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ROBUST STATE ESTIMATION METHOD OF DISTRIBUTION NETWORK FOR STOCHASTIC VARIABLES WITH DYNAMIC INTERVAL VALUES

INTERNATIONAL JOURNAL OF POWER AND ENERGY SYSTEMS 2022年第10期42卷 1页

作者： Li Xin Wu Jiekang Zeng Shunqi Cai Jinjian Guangzhou Power Supply Bur Guangdong Power Grid C Guangzhou Peoples R China Guangdong Univ Technol Sch Automat Guangzhou Peoples R China

The uncertainties in measurement make the estimation accuracy of traditional methods difficult to meet the dispatching requirements of distribution systems. At the same time, the measurement system often has bad data during the acquisition and transmission process, which seriously affects the accuracy of state estimation. The method in this paper is used to improve the influence of the state estimation on measurement uncertainty and bad data, and improve the robustness and accuracy of estimation. Interval analysis method is used to describe the measurement problem with uncertainties, the interval constraint model of state variables in distribution network is established, and the feasible region of state variables is obtained using linear programming method. Based on the measurement uncertainty theory, a robust state estimation optimization model with the highest measurement point accuracy as the objective function is established. The precise value of the state estimation is solved by the interior point method. With the feasible region of state variables as constraints and the median value of the interval as the initial value, it is unnecessary to take the calculation results of power flow as the initial value, thus reducing the scope of solving state variables and reducing the amount of calculation. Compared with the traditional weighted least square, this method has a significant improvement in accuracy and resistance. The feasibility and effectiveness of this method are verified by IEEE30 and IEEE118 systems.

关键词： Distribution network robust state estimation dynamic interval analysis interval value constraint linear programming interior point method

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Efficient Second-Order methods for Second-Order Cone Programs and Continuous Nonlinear Two-Stage Optimization Problems

Efficient Second-Order Methods for Second-Order Cone Program...

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作者： Luo, Xinyi Northwestern University

学位级别：Ph.D., Doctor of Philosophy

This dissertation presents novel advancements in the field of continuous nonlinear optimization, focusing on the development of efficient second-order methods for second-order conic programs (SOCPs) and continuous nonlinear two-stage optimization problems. The primary focus is on the theory and computations of Sequential Quadratic programming (SQP) methods, which are widely used for solving nonlinear optimization problems. In this work, we introduce a new SQP method tailored for linear SOCPs, leveraging the SQP framework for nonlinear optimization. By capitalizing on warm-start capabilities for active-set quadratic programming subproblem solvers and utilizing polyhedral outer approximations of cones, our method achieves a local quadratic rate of convergence. It efficiently identifies the set of cones where the optimal solution lies at extreme points, enabling rapid local convergence. Numerical experiments confirm that the method can take advantage of good starting points and can achieve higher accuracy compared to a state-of-the-art interior point solver. The dissertation also includes two computational projects. Firstly, we introduce Restart-SQP, an open-source C++ software package implementing Fletcher’s Sℓ1QP method and integrating the parametric active-set methods for solving SQP subproblems. RestartSQP offers unique warm-start capabilities, efficiently handling changing constraint values and incorporating additional constraints dynamically. Additionally, it supports crossover from interior-point solvers like Ipopt, enabling the solution of subsequent NLPs using the SQP method. Numerical experiments using the CUTE benchmark NLP test set demonstrate the practical performance of RestartSQP. Furthermore, we develop a two-stage decomposition algorithm for continuous nonlinear two-stage optimization problems and discussed the detailed C++ implementation. The algorithm decomposes large-scale problems into master and independent subproblems, allowing for parallel proce

关键词： Nonlinear optimization Sequential Quadratic programming Two-stage optimization Optimization problems

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On the Convergence of Overlapping Schwarz Decomposition for Nonlinear Optimal Control

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2022年第11期67卷 5996-6011页

作者： Na, Sen Shin, Sungho Anitescu, Mihai Zavala, Victor M. Univ Chicago Dept Stat Chicago IL 60637 USA Univ Wisconsin Dept Chem & Biol Engn Madison WI 53706 USA Argonne Natl Lab Math & Comp Sci Div Lemont IL 60439 USA

