Using sentinels to detect intersections of convex and nonconvex polygons

We describe finite sets of points, called sentinels, which allow us to decide if isometric copies of polygons, convex or not, intersect. As an example of the applicability of the concept of sentinel, we explain how they can be used to formulate an algorithm based on the optimization of differentiable models to pack polygons in convex sets. Mathematical subject classification: 90C53, 65K05

Assessing the reliability of general-purpose Inexact Restoration methods

Inexact Restoration methods have been proved to be effective to solve constrained optimization problems in which some structure of the feasible set induces a natural way of recovering feasibility from arbitrary infeasible points. Sometimes natural ways of dealing with minimization over tangent approximations of the feasible set are also employed. A recent paper Banihashemi and Kaya (2013)] suggests that the Inexact Restoration approach can be competitive with well-established nonlinear programming solvers when applied to certain control problems without any problem-oriented procedure for restoring feasibility. This result motivated us to revisit the idea of designing general-purpose Inexact Restoration methods, especially for large-scale problems. In this paper we introduce affordable algorithms of Inexact Restoration type for solving arbitrary nonlinear programming problems and we perform the first experiments that aim to assess their reliability. Initially, we define a purely local Inexact Restoration algorithm with quadratic convergence. Then, we modify the local algorithm in order to increase the chances of success of both the restoration and the optimization phase. This hybrid algorithm is intermediate between the local algorithm and a globally convergent one for which, under suitable assumptions, convergence to KKT points can be proved28

Special Issue Dedicated To Selected Surveys In Nonlinear Programming

[No abstract available]34337137

Riemannian Sparse Coding for Positive Definite Matrices

International audienceInspired by the great success of sparse coding for vector valued data, our goal is to represent symmetric positive definite (SPD) data matrices as sparse linear combinations of atoms from a dictionary, where each atom itself is an SPD matrix. Since SPD matrices follow a non-Euclidean (in fact a Riemannian) geometry, existing sparse coding techniques for Euclidean data cannot be directly extended. Prior works have approached this problem by defining a sparse coding loss function using either extrinsic similarity measures (such as the log-Euclidean distance) or kernelized variants of statistical measures (such as the Stein divergence, Jeffrey's divergence, etc.). In contrast, we propose to use the intrinsic Riemannian distance on the manifold of SPD matrices. Our main contribution is a novel mathematical model for sparse coding of SPD matrices; we also present a computationally simple algorithm for optimizing our model. Experiments on several computer vision datasets showcase superior classification and retrieval performance compared with state-of-the-art approaches

Implementation of an Optimal First-Order Method for Strongly Convex Total Variation Regularization

We present a practical implementation of an optimal first-order method, due to Nesterov, for large-scale total variation regularization in tomographic reconstruction, image deblurring, etc. The algorithm applies to $\mu$-strongly convex objective functions with $L$-Lipschitz continuous gradient. In the framework of Nesterov both $\mu$ and $L$ are assumed known -- an assumption that is seldom satisfied in practice. We propose to incorporate mechanisms to estimate locally sufficient $\mu$ and $L$ during the iterations. The mechanisms also allow for the application to non-strongly convex functions. We discuss the iteration complexity of several first-order methods, including the proposed algorithm, and we use a 3D tomography problem to compare the performance of these methods. The results show that for ill-conditioned problems solved to high accuracy, the proposed method significantly outperforms state-of-the-art first-order methods, as also suggested by theoretical results.Comment: 23 pages, 4 figure

An artificial fish swarm filter-based Method for constrained global optimization

Ana Maria A.C. Rocha, M. Fernanda P. Costa and Edite M.G.P. Fernandes, An Artificial Fish Swarm Filter-Based Method for Constrained Global Optimization, B. Murgante, O. Gervasi, S. Mirsa, N. Nedjah, A.M. Rocha, D. Taniar, B. Apduhan (Eds.), Lecture Notes in Computer Science, Part III, LNCS 7335, pp. 57–71, Springer, Heidelberg, 2012.An artificial fish swarm algorithm based on a filter methodology for trial solutions acceptance is analyzed for general constrained global optimization problems. The new method uses the filter set concept to accept, at each iteration, a population of trial solutions whenever they improve constraint violation or objective function, relative to the current solutions. The preliminary numerical experiments with a wellknown benchmark set of engineering design problems show the effectiveness of the proposed method.Fundação para a Ciência e a Tecnologia (FCT

The inverse problem of determining the filtration function and permeability reduction in flow of water with particles in porous media

