaluated at x, is positive-definite and x is a stationary point, then x is a local unconstrained minimum of/. 

The condition number of the Hessian matrix of the objective function is an important measure of difficulty in unconstrained optimization. By definition, the smallest a condition number can be is 1.0. A condition number of 105 is moderately large, 109 is large, and 1014 is extremely large. Recall that, if Newton s method is used to minimize a function/, the Newton search direction s is found by solving the linear equations... 

The value objective function is oriented at the company s profit and loss definitions. Guiding principle is to only use value parameters that can be found in the cost controlling of the company signed by controlling. Penalty costs and without currency and weighting factors being applied to steer optimization results but having no actual financial impact - as it can be often found in supply chain optimization models - do not meet this requirement. 

From the definition of the objective functions (Equations 4.88 and 4.89), it can be seen that different scalings of the x-variables would result in different penalization, because only the coefficients themselves but no information about the scale of the x-variables is included in the term for penalization. Therefore the x-variables are usually autoscaled. Note that the intercept b0 is not included in the penalization term in order to make the result not be depending on the origin of the y-variable. 

With these definitions, the mathematical derivation of the FCV family of clustering algorithms depends upon minimizing the generalized weighted sum-of-squared-error objective functional... 

In the definition of the Lagrange function L(x, A, m) (see section 3.2.2) we associated Lagrange multipliers with the equality and inequality constraints only. If, however, a Lagrange multiplier Mo is associated with the objective function as well, the definition of the weak Lagrange function L (x, A, fi) results that is,... 

Remark 3 The objective function is a linear sum of all yij s and simply minimizes the number of potential matches. The energy balances and the definition constraints are linear constraints in the residuals and heat loads. The relations between the continuous and binary variables are also linear since L j and Uij are parameters corresponding to lower and upper bounds, respectively, on the heat exchange of each match (ij). It is important to understand the key role of these constraints which is to make certain that if a match does not exist, then its heat load should be zero while if the match takes place, then its heat load should be between the provided bounds. We can observe that ... 

The objective function represents the total annual cost and consists of the investment and operating cost properly weighted. In the following form of the objective function, we substitute the areas of the heat exchangers via the presented definitions in constraints (B) ... 

From these results, it is found that the proposed multi-criteria synthesis strategy can attain a definite and compromised solution for a problem with assorted conflict objectives. The preference intervals of various objectives have significant effects on final HEN structures. Such acceptable and/or preference intervals can also reflect the importance of different objective functions. Should one specific objective is emphasized, a tighter restriction or shrinking span should be placed on its acceptable ranges. 

The results of the latter two studies described contradict earlier data. Up to this point, an elevated TEWL had been the most widely accepted biophysiological parameter associated with sensitive skin, due to impairment of the skin barrier function or composition. That further studies are imperative to create a consistent and objective operational definition for sensitive skin is affirmed by these conflicting results. 

Optimal Control. Optimal control is extension of the principles of parameter optimization to dynamic systems. In this case one wishes to optimize a scalar objective function, which may be a definite integral of some function of the state and control variables, subject to a constraint, namely a dynamic equation, such as Equation (1). The solution to this problem requires the use of time-varying Lagrange multipliers for a general objective function and state equation, an analytical solution is rarely forthcoming. However, a specific case of the optimal control problem does lend itself to analytical solution, namely a state equation described by Equation (1) and a quadratic objective function given by... 

Mathematical optimization deals with determining values for a set of unknown variables x, X2, , x , which best satisfy (optimize) some mathematical objective quantified by a scalar function of the unknown variables, F(xi, X2, , xn). The function F is termed the objective function bounds on the variables, along with mathematical dependencies between them, are termed constraints. Constraint-based analysis of metabolic systems requires definition of the constraints acting on biochemical variables (fluxes, concentrations, enzyme activities) and determining appropriate objective functions useful in determining the behavior of metabolic systems. 

The general formulation for a dynamic-programming problem, presented in a simplified form, is shown in Fig. 11-11. On the basis of the definitions of terms given in Fig. 11-lla, each of the variables, x1+1, xt, and dt, may be replaced by vectors because there may be several components or streams involved in the input and output, and several decision variables may be involved. The profit or return Pt is a scalar which gives a measure of contribution of stage i to the objective function. 

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