A new search direction of IPM for horizontal linear complementarity problems

This study presents a new search direction for the horizontal linear complementarity problem. A vector-valued function is applied to the system of xy=μe, which defines the central path. Usually, the way to get the equivalent form of the central path is using the square root function. However, in our study, we substitute a new search function formed by a different identity map, which obtains the equivalent shape of the central path using the square root function. We get the new search directions from Newton’s Method. Given this framework, we prove polynomial complexity for the Newton directions. We show that the algorithm’s complexity is O(nlognϵ), which is the same as the best-given algorithms for the horizontal linear complementarity problem

Efficient and Globally Convergent Minimization Algorithms for Small- and Finite-Strain Plasticity Problems

We present efficient and globally convergent solvers for several classes of plasticity models. The models in this work are formulated in the primal form as energetic rate-independent systems with an elastic energy potential and a plastic dissipation component. Different hardening rules are considered, as well as different flow rules. The time discretization leads to a sequence of nonsmooth minimization problems. For small strains, the unknowns live in vector spaces while for finite strains we have to deal with manifold-valued quantities. For the latter, a reformulation in tangent space is performed to end up with the same dissipation functional as in the small-strain case. We present the Newton-type TNNMG solver for convex and nonsmooth minimization problems and a newly developed Proximal Newton (PN) method that can also handle nonconvex problems. The PN method generates a sequence of penalized convex, coercive but nonsmooth subproblems. These subproblems are in the form of block-separable small-strain plasticity problems, to which TNNMG can be applied. Global convergence theorems are available for both methods. In several numerical experiments, both the efficiency and the flexibility of the methods for small-strain and finite-strain models are tested

Quantum Interior Point Methods for Semidefinite Optimization

We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact search direction and is not guaranteed to explore only feasible points; the second scheme uses a nullspace representation of the Newton linear system to ensure feasibility even with inexact search directions. The second is a novel scheme that might seem impractical in the classical world, but it is well-suited for a hybrid quantum-classical setting. We show that both schemes converge to an optimal solution of the semidefinite optimization problem under standard assumptions. By comparing the theoretical performance of classical and quantum interior point methods with respect to various input parameters, we show that our second scheme obtains a speedup over classical algorithms in terms of the dimension of the problem $n$, but has worse dependence on other numerical parameters

Algoritmos cuánticos tolerantes a fallos

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Físicas, leída el 17-01-2023The framework of this thesis is fault-tolerant quantum algorithms, which can roughly be divided into the following non-disjoint families: a) Grover’s algorithm and quantum walks, b) Shor’s algorithm and hidden subgroup problems, c) quantum simulation algorithms, d) quantum linear algebra, and e) variational quantum algorithms. All of them are covered, to some extent, in this thesis. Grover’s algorithm and quantum walks are described in Chapter 2. We start by highlighting the central role that rotations play in quantum algorithms, explaining Grover’s, why it is optimal, and how it may be extended. Key subroutines explained in this area are amplitude amplification and quantum walks, which will constitute useful parts of other algorithms. In this chapter, we present our Ref. [62], where we explore the heuristic use of quantum Metropolis and quantum walk algorithms for solving anNP-hard problem. This method has been suggested as an avenue to digitally simulate quantum annealing and preparing ground states of many-body Hamiltonians. In the third chapter, in contrast, we turn to the exponential advantages promisedby the Fourier transform in the context of the hidden subgroup problem. However, since this application is restricted to cryptography, we later explore its use in quantum linear algebra problems. Here we explain the development of the original quantum linear solver algorithm, its improvements, and finally the dequantization techniques that would often restrict the quantum advantage to polynomial...El marco conceptual de esta tesis son los algoritmos cuánticos tolerantes a fallos, que pueden dividirse aproximadamente en las siguientes clases no mutuamente excluyentes :a) algoritmo de Grover y paseos cuánticos, b) algoritmo de Shor y problemas de subgrupos ocultos, c) algoritmos de simulación cuántica, d) álgebra lineal cuántica, ye) algoritmos cuánticos variacionales. Todos ellos se tratan, en cierta medida, en esta tesis. El algoritmo de Grover y los paseos cuánticos se explican en el capítulo 2. Comenzamos destacando el papel central que juegan las rotaciones en los algoritmos cuánticos, explicando el de Grover, por qué es óptimo, y cómo puede ser extendido. Las subrutinas clave explicadas en esta área son la amplificación de la amplitud y los paseos cuánticos, que serán partes importantes de otros algoritmos. En este capítulo presentamos nuestra Ref. [62], donde exploramos el uso heurístico de los algoritmos de Metrópolis y paseos cuánticos para resolver problemas NP-difíciles. De hecho, este método ha sido sugerido como una vía para simular digitalmente el método conocido como ‘quantum annealing’,y la preparación de estados fundamentales de Hamiltonianos ‘many-body’.En el tercer capítulo, en cambio, nos centramos en las ventajas exponenciales que promete la transformada de Fourier en el contexto del problema de los subgrupos ocultos. Sin embargo, dado que esta aplicación está restringida a la criptografía, más adelante exploramos su uso en problemas de álgebra lineal cuántica. Aquí explicamos el desarrollo del algoritmo cuántico original para la resolución de sistemas lineales de ecuaciones, sus mejoras, y finalmente las técnicas de ‘descuantización’ que a menudo restringen la ventaja cuántica a polinómica...Fac. de Ciencias FísicasTRUEunpu