The original publication can be found at www.springerlink.comDeep bed filtration of particle suspensions in porous media occurs during water injection into oil reservoirs, drilling fluid invasion of reservoir production zones, fines migration in oil fields, industrial filtering, bacteria, viruses or contaminants transport in groundwater etc. The basic features of the process are particle capture by the porous medium and consequent permeability reduction. Models for deep bed filtration contain two quantities that represent rock and fluid properties: the filtration function, which is the fraction of particles captured per unit particle path length, and formation damage function, which is the ratio between reduced and initial permeabilities. These quantities cannot be measured directly in the laboratory or in the field; therefore, they must be calculated indirectly by solving inverse problems. The practical petroleum and environmental engineering purpose is to predict injectivity loss and particle penetration depth around wells. Reliable prediction requires precise knowledge of these two coefficients. In this work we determine these quantities from pressure drop and effluent concentration histories measured in one-dimensional laboratory experiments. The recovery method consists of optimizing deviation functionals in appropriate subdomains; if necessary, a Tikhonov regularization term is added to the functional. The filtration function is recovered by optimizing a non-linear functional with box constraints; this functional involves the effluent concentration history. The permeability reduction is recovered likewise, taking into account the filtration function already found, and the functional involves the pressure drop history. In both cases, the functionals are derived from least square formulations of the deviation between experimental data and quantities predicted by the model.Alvarez, A. C., Hime, G., Marchesin, D., Bedrikovetski, P

Optimizing The Packing Of Cylinders Into A Rectangular Container: A Nonlinear Approach

The container loading problem has important industrial and commercial applications. An increase in the number of items in a container leads to a decrease in cost. For this reason the related optimization problem is of economic importance. In this work, a procedure based on a nonlinear decision problem to solve the cylinder packing problem with identical diameters is presented. This formulation is based on the fact that the centers of the cylinders have to be inside the rectangular box defined by the base of the container (a radius far from the frontier) and far from each other at least one diameter. With this basic premise the procedure tries to find the maximum number of cylinder centers that satisfy these restrictions. The continuous nature of the problem is one of the reasons that motivated this study. A comparative study with other methods of the literature is presented and better results are achieved. © 2003 Elsevier B.V. All rights reserved.16011933Birgin, E.G., Biloti, R., Tygel, M., Santos, L.T., Restricted optimization: A clue to a fast and accurate implementation of the common reflection surface stack method (1999) Journal of Applied Geophysics, 42, pp. 143-155Birgin, E.G., Chambouleyron, I., Martínez, J.M., Estimation of the optical constants and the thickness of thin films using unconstrained optimization (1999) Journal of Computational Physics, 151, pp. 862-880Birgin, E.G., Martínez, J.M., A box constrained optimization algorithm with negative curvature directions and spectral projected gradients (2001) Computing, 15 (SUPPL.), pp. 49-60Birgin, E.G., Martínez, J.M., Large-scale active-set box-constrained optimization method with spectral projected gradients (2002) Computational Optimization and Applications, 23, pp. 101-125Birgin, E.G., Martínez, J.M., Raydan, M., Nonmonotone spectral projected gradient methods on convex sets (2000) SIAM Journal on Optimization, 10, pp. 1196-1211Birgin, E.G., Martínez, J.M., Raydan, M., SPG: Software for convex-constrained optimization (2001) ACM Transactions on Mathematical Software, 27, pp. 340-349Correia, M.H., Oliveira, J.F., Ferreira, J.S., Cylinder packing by simulated annealing (2000) Pesquisa Operacional, 20, pp. 269-284Correia, M.H., Oliveira, J.F., Ferreira, J.S., A new upper bound for the cylinder packing problem (2001) International Transactions in Operational Research, 8, pp. 571-583Dennis Jr., J.E., Schnabel, R.B., (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations, , Englewoods Cliffs, NJ: Prentice-HallDowsland, K.A., Optimising the palletisation of cylinders in cases (1991) OR Spectrum, 13, pp. 204-212Isermann, H., Heuristiken zur Lösung des zweidimensionalen Packproblem für Rundgefäße (1991) OR Spectrum, 54, pp. 213-223Fraser, H.J., George, J.A., Integrated container loading software for pulp and paper industry (1994) European Journal of Operational Research, 77, pp. 466-474Friedman, E., http://www.stetson.edu/~efriedma/packing.htmlGeorge, J.A., George, J.M., Lamar, B.W., Packing different-sized circles into a rectangular container (1995) European Journal of Operational Research, 84, pp. 693-712Graham, R.L., Lubachevsky, B.D., Nurmela, K.J., Östergard, P.R.J., Dense packing of congruent circles in a circle (1998) Discrete Mathematics, 181, pp. 139-154Luenberger, D.G., (1984) Linear and Nonlinear Programming, , Reading, MA: Addison-WesleyPeikert, R., http://www.cg.inf.ethz.ch/~peikert/personal/CirclePackings/Schrage, L., A more portable Fortran random number generator (1979) ACM Transactions on Mathematical Software, 5, pp. 132-138Szabó, P.G., http://www.inf.u-szeged.hu/~pszabo/Pack.htm

Continuous GRASP with a local active-set method for bound-constrained global optimization

Global optimization, Stochastic methods, Active-set methods, Heuristic, CGRASP, GENCAN,