Gradient projection methods

The constrained minimization problem stated above may be transformed into a form well-suited to gradient projection methods of nonlinear programming by making the following substitution ... 

Rosen, J. B. (1961). The gradient projection method for non-linear programming, Part II, Non-linear constraints. J. Soc. Indus. Appl. Math., 9, 414-32. 

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

Methods that use the direction of the function gradient as the projection vector on the active constraints are known as Gradient Projection Methods. 

Several optimization algorithms, such as the feasible direction method, the gradient projection method, and the linearization method are coded in a multi-purpose program ADS (advanced Design Synthesis) [16], which was used in the present work. A modified feasible directions method [17], was selected in the present numerical example. 

Accountability of the variances and covariances of the responses makes this optimization procedure particularly noteworthy. From the formulator s viewpoint, the distance criterion could lead to an unacceptable optimum if the formulation levels at the optimum produce response values in an undesirable property range. Khuri and Conlon mentioned the possible use of this procedure for multiresponse mixture optimization although no elaboration or examples were given. The reliance of this method on the gradient projection technique could present difficulties with component level compensation if applied to formulations. 

The above subproblem can be solved very efficiently for fixed values of the multipliers X and v and penalty parameter p. Here a gradient projection trust region method is applied. Once subproblem (3-104) is solved, the multipliers and penalty parameter are updated in an outer loop and the cycle repeats until the KKT conditions for (3-85) are satisfied. LANCELOT works best when exact second derivatives are available. This promotes a fast convergence rate in solving each... 

In the work of Lindborg et al [119], the resulting linear equation systems were solved with preconditioned Krylov subspace projection methods [166]. The Poisson equation was solved by a conjugate gradient (CG)-solver, while the other transport equations were solved using a bi-conjugate gradient (BCG)-solver which can handle also non-symmetric equations systems. The solvers were preconditioned with a Jacobi preconditioner. 

The acquisition of projections in constant field gradients is the principle of the back-projection method [4]. Here, the spatial resolution is limited by the signal decay in a homogeneous field. But, for liquid samples, the final limit... 

Quadratic programming is discussed in Chapter 11. The quadratic programming methods are implemented by handling the numerical evaluation of Hessian in a novel way. Reduced gradient and gradient projection are described as conventional methods they are then compared to the novel proposed approach based on the Karush-Kuhn-Tucker direction projection method. It represents the basic core for the development of successive quadratic programming (SQP) methods. 

To display properties on molecular surfaces, two different approaches are applied. One method assigns color codes to each grid point of the surface. The grid points are connected to lines chicken-wire) or to surfaces (solid sphere) and then the color values are interpolated onto a color gradient [200]. The second method projects colored textures onto the surface [202, 203] and is mostly used to display such properties as electrostatic potentials, polarizability, hydrophobidty, and spin density. 

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