Interior-Point Methods for Conic-Linear Optimization

Terlaky, Tamás; Boggs, Paul T.

doi:10.1007/978-1-4419-1153-7_475

Cited by 2 publications

(3 citation statements)

References 29 publications

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“…Another commonly used approach is the gradient method. This is one of the ubiquitous method to find the optimization solution Terlaky (2013). Our main objective is to obtain the difference between u k and u k+1 and u * such that the difference between u k and u k+1 is within our acceptance level where u k is the obtained point after the k th loop.…”

Section: Gradient Methodsmentioning

confidence: 99%

Optimal Solution Techniques in Decision Sciences: A Review

2019

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Methods to find the optimization solution are fundamental and extremely crucial for scientists to program computational software to solve optimization problems efficiently and for practitioners to use it efficiently. Thus, it is very essential to know about the idea, origin, and usage of these methods. Although the methods have been used for very long time and the theory has been developed too long, most, if not all, of the authors who develop the theory are unknown and the theory has not been stated clearly and systematically. To bridge the gap in the literature in this area and provide academics and practitioners with an overview of the methods, this paper reviews and discusses the four most commonly used methods to find the optimization solution including the bisection, gradient, Newton-Raphson, and secant methods. We first introduce the origin and idea of the methods and develop all the necessary theorems to prove the existence and convergence of the estimate for each method.We then give two examples to illustrate the approaches. Thereafter, we review the literature of the applications of getting the optimization solutions in some important issues and discuss the advantages and disadvantages of each method. We note that all the theorems developed in our paper could be well-known but, so far, we have not seen any book or paper that discusses all the theorems stated in our paper in detail. Thus, we believe the theorems developed in our paper could still have some contributions to the literature. Our review is useful for academics and practitioners in finding the optimization solutions in their studies.

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Section: Gradient Methodsmentioning

confidence: 99%

Optimal Solution Techniques in Decision Sciences: A Review

2019

ADS

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“…Proof: (Sketch) This is a straightforward corollary of the Chasles's theorem [30]. Analogous to the 2D case, the desired velocity p B " pv B , ω B q can be determined by (23), yielding the spiral trajectory Trjps 0 , p B , tq.…”

Section: ) Computation Complexitymentioning

confidence: 97%

“…The MG-SO method in (13) for mode generation is a LP, with the variable size proportional to the number of all contact points N V . Thus, it has a complexity of OpN 3.5 V q from Terlaky [23]. The time complexity of the GJK algorithm for 2D collision checking from Bergen [3] is OpV `VO q, where V O is the maximum number of vertices for the obstacles.…”

Section: ) Computation Complexitymentioning

confidence: 99%

Internationalization with backward international law

Tang

2018

Peking University Law Journal

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Solving partial differential equations (PDEs) is a central task in scientific computing. Recently, neural network approximation of PDEs has received increasing attention due to its flexible meshless discretization and its potential for high-dimensional problems. One fundamental numerical difficulty is that random samples in the training set introduce statistical errors into the discretization of loss functional which may become the dominant error in the final approximation, and therefore overshadow the modeling capability of the neural network. In this work, we propose a new minmax formulation to optimize simultaneously the approximate solution, given by a neural network model, and the random samples in the training set, provided by a deep generative model. The key idea is to use a deep generative model to adjust random samples in the training set such that the residual induced by the approximate PDE solution can maintain a smooth profile when it is being minimized. Such an idea is achieved by implicitly embedding the Wasserstein distance between the residual-induced distribution and the uniform distribution into the loss, which is then minimized together with the residual. A nearly uniform residual profile means that its variance is small for any normalized weight function such that the Monte Carlo approximation error of the loss functional is reduced significantly for a certain sample size. The adversarial adaptive sampling (AAS) approach proposed in this work is the first attempt to formulate two essential components, minimizing the residual and seeking the optimal training set, into one minmax objective functional for the neural network approximation of PDEs.

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Interior-Point Methods for Conic-Linear Optimization

Cited by 2 publications

References 29 publications

Optimal Solution Techniques in Decision Sciences: A Review

Optimal Solution Techniques in Decision Sciences: A Review

Internationalization with backward international law

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