Data and performance profiles applying an adaptive truncation criterion, within linesearch-based truncated Newton methods, in large scale nonconvex optimization

In this paper, we report data and experiments related to the research article entitled “An adaptive truncation criterion, for linesearch-based truncated Newton methods in large scale nonconvex optimization” by Caliciotti et al. [1]. In particular, in Caliciotti et al. [1], large scale unconstrained optimization problems are considered by applying linesearch-based truncated Newton methods. In this framework, a key point is the reduction of the number of inner iterations needed, at each outer iteration, to approximately solving the Newton equation. A novel adaptive truncation criterion is introduced in Caliciotti et al. [1] to this aim. Here, we report the details concerning numerical experiences over a commonly used test set, namely CUTEst (Gould et al., 2015) [2]. Moreover, comparisons are reported in terms of performance profiles (Dolan and Moré 2002) [3], adopting different parameters settings. Finally, our linesearch-based scheme is compared with a renowned trust region method, namely TRON (Lin and Moré 1999) [4]

A software system for large-scale structural optimization

This work is driven by recent developments in mathematical programming, the state-of-the-art of structural optimization, the spectacular performance of linear programming algorithms, and computer hardware developments which imply that applications of structural optimization might be used commonly in engineering design. Currently, there are few general purpose optimization routines available to the structural engineer and much of the work has addressed specific classes of problems. Further, there is little widespread use of the available routines, partly due to the large amount of familiarity one must have with the specific details of both the problem and the optimization method. In response, it is the intention here to prototype a software system that implements a general approach for structural optimization using the latest in mathematical programming techniques. This work develops a general system that can be used for a variety of structural optimization problems in a manner analogous to the finite element method for structural analysis. The most commonly used structural elements, truss and beam, are included as well as techniques for plate optimization. Consideration is given to the software requirements of a general purpose structural optimization system and the demands of large structural systems typically encountered in design practice. This general approach is aimed at using classical methods taken directly from the area of mathematical programming, specifically linear programming, which has seen considerable change in the last ten years. Here, sequential linear programming (SLP) techniques are shown to handle a wide variety of structural constraints including stress constraints, displacement constraints, buckling, and frequency constraints. It is the purpose of this thesis to bring the latest developments in linear programming to the field of structural optimization in the form of a general purpose, state-of-the-art structural optimization system. The model was tested for sample structures and it was shown to effect a reduction in total structure volume of up to 80%

A Limited-memory Multipoint Symmetric Secant Method For Bound Constrained Optimization

