Trust Region Methods For Unconstrained Optimization Problems
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Trust Region Methods For Unconstrained Optimization Problems
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Author : Mostafa Rezapour
language : en
Publisher:
Release Date : 2020
Trust Region Methods For Unconstrained Optimization Problems written by Mostafa Rezapour and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with Mathematical optimization categories.
We present trust-region methods for the general unconstrained minimization problem. Trust-region algorithms iteratively minimize a model of the objective function within the trust-region and update the size of the region to find a first-order stationary point for the objective function. The radius of the trust-region is updated based on the agreement between the model and the objective function at the new trial point. The efficiency of the trust-region algorithms depends significantly on the size of the trust-region, the agreement between the model and the objective function and the model value reduction at each step. The size of the trust-region at each step plays a key role in the efficiency of the trust-region algorithm, particularly for large scale problems, because constructing and minimizing the model at each step requires gradient and Hessian information of the objective function. If the trust-region is too small or too large, then more models must be constructed and minimized, which is computationally expensive. We propose two adaptive trust-region algorithms that explore beyond the trust region if the boundary of the region prevents the algorithm from accepting a more beneficial point. It occurs when there is very good agreement between the model and the objective function on the trust-region boundary and we can find a step outside the trust-region with smaller model value while maintaining good agreement between the model and the objective function. We also take a different approach to derivative-free unconstrained optimization problems, where the objective function is possibly nonsmooth. We do an exploratory study by using deep neural-networks and their well-known capability as universal function approximator. We propose and investigate two derivative-free trust-region methods for solving unconstrained minimization problems, where we employ artificial neural-networks to construct a model within the trust-region. We directly find an estimate of the objective function minimizer without explicitly constructing a model function through a parent-child neural-network. This approach may provide improved practical performance in cases where the objective function is extremely noisy or stochastic. We provide a framework for future work in this area.
Trust Region Methods
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Author : A. R. Conn
language : en
Publisher: SIAM
Release Date : 2000-01-01
Trust Region Methods written by A. R. Conn and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-01-01 with Mathematics categories.
This is the first comprehensive reference on trust-region methods, a class of numerical algorithms for the solution of nonlinear convex optimization methods. Its unified treatment covers both unconstrained and constrained problems and reviews a large part of the specialized literature on the subject. It also provides an up-to-date view of numerical optimization.
Trust Region Methods For Minimization
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Author : R. B. Schnabel
language : en
Publisher:
Release Date : 1984
Trust Region Methods For Minimization written by R. B. Schnabel and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with categories.
This research project investigated a number of topics in unconstrained optimization, constrained optimization, and solving systems of nonlinear equations. The biggest accomplishment was the development of a new class of of methods, called tensor methods, for solving systems of nonlinear equations. These methods led to large increases in efficiency over standard methods on extensive batteries of test problems, with especially large gains on problems with singular Jacobians at the solution. The other major accomplishment was the development of a unified theory of trust region methods for unconstrained optimization. Our theory shows how line search, dogleg, or optimal step methods can be constructed that satisfy first and second order conditions for convergence. Research was also completed on conic methods for optimization, on secant methods that satisfy multiple secant equations, and on issues concerned with the computation of null space bases in constrained optimization. Research was initiated on computational methods for nonlinear least squares problems with errors in the independent variables, and in parallel algorithms for optimization.
Large Scale Trust Region Methods And Their Application To Primal Dual Interior Point Methods
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Author : Alexander Guldemond
language : en
Publisher:
Release Date : 2023
Large Scale Trust Region Methods And Their Application To Primal Dual Interior Point Methods written by Alexander Guldemond and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023 with categories.
Trust-region methods are amongst the most commonly used methods in unconstrained mathematical optimization. Their impressive performance and sound theoretical guarantees make them suitable for a wide range of problem types. However, the computational complexity of existing methods for solving the trust-region subproblem prevents trust-region methods from being widely used in large-scale problems in both unconstrained and constrained settings. This dissertation introduces and analyzes three novel methods for solving the trust-region subproblem for large-scale constrained optimization problems. Convergence rates and proofs are presented where applicable. Furthermore, a trust-region approach is developed for the recently introduced all-shifted primal-dual penalty-barrier method for solving nonconvex, constrained optimization problems. The three trust-region algorithms introduced are the shifted and inverted generalized Lanczos trust region algorithm, the locally optimal preconditioned conjugate gradient trust region, and the Jacobi-Davidson QZ trust region algorithm. Each new method exhibits improved performance over the existing standard methods and is best suited for problems too large for the traditional methods to handle efficiently. Furthermore, each method exhibits particular benefits for differently scaled problems.
Optimization In Chemical Engineering
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Author : Suman Dutta
language : en
Publisher: Cambridge University Press
Release Date : 2016-03-11
Optimization In Chemical Engineering written by Suman Dutta and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-03-11 with Technology & Engineering categories.
