Constrained extrema

In many apphcations we need to find the maxima or minima of a given function /(xi,X2. x ) subject to some constraints. These constraints are expressed as given relationships between the variables that we express by 

Such constrained extrema can be found by the Lagrange multipliers method One form the Lagrangian  

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Once the objective and the constraints have been set, a mathematical model of the process can be subjected to a search strategy to find the optimum. Simple calculus is adequate for some problems, or Lagrange multipliers can be used for constrained extrema. When a Rill plant simulation can be made, various alternatives can be put through the computer. Such an operation is called jlowsheeting. A chapter is devoted to this topic by Edgar and Himmelblau Optimization of Chemical Processes, McGraw-HiU, 1988) where they list a number of commercially available software packages for this purpose, one of the first of which was Flowtran. 

For the general case Hie foregoing necessary conditions for a constrained extremum follow from the necessary condition for an extremum of the lagrangian,... 

Thomas L. Saaty, Coefficient Perturbation of a Constrained Extremum, Operations Research, 7, No. 3 (1959). 

In some problems the possible region of independent variables is defined by equality or inequality constraints. As you have seen in Section 1.2, such constrained extremum problems are easy to solve if both the constraints and the... 

To obtain a meaningful extremum problem the number of experiments and the set of feasible vectors of the independent variables T are fixed. In most cases T is defined by inequalities x1- < x < x, i = l,2,...,k. Though introducing penalty functions such constrained extremum problems can be solved by the methods and modules described in Section 2.4, this direct approach is usually very inefficient. In fact, experiment design is not easy. The dimensionality of the extremum problem is high, the extrema are partly on the boundaries of the feasible region T, and since the objective functions are... 

The above example clearly shows the difference between the absolute extremum and the constrained extremum. 

In the equilibrium statistical mechanics, the unknown probabilities of microstates p, are found from the second part of the second law of thermodynamics, i.e., from the constrained extremum of the thermodynamic potential (Eq. (29)) as a function of the variables (pv pw) under the condition that the variables (pv. .., pw) satisfy Eq. (27). Moreover, it is supposed that the value of the entropy in the i th microstate of the system is a function of the probability pt of this microstate, i.e., =Sf=Sf(pf). Then to determine the unknown probabilities [pt] at... 

The procedure, used so far to adjust parameters for the mixture to (V = L) critical data, outlined in par. 4 can be replaced by the following. We are dealing here with the problem to find the (absolute) extreme of the pressure (eq. 2) on the spinodal curve at constant temperature. This leads to the so called constrained extremum problem, which can be solved by the method of Lagrange multipliers. Therefore it is required to construct a helping function Z of the independant concentration variables and 2 ... 

Straight forward algebra and elimination of X from (5) and (6) leads to the solution of the constrained extremum problem ... 

We solve this constrained extremum problem by Lagrange s method of multipliers using the... 

This result shows how the subsystems share the thermodynamic forces. What is the situation with the extremum property of GEP The equations to solve this problem could be derived from the following constrained extremum-task... 

Hence the free forces could be determined by the following constrained extremum- task ... 

According to the maximum entropy principle, the probability density p describing the macroscopic state of our system must represent all available information, OjYJ = 0,..., n, and correspond to a maximum in Gibbs entropy S. The actual constraction of p can be accomplished by adopting a procedure due to Lagrange that allows one to change a constrained extremum problem into an unconstrained extremum problem. 

Furthermore, this shows that at the rainbow, j is a constrained extremum, the constraint being that 0 is constant. 

We could now reverse the roles of j and 0 in the above paragraph and arrive at another set of rainbow points at which the other two partial derivatives were pair-wise singular. At each of these points 0 would be a constrained extremum with the constraint that j is constant. In fact, however, the set of rainbow points found in this way would be identical to the set found in the above paragraph. That is, at each rainbow point (0r>Jr) four of the partial derivatives... 

Because of the dependence of the determinants forming this set on the orbital expansion coefficients as well as on the orbital exponents, it is clear that the optimal determinant which minimizes the kinetic energy for the noninteracting system can be reached by varying these orbital parameters. At the extremum of this constrained variation we have, therefore, the following inequality for the kinetic energy functional ... 

Sometimes we must find a maximum or a minimum value of a function subject to some condition, which is called a constraint. Such an extremum is called a constrained maximum or a constrained minimum. Generally, a constrained maximum is smaller than the unconstrained maximum of the function, and a constrained minimum is larger than the unconstrained minimum of the function. Consider the following example ... 

One application of partial derivatives is in the search for minimum and maximum values of a function. An extremum (minimum or maximum) of a function in a region is found either at a boundary of the region or at a point where all of the partial derivatives vanish. A constrained maximum or minimum is found by the method of Lagrange, in which a particular augmented function is maximized or minimized. 

There are certain rules, due to thermodynamic constrains, that these extrema (minima or maxima) lines must satisfy. For example, the and Cp lines must emerge from the LLCP. However, this is not necessarily the case for other extrema lines such as the p line (e.g., see Fig. 5a). In fact, thermodynamic arguments [45,46] indicate that the line must either merge with the p""" line or end at a spinodal line. In the first case, the line must merge with the p " line at a point in the P-T plane where the line has an extremum point see... 

If the extremum of a function such as 5 ((/) predicts equilibrium, the variable U is called the natural variable of 5. T is not a natural variable of 5. Now we show that (T,V,N) are natural variables of a function F, the Helmholtz free energy. An extremum in f (T, V,N) predicts equilibria in systems that are constrained to constant temperature at their boundaries. 

Before taking up the specific discussion on our results, let us highlight that the existence of extremum points in the SC is a consequence of the confinement effects brought about by the presence of external fields. This is in line with the recent finding of Patil et al. [25], who show for some specific constrained Coulomb potentials that the simplest composite uncertainty measure, the Heisenberg uncertainty product, of the electron density presents an extremum located at a critical position which scales as the reciprocal value of the potential strength. 

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