Projection direction search method

The proposed method can be called the Projection Direction Search Method or Reduced Direction Search Method since whatever search direction is selected, the projection on a bound constraint corresponds to the removal of such a constraint from the same direction. 

Projection or Reduced Direction Search Methods for Bound-Constrained Problems 1407... 

Chapter 13 illustrates the problem of constrained optimization by introducing the active set methods. Successive linear programming (SLP), projection, reduced direction search, SQP methods are described, implemented, and adopted to solve several practical examples of constrained linear/nonlinear optimization, including the solution of the Maratos effect. 

Figure 6.1 Search for the minimum of the Gibbs function in a two-component space (nn and ni2 are mole numbers) with the mass conservation constraints Bn = q. The search direction is the projection of the gradient onto the constraint subspace. Minimum is attained when the gradient is orthogonal to the constraint direction, which is the geometrical expression of the Lagrange multiplier methods.

Methods of robust PCA are less sensitive to outliers and visualize the main data structure one approach for robust PCA uses a robust estimation of the covariance matrix, another approach searches for a direction which has the maximum of a robust variance measure (projection pursuit). 

Another technique for handling the mass balances was introduced by Castillo and Grossmann (1). Rather than convert the objective function into an unconstrained form, they implemented the Variable Metric Projection method of Sargent and Murtagh (32) to minimize Gibbs s free energy. This is a quasi-Newton method which uses a rank-one update to the approximation of H l, with the search direction "projected" onto the intersection of hyperplanes defined by linear mass balances. 

In reference point based methods, the DM first specifies a reference point z S consisting of desirable aspiration levels for each objective and then this reference point is projected onto the Pareto optimal set. That is, a Pareto optimal solution closest to the reference point is found. The distance can be measured in different ways. Specifying a reference points is an intuitive way for the DM to direct the search of the most preferred solution. It is straightforward to compare the point specified and the solution obtained without artificial concepts. Examples of methods of this t rpe are the reference point method and the light beam search . 

It is worth pointing out that the idea of searching directly for a state-specific solution for the wavefunctions of multiply excited states (MES) implies projections on distinct function spaces with separate optimization of some type, thereby avoiding serious problems having to do with the undue mixing of states and channels of the same symmetry. This idea has since been a central element of our analyses and state-specific computations. In fact, in recent years, such concerns have led to appropriate modifications of conventional methods of quantum chemistry, such as perturbation or coupled-cluster. 

One problem, related to the search for a point that fulfills a set of equations, is projecting a vector in the space of some constraints. As will be discussed in the following chapters, many optimization methods adopt this technique. To understand what projecting a vector means, it is useful to consider a simple problem with one single linear equality constraint. Suppose we know a point, Xj, that fulfills the constraint and we have a direction d that passes through the point x such that along it a certain function decreases. If d does not lie on the plane of the constraint, the step... 

If the search direction is the gradient direction only, the method joins the family of methods known as the Gradient Projection Methods. 

Since an exhaustive search algorithm is required for image segmentation and searching of the best directions on each scale, the computational time of the bandelet-based method is relatively high compared with other compression methods. Since the Radon transform uses fast computational algorithm which computes pair-by-pair differences between all projections, the computational time is relatively very low when compared to existing methods. 

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