Optimization is used to determine the most appropriate value of variables under given conditions. The primary focus of using optimisation techniques is to measure the maximum or minimum value of a function depending on the circumstances. This book discusses problem formulation and problem solving with the help of algorithms such as secant method, quasi-Newton method, linear programming and dynamic programming. It also explains important chemical processes such as fluid flow systems, heat exchangers, chemical reactors and distillation systems using solved examples. The book begins by explaining the fundamental concepts followed by an elucidation of various modern techniques including trust-region methods, Levenberg–Marquardt algorithms, stochastic optimization, simulated annealing and statistical optimization. It studies the multi-objective optimization technique and its applications in chemical engineering and also discusses the theory and applications of various optimization software tools including LINGO, MATLAB, MINITAB and GAMS.
Combination Trust Region Line Search Methods For Unconstrained Optimization
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Author : Edward Michael Gertz
language : en
Publisher:
Release Date : 1998
Combination Trust Region Line Search Methods For Unconstrained Optimization written by Edward Michael Gertz and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with categories.
Numerical Methods For Unconstrained Optimization And Nonlinear Equations
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Author : J. E. Dennis, Jr.
language : en
Publisher: SIAM
Release Date : 1996-12-01
Numerical Methods For Unconstrained Optimization And Nonlinear Equations written by J. E. Dennis, Jr. and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-12-01 with Mathematics categories.
A complete, state-of-the-art description of the methods for unconstrained optimization and systems of nonlinear equations.
Efficient Trust Region Subproblem Algorithms
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Author : Heng Ye
language : en
Publisher:
Release Date : 2011
Efficient Trust Region Subproblem Algorithms written by Heng Ye and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with categories.
The Trust Region Subproblem (TRS) is the problem of minimizing a quadratic (possibly non-convex) function over a sphere. It is the main step of the trust region method for unconstrained optimization problems. Two cases may cause numerical difficulties in solving the TRS, i.e., (i) the so-called hard case and (ii) having a large trust region radius. In this thesis we give the optimality characteristics of the TRS and review the major current algorithms. Then we introduce some techniques to solve the TRS efficiently for the two difficult cases. A shift and deflation technique avoids the hard case; and a scaling can adjust the value of the trust region radius. In addition, we illustrate other improvements for the TRS algorithm, including: rotation, approximate eigenvalue calculations, and inverse polynomial interpolation. We also introduce a warm start approach and include a new treatment for the hard case for the trust region method. Sensitivity analysis is provided to show that the optimal objective value for the TRS is stable with respect to the trust region radius in both the easy and hard cases. Finally, numerical experiments are provided to show the performance of all the improvements.
Unconstrained Optimization Methods
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Author : Snezana S. Djordjevic
language : en
Publisher:
Release Date : 2019
Unconstrained Optimization Methods written by Snezana S. Djordjevic and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with Electronic books categories.
Here, we consider two important classes of unconstrained optimization methods: conjugate gradient methods and trust region methods. These two classes of methods are very interesting; it seems that they are never out of date. First, we consider conjugate gradient methods. We also illustrate the practical behavior of some conjugate gradient methods. Then, we study trust region methods. Considering these two classes of methods, we analyze some recent results.
A Derivative Free Two Level Random Search Method For Unconstrained Optimization
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Author : Neculai Andrei
language : en
Publisher: Springer Nature
Release Date : 2021-03-31
A Derivative Free Two Level Random Search Method For Unconstrained Optimization written by Neculai Andrei and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-03-31 with Mathematics categories.
The book is intended for graduate students and researchers in mathematics, computer science, and operational research. The book presents a new derivative-free optimization method/algorithm based on randomly generated trial points in specified domains and where the best ones are selected at each iteration by using a number of rules. This method is different from many other well established methods presented in the literature and proves to be competitive for solving many unconstrained optimization problems with different structures and complexities, with a relative large number of variables. Intensive numerical experiments with 140 unconstrained optimization problems, with up to 500 variables, have shown that this approach is efficient and robust. Structured into 4 chapters, Chapter 1 is introductory. Chapter 2 is dedicated to presenting a two level derivative-free random search method for unconstrained optimization. It is assumed that the minimizing function is continuous, lower bounded and its minimum value is known. Chapter 3 proves the convergence of the algorithm. In Chapter 4, the numerical performances of the algorithm are shown for solving 140 unconstrained optimization problems, out of which 16 are real applications. This shows that the optimization process has two phases: the reduction phase and the stalling one. Finally, the performances of the algorithm for solving a number of 30 large-scale unconstrained optimization problems up to 500 variables are presented. These numerical results show that this approach based on the two level random search method for unconstrained optimization is able to solve a large diversity of problems with different structures and complexities. There are a number of open problems which refer to the following aspects: the selection of the number of trial or the number of the local trial points, the selection of the bounds of the domains where the trial points and the local trial points are randomly generated and a criterion for initiating the line search.