Maximum constrained

In the hope that additional constraints such as positivity (which must hold for the restored brightness distribution) may avoid noise amplification, we can seek for the constrained maximum likelihood (CML) solution ... 

Image Space Reconstruction Algorithm. ISRA (Daube-Witherspoon and MuehUehner, 1986) is a multiplicative and iterative method which yields the constrained maximum likelihood in the case of Gaussian noise. The ISRA solution is obtained using the recursion ... 

Figures 4b and 4c show that neither unconstrained nor non-negative maximum likelihood approaches are able to recover a usable image. Deconvolution by unconstrained/constrained maximum likelihood yields noise amplification — in otfier words, the maximum likelihood solution remains iU-conditioned (i.e. a small change in the data due to noise can produce arbitrarily large changes in the solution) regularization is needed.

The conditions yielding the unconstrained maximum centerline deposition rate give a deposition uniformity of only about 25%. While this may well be acceptable for some fiber coating processes, there are likely applications for which it is not. We now consider the problem of maximizing the centerline deposition rate, subject to an additional constraint that the deposition uniformity satisfies some minimum requirement. Assuming that the required uniformity is better than that obtained in the unconstrained case, the constrained maximum centerline deposition rate should occur when the uniformity constraint is just marginally satisfied. This permits replacing the inequality constraint of a minimum uniformity by an equality constraint that is satisfied exactly. 

The constrained maximum deposition rate is defined by the requirement that the variation in rate with respect to all independent variables is zero along the direction of constant uniformity. To obtain this maximum we employ the method of Lagrange multipliers. This technique introduces one new unknown... 

Is nondegenerate and corresponds to S = 0 and T = 0. The nondegeneracy of the ground state is a consequence of the third law of classical thermodynamics. The boundary EgAgAg represents the stable equilibrium states of the system, which may be treated by classical thermodynamics. Thus, stable-equilibrium-state quantum mechanics is constrained-maximum-entropy physics. 

Figure 7.8 The surface representing a function of x and y with the absolute maximum and a constrained maximum shown.

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 ... 

This function is given by the line in the x-y plane of the figure. We are now looking for the place along this curve at which the function has a larger value than at any other place on the curve. Unless the curve happens to pass through the unconstrained maximum, the constrained maximum will be smaller than the unconstrained maximum. 

If we have a constrained maximum or minimum problem with more than two variables, the direct method of substituting the constraint relation into the function is usually not practical. Lagrange s method finds a constrained maximum or minimum without substituting the constraint relation into the function. If the constraint is written in the form g(x, y) = 0, the method for finding the constrained maximum or minimum in f x, y) is as follows ... 

EXAMPLE 7.32 Find the constrained maximum of Example 7.31 by the method of Lagrange. 

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. 

The siHailest value of for a given pressure difference iPi — Po) obviopsly exists when P has a tnajaihihit valhe. The method of Lagrange multipliers may be uwd for determining such a constrained maximum (34). This method indicates that the maximum value of F exists when... 

Ziegler s principle. If the forces X, are prescribed, the actual fluxes /, (which are satisfied the entropy production c(/, ) = /,X,), maximize the function of entropy production (304), when the possible form of constitutive equations (303) are fulfilled as well. This principle can be formulated as a constrained maximum task, i.e. 

In order to solve for the two Lagrange multipliers, SMO first computes the constraints on these multipliers and then solves for the constrained maximum. 

The next step of SMO is to compute the location of the constrained maximum of the objective function in the following equation while allowing only two Lagrange multipliers to change. Under normal circumstances (it 9 0), there will be a maximum along the direction of the linear equality constrain, and k will be less than zero. In this case, SMO computes the maximum along the direction of the constraint. 

The figure also shows a curve at which the surface intersects with a plane representing the equation y = 1 — x. On this curve there is also a maximum, which has a smaller value than the maximum at the peak. We call this value a constrained maximum subject to the constraint that y = 1 — X. We discuss the constrained maximum later. 

---