Quadratic algorithm

For systems smaller than Ni, the most efficient method is the quartic one, the quadratic algorithm is the most efficient for systems sizes between Ni and N2, while the linear scaling method becomes the most efficient beyond N2. Note that N2 may be so large that the total computational resources may be exhausted before the cross-over point is reached. 

Simulations of the adaptive reconstruction have been performed for a single slice of a porosity in ferritic weld as shown in Fig. 2a [11]. The image matrix has the dimensions 230x120 pixels. The number of beams in each projection is M=131. The total number of projections K was chosen to be 50. For the projections the usual CT setup was used restricted to angels between 0° and 180° with the uniform step size of about 3.7°. The diagonal form of the quadratic criteria F(a,a) and f(a,a) were used for the reconstruction algorithms (5) and (6). 

Baker J, KInghorn D and Pulay P 1999 Geometry optimization In delocalized Internal coordinates An efficient quadratically scaling algorithm for large molecules J. Chem. Phys. 110 4986... 

IlyperChem supplies two differeiii types or algorithms for transition state search eigenvector I ollowing and synchronous transit (linear and quadratic search ). 

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. 

Reformulating the necessaiy conditions as a linear quadratic program has an interesting side effect. We can simply add linearizations of the inactive inequalities to the problem ana let the ac tive set be selected by the algorithm used to solve the linear quadratic program. 

One important class of nonlinear programming techniques is called quadratic programming (QP), where the objective function is quadratic and the constraints are hnear. While the solution is iterative, it can be obtained qmckly as in linear programming. This is the basis for the newest type of constrained multivariable control algorithms called model predic tive control. The dominant method used in the refining industiy utilizes the solution of a QP and is called dynamic matrix con-... 

There are several reasons that Newton-Raphson minimization is rarely used in mac-romolecular studies. First, the highly nonquadratic macromolecular energy surface, which is characterized by a multitude of local minima, is unsuitable for the Newton-Raphson method. In such cases it is inefficient, at times even pathological, in behavior. It is, however, sometimes used to complete the minimization of a structure that was already minimized by another method. In such cases it is assumed that the starting point is close enough to the real minimum to justify the quadratic approximation. Second, the need to recalculate the Hessian matrix at every iteration makes this algorithm computationally expensive. Third, it is necessary to invert the second derivative matrix at every step, a difficult task for large systems. 

It uses a linear or quadratic synchronous transit approach to get closer to the quadratic region of the transition state and then uses a quasi-Newton or eigenvalue-following algorithm to complete the optimization. 

In the above equation, the norm is usually the Euclidean norm. We have a linear convergence rate when 0 is equal to 1. Superlinear convergence rate refers to the case where 0=1 and the limit is equal to zero. When 0=2 the convergence rate is called quadratic. In general, the value of 0 depends on the algorithm while the value of the limit depends upon the function that is being minimized. 

Minimization of S(k) can be accomplished by using almost any technique available from optimization theory, however since each objective function evaluation requires the integration of the state equations, the use of quadratically convergent algorithms is highly recommended. The Gauss-Newton method is the most appropriate one for ODE models (Bard, 1970) and it presented in detail below. 

The above method is the well-known Gauss-Newton method for differential equation systems and it exhibits quadratic convergence to the optimum. Computational modifications to the above algorithm for the incorporation of prior knowledge about the parameters (Bayessian estimation) are discussed in detail in Chapter 8. 

Problem 4.1 is nonlinear if one or more of the functions/, gv...,gm are nonlinear. It is unconstrained if there are no constraint functions g, and no bounds on the jc,., and it is bound-constrained if only the xt are bounded. In linearly constrained problems all constraint functions g, are linear, and the objective/is nonlinear. There are special NLP algorithms and software for unconstrained and bound-constrained problems, and we describe these in Chapters 6 and 8. Methods and software for solving constrained NLPs use many ideas from the unconstrained case. Most modem software can handle nonlinear constraints, and is especially efficient on linearly constrained problems. A linearly constrained problem with a quadratic objective is called a quadratic program (QP). Special methods exist for solving QPs, and these iare often faster than general purpose optimization procedures. 

If the BFGS algorithm is applied to a positive-definite quadratic function of n variables and the line search is exact, it will minimize the function in at most n iterations (Dennis and Schnabel, 1996, Chapter 9). This is also true for some other updating formulas. For nonquadratic functions, a good BFGS code usually requires more iterations than a comparable Newton implementation and may not be as accurate. Each BFGS iteration is generally faster, however, because second derivatives are not required and the system of linear equations (6.15) need not be solved. 

LP software includes two related but fundamentally different kinds of programs. The first is solver software, which takes data specifying an LP or MILP as input, solves it, and returns the results. Solver software may contain one or more algorithms (simplex and interior point LP solvers and branch-and-bound methods for MILPs, which call an LP solver many times). Some LP solvers also include facilities for solving some types of nonlinear problems, usually quadratic programming problems (quadratic objective function, linear constraints see Section 8.3), or separable nonlinear problems, in which the objective or some constraint functions are a sum of nonlinear functions, each of a single variable, such as... 

Note that there are n + m equations in the n + m unknowns x and A. In Section 8.6 we describe an important class of NLP algorithms called successive quadratic programming (SQP), which solve (8.17)—(8.18) by a variant of Newton s method. 

Successive quadratic programming (SQP) methods solve a sequence of quadratic programming approximations to a nonlinear programming problem. Quadratic programs (QPs) have a quadratic objective function and linear constraints, and there exist efficient procedures for solving them see Section 8.3. As in SLP, the linear constraints are linearizations of the actual constraints about the selected point. The objective is a quadratic approximation to the Lagrangian function, and the algorithm is simply Newton s method applied to the KTC of the problem. 

Fan, Y. S. Sarkar and L. Lasdon. Experiments with Successive Quadratic Programming Algorithms. J Optim Theory Appli 56 (3), 359-383 (March 1988). 

Temet, D. J. and L. T. Biegler. Recent Improvements to a Multiplier-free Reduced Hessian Successive Quadratic Programming Algorithm. Comp Chem Engin 22 963 (1998). 

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