Unconstrained

Exactly the same types of step as for an unconstrained optimization can then be taken, using the modified as opposed to the regular gradient. 

Fletcher R 1981 Practical Methods of Optimization Vol 1—Unconstrained Optimization (New York Wley)... 

Dennis J E and Schnabel R B 1983 Numerical Methods for Unconstrained Optimization and Non-linear Equations (Englewood Cliffs, NJ Prentice-Hall)... 

Powell M J D 1971 Recent advances in unconstrained optimization Math. Prog. 1 26... 

The two /3-turn structures, pc and Pe are the most stable among those considered. This is in accord with the unconstrained nanosecond simulations of linear DPDPE, which converged to these conformers [14]. Because the cyclic form is relatively rigid, it is assumed that the conformation it adopts in solution is the biologically active one, responsible for its high affinity and specificity towards the 5 opioid receptor. The relatively low population of the cyclic-like structure for the linear peptide thus agrees qualitatively with the... 

The free energy differences obtained from our constrained simulations refer to strictly specified states, defined by single points in the 14-dimensional dihedral space. Standard concepts of a molecular conformation include some region, or volume in that space, explored by thermal fluctuations around a transient equilibrium structure. To obtain the free energy differences between conformers of the unconstrained peptide, a correction for the thermodynamic state is needed. The volume of explored conformational space may be estimated from the covariance matrix of the coordinates of interest, = ((Ci [13, lOj. For each of the four selected conform-... 

Although unconstrained dynamics is being considered here, the ideas extend to the case where bond lengths (and bond angles) are constrained. Also, the ideas are applicable to other than constant NVE simulations. 

The method presented in the next section is an attempt to overcome the barrier due to the highest frequencies whatever their origin. Although it has been implemented and tested for unconstrained dynamics only, there is no fundamental reason why it cannot be applied to overcome the less restrictive time step barrier arising in constrained dynamics. 

It has been observed by [27, 24] that the equations of motion of a free rigid body are subject to reduction. (For a detailed discussion of this interesting topic, see [23].) This leads to an unconstrained Lie-Poisson system which is directly solvable by splitting, i.e. the Euler equations in the angular momenta ... 

HyperChem includes only unconstrained optimization. That is, given the coordinates ( X of a set of atoms. A (the inde-... 

Unconstrained optimization methods [W. II. Press, et. ah, Numerical Recipes The An of Scieniific Compulime.. Cambridge University Press, 1 9H6. Chapter 101 can use values of only the objective function, or of first derivatives of the objective function. second derivatives of the objective function, etc. llyperChem uses first derivative information and, in the Block Diagonal Newton-Raphson case, second derivatives for one atom at a time. TlyperChem does not use optimizers that compute the full set of second derivatives (th e Hessian ) because it is im practical to store the Hessian for mac-romoleciiles with thousands of atoms. A future release may make explicit-Hessian meth oils available for smaller molecules but at this release only methods that store the first derivative information, or the second derivatives of a single atom, are used. 

The temperature of the system is related to the time average of the kinetic energy, which fc an unconstrained system is given by ... 

In Figure 5.23 the finite element model predictions based on with constraint and unconstrained boundary conditions for the modulus of a glass/epoxy resin composite for various filler volume fractions are shown. 

Figure 5.23 Composite modulus obtained using constrained and unconstrained boundary conditions...

In Figure 5.24 the predicted direct stress distributions for a glass-filled epoxy resin under unconstrained conditions for both pha.ses are shown. The material parameters used in this calculation are elasticity modulus and Poisson s ratio of (3.01 GPa, 0.35) for the epoxy matrix and (76.0 GPa, 0.21) for glass spheres, respectively. According to this result the position of maximum stress concentration is almost directly above the pole of the spherical particle. Therefore for a... 

The response surfaces in Figure 14.2 are plotted for a limited range of factor levels (0 < A < 10, 0 < B < 10), but can be extended toward more positive or more negative values. This is an example of an unconstrained response surface. Most response surfaces of interest to analytical chemists, however, are naturally constrained by the nature of the factors or the response or are constrained by practical limits set by the analyst. The response surface in Figure 14.1, for example, has a natural constraint on its factor since the smallest possible concentration for the analyte is zero. Furthermore, an upper limit exists because it is usually undesirable to extrapolate a calibration curve beyond the highest concentration standard. 

The distance from the crack tip, along the X-axis, at which the von Mises equivalent stress falls below the yield stress, defines the size of the plastic zone, r. For the plane stress case of unconstrained yielding, which corresponds to the free surface of the specimen in Figure 4, this gives... 

J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimi tion and Non-Uinear Equations, Prentice-Hall, Englewood Chffs, N.J., 1983. 

Local Minimum Point for Unconstrained Problems Consider the following unconstrained optimization problem ... 

Optimization of Unconstrained Olnective Assume the objective Func tion F is a function of independent variables i = r. A computer program, given the values for the independent variables, can calculate F and its derivatives with respect to each Uj. Assume that F is well approximated as an as-yet-unknown quadratic function in u. 

The calculations begin with given values for the independent variables u and exit with the (constrained) derivatives of the objective function with respec t to them. Use the routine described above for the unconstrained problem where a succession of quadratic fits is used to move toward the optimal point for an unconstrained problem. This approach is a form or the generahzed reduced gradient (GRG) approach to optimizing, one of the better ways to cany out optimization numerically. 

Fluidized Beds When gas or liquid flows upward through a vertically unconstrained bed of particles, there is a minimum fluid velocity at which the particles will begin to move. Above this minimum velocity, the bed is said to be fluidized. Fluidized beds are widely used, in part because of their excellent mixing and heat and mass transfer characteristics. See Sec. 17 of this Handbook for detailed information. 

Unconstrained Optimization Unconstrained optimization refers to the case where no inequahty constraints are present and all equahty constraints can be eliminated by solving for selected dependent variables followed by substitution for them in the objec tive func tion. Veiy few reahstic problems in process optimization are unconstrained. However, it is desirable to have efficient unconstrained optimization techniques available since these techniques must be applied in real time and iterative calculations cost computer time. The two classes of unconstrained techniques are single-variable optimization and multivariable optimization. 

There are two basic types of unconstrained optimization algorithms (I) those reqmring function derivatives and (2) those that do not. The nonderivative methods are of interest in optimization applications because these methods can be readily adapted to the case in which experiments are carried out directly on the process. In such cases, an ac tual process measurement (such as yield) can be the objec tive function, and no mathematical model for the process is required. Methods that do not reqmre derivatives are called direc t methods and include sequential simplex (Nelder-Meade) and Powell s method. The sequential simplex method is quite satisfac tory for optimization with two or three independent variables, is simple to understand, and is fairly easy to execute. Powell s method is more efficient than the simplex method and is based on the concept of conjugate search directions. 

Constrained Optimization When constraints exist and cannot be eliminated in an optimization problem, more general methods must be employed than those described above, since the unconstrained optimum may correspond to unrealistic values of the operating variables. The general form of a nonhuear programming problem allows for a nonlinear objec tive function and nonlinear constraints, or... 

Nonlinear Programming The most general case for optimization occurs when both the objective function and constraints are nonlinear, a case referred to as nonlinear programming. While the idea behind the search methods used for unconstrained multivariable problems are applicable, the presence of constraints complicates the solution procedure. 

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