Abstract:Adequate initial configurations for molecular dynamics simulations consist of arrangements of molecules distributed in space in such a way to approximately represent the system's overall structure. In order that the simulations are not disrupted by large van der Waals repulsive interactions, atoms from different molecules must keep safe pairwise distances. Obtaining such a molecular arrangement can be considered a packing problem: Each type molecule must satisfy spatial constraints related to the geometry of the system, and the distance between atoms of different molecules must be greater than some specified tolerance. We have developed a code able to pack millions of atoms, grouped in arbitrarily complex molecules, inside a variety of three-dimensional regions. The regions may be intersections of spheres, ellipses, cylinders, planes, or boxes. The user must provide only the structure of one molecule of each type and the geometrical constraints that each type of molecule must satisfy. Building complex mixtures, interfaces, solvating biomolecules in water, other solvents, or mixtures of solvents, is straightforward. In addition, different atoms belonging to the same molecule may also be restricted to different spatial regions, in such a way that more ordered molecular arrangements can be built, as micelles, lipid double-layers, etc. The packing time for state-of-the-art molecular dynamics systems varies from a few seconds to a few minutes in a personal computer. The input files are simple and currently compatible with PDB, Tinker, Molden, or Moldy coordinate files. The package is distributed as free software and can be downloaded from

Abstract. Nonmonotone projected gradient techniques are considered for the minimization of differentiable functions on closed convex sets. The classical projected gradient schemes are extended to include a nonmonotone steplength strategy that is based on the Grippo-Lampariello-Lucidi nonmonotone line search. In particular, the nonmonotone strategy is combined with the spectral gradient choice of steplength to accelerate the convergence process. In addition to the classical projected gradient nonlinear path, the feasible spectral projected gradient is used as a search direction to avoid additional trial projections during the one-dimensional search process. Convergence properties and extensive numerical results are presented.Key words. projected gradients, nonmonotone line search, large-scale problems, bound constrained problems, spectral gradient method AMS subject classifications. 49M07, 49M10, 65K, 90C06, 90C20 PII. S10526234973309631. Introduction. We consider the projected gradient method for the minimization of differentiable functions on nonempty closed and convex sets. Over the last few decades, there have been many different variations of the projected gradient method that can be viewed as the constrained extensions of the optimal gradient method for unconstrained minimization. They all have the common property of maintaining feasibility of the iterates by frequently projecting trial steps on the feasible convex set. This process is in general the most expensive part of any projected gradient method. Moreover, even if projecting is inexpensive, as in the box-constrained case, the method is considered to be very slow, as is its analogue, the optimal gradient method (also known as steepest descent), for unconstrained optimization. On the positive side, the projected gradient method is quite simple to implement and very effective for large-scale problems.This state of affairs motivates us to combine the projected gradient method with two recently developed ingredients in optimization. First we extend the typical globalization strategies associated with these methods to the nonmonotone line search schemes developed by Grippo, Lampariello, and Lucidi [17] for Newton's method. Second, we propose to associate the spectral steplength, introduced by Barzilai and Borwein [1] and analyzed by Raydan [26]. This choice of steplength requires little computational work and greatly speeds up the convergence of gradient methods. In fact, while the spectral gradient method appears to be a generalized steepest descent method, it is clear from its derivation that it is related to the quasi-Newton family of methods through an approximated secant equation. The fundamental difference is

Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lower-level type. Two methods of this class are introduced and analyzed. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved using the Constant Positive Linear Dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The reliability of the approach is tested by means of an exhaustive comparison against Lancelot . All the problems of the Cute collection are used in this comparison. Moreover, the resolution of location problems in which many constraints of the lower-level set are nonlinear is addressed, employing the Spectral Projected Gradient method for solving the subproblems. Problems of this type with more than 3 × 10 6 variables and 14 × 10 6 constraints are solved in this way, using moderate computer time.

Molecular Dynamics is a powerful methodology for the comprehension at molecular level of many chemical and biochemical systems. The theories and techniques developed for structural and thermodynamic analyses are well established, and many software packages are available. However, designing starting configurations for dynamics can be cumbersome. Easily generated regular lattices can be used when simple liquids or mixtures are studied. However, for complex mixtures, polymer solutions or solid adsorbed liquids (for example) this approach is inefficient, and it turns out to be very hard to obtain an adequate coordinate file. In this article, the problem of obtaining an adequate initial configuration is treated as a "packing" problem and solved by an optimization procedure. The initial configuration is chosen in such a way that the minimum distance between atoms of different molecules is greater than a fixed tolerance. The optimization uses a well-known algorithm for box-constrained minimization. Applications are given for biomolecule solvation, many-component mixtures, and interfaces. This approach can reduce the work of designing starting configurations from days or weeks to few minutes or hours, in an automated fashion. Packing optimization is also shown to be a powerful methodology for space search in docking of small ligands to proteins. This is demonstrated by docking of the thyroid hormone to its nuclear receptor.

Abstract. A fully derivative-free spectral residual method for solving largescale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonotone behavior. The global convergence analysis of the combined scheme is presented. An extensive set of numerical experiments that indicate that the new combination is competitive and frequently better than well-known Newton-Krylov methods for large-scale problems is also presented.

A new method is introduced for large scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches. A specific algorithm, the Inexact Spectral Projected Gradient method (ISPG), is implemented using inexact projections computed by Dykstra's alternating projection method and generates interior iterates. The ISPG method is a generalization of the Spectral Projected Gradient method (SPG), but can be used when projections are difficult to compute. Numerical results for constrained least-squares rectangular matrix problems are presented.

A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the ε k -global minimization of the Augmented Lagrangian with simple constraints, where ε k → ε. Global convergence to an ε-global minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.

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