We study the convergence properties of an overlapping Schwarz decomposition algorithm for solving nonlinear optimal control problems (OCPs). The algorithm decomposes the time domain into a set of overlapping subdomains, and solves all subproblems defined over subdomains in parallel. The convergence is attained by updating primal-dual information at the boundaries of overlapping subdomains. We show that the algorithm exhibits local linear convergence, and that the convergence rate improves exponentially with the overlap size. We also establish global convergence results for a general quadratic programming, which enables the application of the Schwarz scheme inside second-order optimization algorithms (e.g., sequential quadratic programming). The theoretical foundation of our convergence analysis is a sensitivity result of nonlinear OCPs, which we call "exponential decay of sensitivity" (EDS). Intuitively, EDS states that the impact of perturbations at domain boundaries (i.e., initial and terminal time) on the solution decays exponentially as one moves into the domain. Here, we expand a previous analysis available in the literature by showing that EDS holds for both primal and dual solutions of nonlinear OCPs, under uniform second-order sufficient condition, controllability condition, and boundedness condition. We conduct experiments with a quadrotor motion planning problem and a partial differential equations (PDE) control problem to validate our theory, and show that the approach is significantly more efficient than alternating direction method of multipliers and as efficient as the centralized interior-point solver.

关键词： Decomposition methods nonlinear programming optimal control overlapping parallel algorithms

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Faster randomized interior point methods for tall/wide linear programs

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The Journal of Machine Learning Research 2022年第1期23卷 15156-15203页

作者： Agniva Chowdhury Gregory Dexter Palma London Haim Avron Petros Drineas Computer Science and Mathematics Division Oak Ridge National Laboratory Oak Ridge TN Department of Computer Science Purdue University West Lafayette IN Operations Research and Information Engineering Cornell University Ithaca NY School of Mathematical Sciences Tel Aviv University Tel Aviv Israel

linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such as combinatorics. It is also used in many machine learning applications, such as ℓ1-regularized SVMs, basis pursuit, nonnegative matrix factorization, etc. interior point methods (IPMs) are one of the most popular methods to solve LPs both in theory and in practice. Their underlying complexity is dominated by the cost of solving a system of linear equations at each iteration. In this paper, we consider both feasible and infeasible IPMs for the special case where the number of variables is much larger than the number of constraints. Using tools from Randomized linear Algebra, we present a preconditioning technique that, when combined with the iterative solvers such as Conjugate Gradient or Chebyshev Iteration, provably guarantees that IPM algorithms (suitably modified to account for the error incurred by the approximate solver), converge to a feasible, approximately optimal solution, without increasing their iteration complexity. Our empirical evaluations verify our theoretical results on both real-world and synthetic data.

关键词： randomized linear algebra linear programming interior point methods

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What Tropical Geometry Tells Us about the Complexity of linear programming

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SIAM REVIEW 2021年第1期63卷 123-164页

作者： Allamigeon, Xavier Benchimol, Pascal Gaubert, Stephane Joswig, Michael Ecole Polytech Inst Polytech Paris INRIA F-91128 Palaiseau France Ecole Polytech Inst Polytech Paris CMAP CNRS F-91128 Palaiseau France Kisio Digital 20 Rue Hector Malot F-75012 Paris France INRIA Saclay Ile de France Ile De France France Ecole Polytech CMAP CNRS UMR 7641 Palaiseau France Tech Univ Berlin Chair Discrete Math Geometry Berlin Germany MPI MiS Leipzig Germany

Tropical geometry has been recently used to obtain new complexity results in convex optimization and game theory. In this paper, we present an application of this approach to a famous class of algorithms for linear programming, i.e., log-barrier interior point methods. We show that these methods are not strongly polynomial by constructing a family of linear programs with 3r + 1 inequalities in dimension 2r for which the number of iterations performed is in Omega(2(r)). The total curvature of the central path of these linear programs is also exponential in r, disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky, and Zinchenko. These results are obtained by analyzing the tropical central path, which is the piecewise linear limit of the central paths of parameterized families of classical linear programs viewed through "logarithmic glasses."" This allows us to provide combinatorial lower bounds for the number of iterations and the total curvature in a general setting.