A new algorithm for solving smooth large-scale minimization problems with bound constraints is introduced. The way of dealing with active constraints is similar to the one used in some recently introduced quadratic solvers. A limited-memory multipoint symmetric secant method for approximating the Hessian is presented. Positive-definiteness of the Hessian approximation is not enforced. A combination of trust-region and conjugate-gradient approaches is used to explore a useful negative curvature information. Global convergence is proved for a general model algorithm. Results of numerical experiments are presented.1171-45170Andreani, R., Friedlander, A., Martínez, J.M., On the solution of finite-dimensional variational inequalities using smooth optimization with simple bounds (1997) Journal on Optimization Theory and Applications, 94, pp. 635-657Andreani, R., Martínez, J.M., On the solution of the extended linear complementarity problem (1998) Linear Algebra and Its Applications, 281, pp. 247-257Andreani, R., Martínez, J.M., On the reformulation of nonlinear complementarity problems using the Fischer Burmeister function (1999) Applied Mathematics Letters, 12, pp. 7-12Andreani, R., Martínez, J.M., Reformulation of variational inequalities on a simplex and the compactification of complementarity problems (2000) SIAM Journal on Optimization, 10, pp. 878-895Bielschowsky, R., Friedlander, A., Gomes, F.A.M., Martínez, J.M., Raydan, M., An adaptive algorithm for bound constrained quadratic minimization (1998) Investigación Operativa, 7, pp. 67-102Biryukov, A.G., On the difference-approximation approach to the solution of systems of nonlinear equations (1983) Soviet Mathematics. Doklady, 17, pp. 660-664Bjorck, A., (1996) Numerical Methods for Least-Squares Problems, , SIAM, PhiladelphiaBongartz, I., Conn, A.R., Gould, N.I.M., Toint, Ph.L., CUTE: Constrained and unconstrained testing environment (1995) ACM Transactions on Mathematical Software, 21, pp. 123-160Burdakov, O.P., Methods of secant type for systems of equations with symmetric Jacobian matrix (1983) Numerical Functional Analysis and Optimization, 6, pp. 183-195Burdakov, O.P., Stable versions of the secant method for solving systems of equations (1983) USSR Computational Mathematics and Mathematical Physics, 23, pp. 1-10Burdakov, O.P., On superlinear convergence of some stable variants of the secant method (1986) ZAMM, 66, pp. 615-622Burdakov, O.P., Stable symmetric secant methods with restarts (1991) Cybernetics, 27, pp. 390-396Byrd, R.H., Lu, P., Nocedal, J., Zhu, C., A limited memory algorithm for bound constrained minimization (1995) SIAM Journal on Scientific Computing, 16, pp. 1190-1208Byrd, R.H., Nocedal, J., Schnabel, R.B., Representations of quasi-Newton matrices and their use in limited memory methods (1994) Mathematical Programming, 63, pp. 129-156Calamai, P.H., Morá, J.J., Projected gradient methods for linearly constrained problems (1987) Mathematical Programming, 39, pp. 93-116Conn, A.R., Gould, N.I.M., Toint, Ph.L., Global convergence of a class of trust region algorithms for optimization with simple bounds (1988) SIAM Journal on Numerical Analysis, 25, pp. 433-460Conn, A.R., Gould, N.I.M., Toint, Ph.L., A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds (1991) SIAM Journal on Numerical Analysis, 28, pp. 545-572Conn, A.R., Gould, N.I.M., Toint, Ph.L., (1992) LANCELOT: A Fortran Package for Large-Scale Nonlinear Optimization (Release A), , Springer, BerlinConn, A.R., Gould, N.I.M., Toint, Ph.L., (2000) Trust-Region Methods, , SIAM-MPSDax, A., A modified Gram Schmidt algorithm with iterative orthogonalization and column pivoting (2000) Linear Algebra and Its Applications, 310, pp. 25-42Dembo, R.S., Eisenstat, S.C., Steihaug, T., Inexact Newton methods (1982) SIAM Journal on Numerical Analysis, 19, pp. 400-408Dennis, J.E., Schnabel, R.B., (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations, , Prentice-Hall, New YorkDostál, Z., Box constrained quadratic programming with proportioning and projections (1997) SIAM Journal on Optimization, 7, pp. 871-887Dostál, Z., Friedlander, A., Santos, S.A., Solution of contact problems using subroutine BOX-QUACAN (1997) Investigación Operativa, 7, pp. 13-22Dostál, Z., Friedlander, A., Santos, S.A., Analysis of block structures by augmented Lagrangians with adaptive precision control (1997) Proceedings of GEOMECHANICS'96, pp. 175-180. , ed. Z. Rakowski (A.A. Balkema, Rotterdam)Dostál, Z., Friedlander, A., Santos, S.A., Solution of coercive and semicoercive contact problems by FETI domain decomposition (1998) The Tenth International Conference on Domain Decomposition Methods, 218, pp. 82-93. , Boulder, CO, Contemporary Mathematics, eds. J. Mandel, C. Farhat and X. CaiDostál, Z., Friedlander, A., Santos, S.A., Augmented Lagrangians with adaptive precision control for quadratic programming with equality constraints (1999) Computational Optimization and Applications, 14, pp. 1-17Dostál, Z., Friedlander, A., Santos, S.A., Malík, J., Analysis of semicoercive contact problems using symmetric BEM and augmented Lagrangians (1997) Engineering Analysis with Boundary Elements, 18, pp. 195-201Dostál, Z., Gomes, F.A.M., Santos, S.A., Duality-based domain decomposition with natural coarse-space for variational inequalities (2000) Journal of Computational and Applied Mathematics, 126, pp. 397-415Facchinei, J., Júdice, J., Soares, J., An active set Newton algorithm for large scale nonlinear programs with box constraints (1998) SIAM Journal on Optimization, 8, pp. 158-186Friedlander, A., Martínez, J.M., On the numerical solution of bound constrained optimization problems (1989) RAIRO Operations Research, 23, pp. 319-341Friedlander, A., Martínez, J.M., New algorithms for maximization of concave functions with box constraints (1992) RAIRO Operations Research, 26, pp. 209-236Friedlander, A., Martínez, J.M., On the maximization of a concave quadratic function with box constraints (1994) SIAM Journal on Optimization, 4, pp. 177-192Friedlander, A., Martínez, J.M., Santos, S.A., A new trust region algorithm for bound constrained minimization (1994) Applied Mathematics and Optimization, 30, pp. 235-266Gill, P.E., Leonard, M.W., Reduced-Hessian quasi-Newton methods for unconstrained optimization (2001) SIAM Journal on Optimization, 12, pp. 209-237Golub, G.H., Van Loan, C.F., (1989) Matrix Computations, , Johns Hopkins University PressKrejić, N., Martínez, J.M., Mello, M.P., Pilotta, E.A., Validation of an augmented Lagrangian algorithm with a Gauss-Newton Hessian approximation using a set of hard-spheres problems (2000) Computational Optimization and Applications, 16, pp. 247-263Lin, C.J., Moré, J.J., Newton's method for large bound-constrained optimization problems (1999) SIAM Journal on Optimization, 9, pp. 1100-1127Martínez, J.M., Three new algorithms based on the sequential secant method (1979) BIT, 19, pp. 236-243Martínez, J.M., Pilotta, E.A., Raydan, M., Spectral gradient method for linearly constrained optimization (1999) Technical Report 22/99, , IMECC, University of Campinas (UNICAMP), Campinas, BrazilMoré, J.J., Thuente, D.J., Line search algorithms with guaranteed sufficient decrease (1994) ACM Transactions on Mathematical Software, 20, pp. 286-307Ortega, J.M., Rheinboldt, W., (1970) Iterative Solution of Nonlinear Equations in Several Variables, , Academic Press, New YorkRaydan, M., On the Barzilai and Borwein choice of steplength for the gradient method (1993) SIAM Journal on Optimization, 7, pp. 26-33Schnabel, R.B., Quasi-Newton methods using multiple secant equations (1983) Technical Report CU-CS-247-83, , Department of Computer Science, University of Colorado, Boulder, COSteihaug, T., The conjugate gradient method and trust regions in large scale optimization (1983) SIAM Journal on Numerical Analysis, 20, pp. 626-63