Unconstrained minimization

Residual minimization method (RMM-DIIS). Wood and Zunger [27] proposed lo minimize the norm of the residual vector instead of the Rayleigh quotient. This is an unconstrained minimization condition. Each minimization step starts with the evaluation of the preconditioned residual vector K for the approximate eigenstate... 

The point where the constraint is satisfied, (x0,yo), may or may not belong to the data set (xj,yj) i=l,...,N. The above constrained minimization problem can be transformed into an unconstrained one by introducing the Lagrange multiplier, to and augmenting the least squares objective function to form the La-grangian,... 

The problem of minimizing Equation 14.24 subject to the constraint given by Equation 14.26 or 14.28 is transformed into an unconstrained one by introducing the Lagrange multiplier, to, and augmenting the LS objective function, SLS(k), to yield... 

Copp and Everet (1953) have presented 33 experimental VLE data points at three temperatures. The diethylamine-water system demonstrates the problem that may arise when using the simplified constrained least squares estimation due to inadequate number of data. In such case there is a need to interpolate the data points and to perform the minimization subject to constraint of Equation 14.28 instead of Equation 14.26 (Englezos and Kalogerakis, 1993). First, unconstrained LS estimation was performed by using the objective function defined by Equation 14.23. The parameter values together with their standard deviations that were obtained are shown in Table 14.5. The covariances are also given in the table. The other parameter values are zero. 

Fiacco, A. V. McCormick, G. P. "Nonlinear Sequential Unconstrained Minimization Techniques" John Wiley Sons ... 

Ordejon, P., D. A. Drabold, M. P. Grumback, and R. M. Martin. 1993. Unconstrained Minimization Approach for Electronic Computations Which Scales Linearly with System Size. Phys. Rev. B 14646. 

Banga et al. [in State of the Art in Global Optimization, C. Floudas and P. Pardalos (eds.), Kluwer, Dordrecht, p. 563 (1996)]. All these methods require only objective function values for unconstrained minimization. Associated with these methods are numerous studies on a wide range of process problems. Moreover, many of these methods include heuristics that prevent premature termination (e.g., directional flexibility in the complex search as well as random restarts and direction generation). To illustrate these methods, Fig. 3-58 illustrates the performance of a pattern search method as well as a random search method on an unconstrained problem. 

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

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... 

Despite the exactness feature of Pv no general-purpose, widely available NLP solver is based solely on the Lx exact penalty function Pv This is because Px also has a negative characteristic it is nonsmooth. The term hj(x) has a discontinuous derivative at any point x where hj (x) = 0, that is, at any point satisfying the y th equality constraint in addition, max 0, gj (x) has a discontinuous derivative at any x where gj (x) = 0, that is, whenever the yth inequality constraint is active, as illustrated in Figure 8.6. These discontinuities occur at any feasible or partially feasible point, so none of the efficient unconstrained minimizers for smooth problems considered in Chapter 6 can be applied, because they eventually encounter points where Px is nonsmooth. 

This backtracking line search tries a = 1.0 first and accepts it if the sufficient decrease criterion (8.78) is met. This criterion is also used in unconstrained minimization, as discussed in Section 6.3.2. If a = 1.0 fails the test (8.78), a safe-... 

Figure 3.13 Constrained minimization the minimum of a function f x) submitted to the constraint g(x)=0 occurs at M on the constraint subspace, here on the curve (x)=0 where Vf(x)+XVg(x)=0. P is the unconstrained minimum of/( ). This principle is the base for the method of Lagrange multipliers.

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

We conclude that a) all degrees of freedom should be unconstrained (including those related to the shape of the sugar rings)/ b) the packing forces can significantly influence the conformation even of a polymer main chain and c) a suitable potential set (with all atoms), used in total energy minimization, can account for these effects. With these considerations in mind, we attempted to model the crystal structures of DeS and an isolated chain of Hep. 

Rather than minimize the energy function o(P) = (P, H) by varying over the set of fe-matrices, there is a dual formulation where the bottom eigenvalue /lo(H + S) of the matrix H + S is maximized over the set of Pauli matrices S e The dual formulation can be derived using Lagrange s method, which requires converting the constrained energy problem to an unconstrained one. If... 

It is easy to modify the program to solve the problem assuming 57. and 10 /. errors in observations. Results are summarized in Table 1.2. The table also includes the unconstrained least squares estimates of x, i.e., the values minimizing (1.86) with n = 3 and m = 20. This latter result was obtained by inserting the appropriate data into the main program of Section 3.2. ... 

A. V. Fiacco and G. R McCormick. Nonlinear Programming Sequential Unconstrained Minimization Techniques. Society for Industrial and Applied Mathematics, 1968. 

Another useful program (E04HAA) provides constrained optimization with bounds for each parameter using a sequential penalty function technique, which effectively operates around unconstrained minimization cycles. 

Internal ligand relaxation allows the removal of strain possibly imposed on the ligands by the receptor during correlation-coupled refinement but usually yields suboptimal models. Therefore, correlation-coupled receptor minimization followed by unconstrained ligand relaxation is repeated several times until a highly correlated pseudoreceptor model is obtained in the relaxed state (designated ligand equilibration). 

The majority of minimization routines are designed for unconstrained optimization, in which the control algorithm is free to select any parameters it wishes. Only a minority can handle constrained optimization. 

Because (4> ff S) is not in itself a variational expression, its unconstrained minimum value is not simply related to an eigenstate of the Hamiltonian Hv defined by v in Eq.(3), whereas Eq.(2) defines F[p only for such eigenstates. Any arbitrary trial function J —> can be expressed in the form + Aca with ca = 1. If the minimizing trial function in Eq.(3) were not an eigenfunction of Hv, then for some subset of trial functions, using the Brueckner-Brenig condition,... 

In conventional variance minimization calculations (14) (i.e. the unconstrained A = 1 case), the above property is used to find an overall fit to IP (we drop the A superscript for simplicity). The procedure is to determine the parameters a in the trial function vPt(R-5 a ) by minimizing the variance of local energy a2... 

When unconstrained energy minimization leads to a symmetry-broken single-configuration solution, one may obviously solve the problem by imposing symmetry constraints to force it to assume the required symmetry properties. 

There is no need for orbital constraints to enforce the fully symmetric nature of the two inner orbitals or the symmetry relations between the three pairs of valence orbitals. The fully-symmetric SC solution corresponds to a proper minimum in the unconstrained SC optimization space. It has been verified to be stable against symmetry-breaking perturbations, including the admixture of n basis functions into the orbitals, in the sense that energy minimization from such a perturbed initial guess spontaneously restores the orbitals to purely a character and to full symmetry, converging back onto the unperturbed solution. 

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