关键词： linear programming central path strongly polynomial complexity continuous analogue of the Hirsch conjecture tropical geometry

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Exact algorithms for semidefinite programs with degenerate feasible set

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JOURNAL OF SYMBOLIC COMPUTATION 2021年 104卷 942-959页

作者： Henrion, Didier Naldi, Simone El Din, Mohab Safey Univ Toulouse LAAS CNRS Toulouse France Czech Tech Univ Fac Elect Engn Prague Czech Republic Univ Limoges XLIM UMR 7252 F-87000 Limoges France Sorbonne Univ CNRS Equipe PoLSys LIP6 4 Pl Jussieu F-75252 Paris 05 France

Given symmetric matrices A(0), A(1), ..., A(n) of size m with rational entries, the set of real vectors x = (x(1), ..., x(n)) such that the matrix A(0) + x(1)A(1) + ... + x(n)A(n) has non-negative eigenvalues is called a spectrahedron. Minimization of linear functions over spectrahedra is called semidefinite programming. Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares. Numerical software for semidefinite programming are mostly based on interior point methods, assuming non-degeneracy properties such as the existence of an interior point in the spectrahedron. In this paper, we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice. However, we prove that solving such problems can be done in polynomial time if either nor m is fixed. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Semidefinite programming Polynomial optimization Exact computation Homotopy

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Solving the dose distribution problem in the radiotherapy planning applying the interior point methods

Solving the dose distribution problem in the radiotherapy pl...

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International Conference on Computational Science and Computational Intelligence (CSCI)

作者： Huaccha Neyra, Jackeline del Carmen Leite de Oliveira, Aurelio Ribeiro Univ Estadual Campinas Dept Appl Math Campinas Brazil

ISBN: (纸本)9781665458412

The dose distribution problem in the Radiation Therapy planning consists in optimize the total radiation dosage delivered into the patient. Its main aim is to attack the tumor delivering as radiation as possible, avoiding severe damage the nearby tissues and organs, for that we use a linear programming model to formulate this problem and the Intensity-modulated radiation therapy (IMRT) technique that allows us to deliver nonuniform radiation flow obtaining a best adjustment of radiation doses to the tumor cells. However, in practice, the oncologist can vary the dose according to his/her expertise making it difficult to guarantee an exact quantity of radiation. Therefore in this paper, the doses are considered as triangular fuzzy numbers and we use the surprise function, that can be understood as a penalty for the violation of each constraint, translate these fuzzy constraints in a non-linear function that must be minimized. Thus we present a mathematical model with non-linear and convex objective function with linear constraints for bounding the radiation flow. We propose an algorithm that was designed using a specially tailored Primal-Dual interior-point Method to solve this problem. Our algorithm is implemented in MATLAB and tested in real world large-scale problems in which the tumor is localized on the Head and Neck area. The numerical experiments show the amount of radiation delivered in each region of patient and, in general, the developed approach provides favorable solutions for the dose distribution problem.

关键词： interior-point methods radiation therapy fuzzy theory

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A LINMAP Method Based on the Bounded Rationality of Evaluators for Property Service Quality Evaluation

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IEEE ACCESS 2021年 9卷 122668-122684页

作者： Zuo, Wenjin Zhang, Xingxian Zeng, Shouzhen Liu, Lijun Shanghai Univ Finance & Econ Zhejiang Coll Jinhua 321013 Zhejiang Peoples R China Tongling Univ Sch Architecture & Engn Tongling 244061 Anhui Peoples R China Ningbo Univ Business Sch Ningbo 315211 Zhejiang Peoples R China

Property service quality (PSQ) evaluation is the key means of property management. Traditional PSQ evaluation is usually based on the rational hypothesis of the evaluators. The evaluators' behaviour with bounded rationality is more in line with the characteristics of modern PSQ evaluation practice. At this point, evaluation information is usually incomplete information and heterogeneous data. The existing LINMAP methods have the ability to process incomplete information and heterogeneous information, but cannot adapt to evaluation information processing under bounded rational behaviour. As one of the major achievements of behavioural economics, prospect theory can explain people's bounded behaviour when making decisions. The purpose of this paper was to develop a linear programming technique for multidimensional analysis of preference (LINMAP) model based on prospect theory for modern PSQ evaluation. First, a data preprocessing method was designed by referring to the TOPSIS method. Then, the value function was used to construct the dominance matrix, and the LINMAP model based on prospect theory was constructed. Finally, all alternatives were ranked in order according to overall performance value, which was solved by the LINMAP model. The feasibility and effectiveness of the proposed method were demonstrated by example analysis. Based on the evaluation results of PSQ of public projects, the important items and excellent property service projects and their reasons were determined, and some suggestions were proposed to improve PSQ.