A Primal-Dual Augmented Lagrangian Penalty-Interior-Point Algorithm for Nonlinear Programming

This thesis treats a new numerical solution method for large-scale nonlinear optimization problems. Nonlinear programs occur in a wide range of engineering and academic applications like discretized optimal control processes and parameter identification of physical systems. The most efficient and robust solution approaches for this problem class have been shown to be sequential quadratic programming and primal-dual interior-point methods. The proposed algorithm combines a variant of the latter with a special penalty function to increase its robustness due to an automatic regularization of the nonlinear constraints caused by the penalty term. In detail, a modified barrier function and a primal-dual augmented Lagrangian approach with an exact l2-penalty is used. Both share the property that for certain Lagrangian multiplier estimates the barrier and penalty parameter do not have to converge to zero or diverge, respectively. This improves the conditioning of the internal linear equation systems near the optimal solution, handles rank-deficiency of the constraint derivatives for all non-feasible iterates and helps with identifying infeasible problem formulations. Although the resulting merit function is non-smooth, a certain step direction is a guaranteed descent. The algorithm includes an adaptive update strategy for the barrier and penalty parameters as well as the Lagrangian multiplier estimates based on a sensitivity analysis. Global convergence is proven to yield a first-order optimal solution, a certificate of infeasibility or a Fritz-John point and is maintained by combining the merit function with a filter or piecewise linear penalty function. Unlike the majority of filter methods, no separate feasibility restoration phase is required. For a fixed barrier parameter the method has a quadratic order of convergence. Furthermore, a sensitivity based iterative refinement strategy is developed to approximate the optimal solution of a parameter dependent nonlinear program under parameter changes. It exploits special sensitivity derivative approximations and converges locally with a linear convergence order to a feasible point that further satisfies the perturbed complementarity condition of the modified barrier method. Thereby, active-set changes from active to inactive can be handled. Due to a certain update of the Lagrangian multiplier estimate, the refinement is suitable in the context of warmstarting the penalty-interior-point approach. A special focus of the thesis is the development of an algorithm with excellent performance in practice. Details on an implementation of the proposed primal-dual penalty-interior-point algorithm in the nonlinear programming solver WORHP and a numerical study based on the CUTEst test collection is provided. The efficiency and robustness of the algorithm is further compared to state-of-the-art nonlinear programming solvers, in particular the interior-point solvers IPOPT and KNITRO as well as the sequential quadratic programming solvers SNOPT and WORHP

2nd International Conference on Numerical and Symbolic Computation

The Organizing Committee of SYMCOMP2015 – 2nd International Conference on Numerical and Symbolic Computation: Developments and Applications welcomes all the participants and acknowledge the contribution of the authors to the success of this event. This Second International Conference on Numerical and Symbolic Computation, is promoted by APMTAC - Associação Portuguesa de Mecânica Teórica, Aplicada e Computacional and it was organized in the context of IDMEC/IST - Instituto de Engenharia Mecânica. With this ECCOMAS Thematic Conference it is intended to bring together academic and scientific communities that are involved with Numerical and Symbolic Computation in the most various scientific area

Snapshot-Based Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science