We consider solving unconstrained optimization problems by means of two popular globalization techniques: trust-region (TR) algorithms and adaptive regularized framework using cubics (ARC). Both techniques require the...
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We consider solving unconstrained optimization problems by means of two popular globalization techniques: trust-region (TR) algorithms and adaptive regularized framework using cubics (ARC). Both techniques require the solution of a so-called "subproblem" in which a trial step is computed by solving an optimization problem involving an approximation of the objective function, called "the model". The latter is supposed to be adequate in a neighborhood of the current iterate. In this paper, we address an important practical question related with the choice of the norm for defining the neighborhood. More precisely, assuming here that the Hessian B of the model is symmetric positive definite, we propose the use of the so-called "energy norm"-defined by vertical bar vertical bar x vertical bar vertical bar(B) = root x(T) B-x for all x is an element of R-n -in both TR and ARC techniques. We show that the use of this norm induces remarkable relations between the trial step of both methods that can be used to obtain efficient practical algorithms. We furthermore consider the use of truncated Krylov subspace methods to obtain an approximate trial step for large scale optimization. Within the energy norm, we obtain line search algorithms along the Newton direction, with a special backtracking strategy and an acceptability condition in the spirit of TR/ARC methods. The new line search algorithm, derived by ARC, enjoys a worst-case iteration complexity of O(is an element of(-3/2)) We show the good potential of the energy norm on a set of numerical experiments.
In this work, we present an extension of the spectral conjugate gradient (SCG) methods for solving unconstrained vector optimization problems, with respect to the partial order induced by a pointed, closed and convex ...
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In this work, we present an extension of the spectral conjugate gradient (SCG) methods for solving unconstrained vector optimization problems, with respect to the partial order induced by a pointed, closed and convex cone with a nonempty interior. We first study the direct extension version of the SCG methods and its global convergence without imposing an explicit restriction on parameters. It shows that the methods may lose their good scalar properties, like yielding descent directions, in the vector setting. By using a truncation technique, we then propose a modified self-adjusting SCG algorithm which is more suitable for various parameters. Global convergence of the new scheme covers the vector extensions of three different spectral parameters and the corresponding Perry, Andrei, and Dai-Kou conjugate parameters (SP, N, and JC schemes, respectively) without regular restarts and any convex assumption. Under inexact linesearches, we prove that the sequences generated by the proposed methods find points that satisfy the first-order necessary condition for Pareto-optimality. Finally, numerical experiments illustrating the practical behavior of the methods are presented.
In this paper, using sunny generalized nonexpansive retractions which are different from the metric projection and generalized metric projection in Banach spaces, we present new extragradient and line search algorithm...
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In this paper, using sunny generalized nonexpansive retractions which are different from the metric projection and generalized metric projection in Banach spaces, we present new extragradient and line search algorithms for finding the solution of a J-variational inequality whose constraint set is the common elements of the set of fixed points of a family of generalized nonexpansive mappings and the set of solutions of a pseudomonotone J-equilibrium problem for a J -alpha-inverse-strongly monotone operator in a Banach space. To prove strong convergence of generated iterates in the extragradient method, we introduce a I center dot (au)-Lipschitz-type condition and assume that the equilibrium bifunction satisfies this condition. This condition is unnecessary when the linesearch method is used instead of the extragradient method. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in literature to illustrate the usability of our results.
This paper considers sufficient descent Riemannian conjugate gradient methods with line search algorithms. We propose two kinds of sufficient descent nonlinear conjugate gradient method and prove that these methods sa...
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This paper considers sufficient descent Riemannian conjugate gradient methods with line search algorithms. We propose two kinds of sufficient descent nonlinear conjugate gradient method and prove that these methods satisfy the sufficient descent condition on Riemannian manifolds. One is a hybrid method combining a Fletcher-Reeves-type method with a Polak-Ribiere-Polyak-type method, and the other is a Hager-Zhang-type method, both of which are generalizations of those used in Euclidean space. Moreover, we prove that the hybrid method has a global convergence property under the strong Wolfe conditions and the Hager-Zhang-type method has the sufficient descent property regardless of whether a linesearch is used or not. Further, we review two kinds of line search algorithm on Riemannian manifolds and numerically compare our generalized methods by solving several Riemannian optimization problems. The results show that the performance of the proposed hybrid methods greatly depends on the type of linesearch used. Meanwhile, the Hager-Zhang-type method has the fast convergence property regardless of the type of linesearch used.
A numerical method for finding the roots of any function is developed. This method considers a circle using the concept of curvature instead of the tangential line in the Newton-Raphson method. The compared results be...
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A numerical method for finding the roots of any function is developed. This method considers a circle using the concept of curvature instead of the tangential line in the Newton-Raphson method. The compared results between the proposed method and the Newton-Raphson method are listed. The proposed method has a wider convergent region of initial points and finds more proper solutions than the Newton-Raphson method. In particular, the paper proposes that the curvature method is replaced by the modified Newton method discussed by Ralston in dealing with multiple roots.
Alternating current optimal power flow (ACOPF) problem is a non-convex and a nonlinear optimization problem. Similar to most nonlinear optimization problems, ACOPF is an NP-hard problem. On the other hand, Utilities a...
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ISBN:
(纸本)9781665449199
Alternating current optimal power flow (ACOPF) problem is a non-convex and a nonlinear optimization problem. Similar to most nonlinear optimization problems, ACOPF is an NP-hard problem. On the other hand, Utilities and independent service operators (ISO) require the problem to be solved in almost real-time. The real-world networks are often large in size and developing an efficient and tractable algorithm is critical to many decision-making processes in electricity markets. Interior-point methods (IPMs) for nonlinear programming are considered one of the most powerful algorithms for solving large-scale nonlinear optimization problems. However, the performance of these algorithms is significantly impacted by the optimization structure of the problem. Thus, the choice of the formulation is as important as choosing the algorithm for solving an ACOPF problem. Different ACOPF formulations are evaluated in this paper for computational viability and best performance using the interior-point linesearch (IPLS) algorithm. Different optimization structures are used in these formulations to model the ACOPF problem representing a range of varying sparsity. The numerical experiments suggest that the least sparse ACOPF formulation with polar voltages yields the best computational results. A wide range of test cases, ranging from 500-bus systems to 9591-bus systems, are used to verify the test results.
Accurate range prediction is crucial for electric vehicle (EV) adoption, mitigating range anxiety and optimizing energy usage. This paper presents a novel approach for EV range prediction using a physics-based formula...
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ISBN:
(纸本)9798350350661;9798350350654
Accurate range prediction is crucial for electric vehicle (EV) adoption, mitigating range anxiety and optimizing energy usage. This paper presents a novel approach for EV range prediction using a physics-based formula and optimization algorithms. Unlike existing range planning software, our method focuses solely on prediction, employing Particle Swarm Optimization (PSO) and line search algorithm to optimize trip-specific variables within a range, providing best-case and worst-case range scenarios. This paper details the methodology, underlying formula, and future research directions.
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