关键词： Decision making Economics Psychology linear programming Industries Standards Licenses Bounded rationality linear programming technique for multidimensional analysis of preference prospect theory property service quality evaluation

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End-To-End Resource Analysis for Quantum interior-point methods and Portfolio Optimization

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PRX Quantum 2023年第4期4卷 040325-040325页

作者： Alexander M. Dalzell B. David Clader Grant Salton Mario Berta Cedric Yen-Yu Lin David A. Bader Nikitas Stamatopoulos Martin J. A. Schuetz Fernando G. S. L. Brandão Helmut G. Katzgraber William J. Zeng AWS Center for Quantum Computing Pasadena California 91106 USA California Institute of Technology Pasadena California 91125 USA Goldman Sachs New York New York 10282 USA Amazon Quantum Solutions Lab Seattle Washington 98121 USA Department of Computing Imperial College London London SW7 2AZ United Kingdom Institute for Quantum Information RWTH Aachen University 52056 Aachen Germany AWS Quantum Technologies Seattle WA 98170 USA 98170 New Jersey Institute of Technology Newark New Jersey 07102 USA University of Washington Seattle Washington 98195 USA

We study quantum interior-point methods (QIPMs) for second-order cone programming (SOCP), guided by the example use case of portfolio optimization (PO). We provide a complete quantum circuit-level description of the algorithm from problem input to problem output, making several improvements to the implementation of the QIPM. We report the number of logical qubits and the quantity and/or depth of non-Clifford T gates needed to run the algorithm, including constant factors. The resource counts we find depend on instance-specific parameters, such as the condition number of certain linear systems within the problem. To determine the size of these parameters, we perform numerical simulations of small PO instances, which lead to concrete resource estimates for the PO use case. Our numerical results do not probe large enough instance sizes to make conclusive statements about the asymptotic scaling of the algorithm. However, already at small instance sizes, our analysis suggests that, due primarily to large constant prefactors, poorly conditioned linear systems, and a fundamental reliance on costly quantum state tomography, fundamental improvements to the QIPM are required for it to lead to practical quantum advantage.

关键词： Quantum algorithms & computation Quantum circuits Quantum computation Quantum information processing Quantum information theory

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interior point methods for an algebraic system involving complementarity equations for geomechanical fractures

arXiv

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arXiv 2024年

作者： Hoang, Trung Hau

Many applications like subseismic fault modeling, fractured reservoirmodeling and interpretation/validation of fault connectivity involve thesolution to an elliptic boundary value problem in a background medium perturbedby the presence of cracks that take the form of one or many pieces of surface(with boundary). When the background medium can be considered as homogeneous,boundary integral equations appear as a method of choice for the numericalsolution to fractures problems. With such an approach, the problem isreformulated as a fully non-local equation posed at the surface of *** of boundary integral resulting in the so-called Boundary ElementMethod (BEM) leads to densely populated matrices due to the full non-localityof the operators under consideration. After the discretization process,geologists are faced with a system of equations that turns out difficult tosolve numerically. Many empirical algorithms have been proposed by geologiststo solve this system of equations. Unfortunately, none of them is guaranteed toconverge in theory (in particular when faults (fractures) intersect each otherforming a geometrically highly irregular structure). In practice, none of themappears to be either robust or efficient. We investigate another approach,referred to as interior point methods, for which convergence can be ensured(even if faults are too close). interior point methods have proved theirefficiency in a wide variety of domains, most notably for linear ***, even though we do not have any optimization problem, we can adapt ideasfrom interior point methods for the numerical resolution of the systemconsidered. The numerical results obtained demonstrate computational efficiencyand accuracy, highlighting the robustness and effectiveness of the implementedmethods. © 2024, CC BY-NC-ND.

关键词： Cracks

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