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New insights and augmented lagrangian algorithm for optimal portfolio liquidation with market impact

INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH 2023年第5期30卷 2640-2664页

作者： Xu, Fengmin Li, Xuepeng Dai, Yu-Hong Wang, Meihua Xi An Jiao Tong Univ Sch Econ & Finance Xian 710061 Peoples R China Chinese Acad Sci LSEC Acad Math Syst Sci Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Xidian Univ Sch Econ & Management Xian 710126 Peoples R China

In financial markets, investors may be forced to unwind their portfolios to meet a determined leverage ratio within regulatory policy or risk management requirements. This paper studies the optimal portfolio liquidation problem with market impact. Some new insights are given into this problem, and trading orders are discussed for various financial parameters. Specifically, we establish the equivalence between equity and liability maximization. This means if one wants to maximize the equity, it is to maximize the liability and vice versa. The computational complexity of the problem is examined to be NP-hard. We expose the hidden convexity through monotonicity analysis and linearization techniques. Although good properties are established for the lagrangian algorithm, a counter-example is constructed to show one deficiency of this algorithm. Therefore, we propose an augmented lagrangian algorithm for solving the problem. The inverse Hessian of the augmented lagrangian function is explicitly calculated for the subproblem solved by the projected Newton method. Meanwhile, we consider how to choose a good initial point, which is essential for seeking a high-quality solution. Some numerical results are presented, which validate the usefulness of the augmented lagrangian algorithm.

关键词： augmented lagrangian algorithm equity and liability optimal portfolio liquidation price impact

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A Semismooth Newton-based augmented lagrangian algorithm for Density Matrix Least Squares Problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2022年第3期195卷 749-779页

作者： Liu, Yong-Jin Yu, Jing Fuzhou Univ Ctr Appl Math Fujian Prov Sch Math & Stat Fuzhou 350108 Fujian Peoples R China Fuzhou Univ Sch Math & Stat Fuzhou 350108 Fujian Peoples R China

The density matrix least squares problem arises from the quantum state tomography problem in experimental physics and has many applications in signal processing and machine learning, mainly including the phase recovery problem and the matrix completion problem. In this paper, we first reformulate the density matrix least squares problem as an equivalent convex optimization problem and then design an efficient semismooth Newton-based augmented lagrangian (Ssnal) algorithm to solve the dual of its equivalent form, in which an inexact semismooth Newton (Ssn) algorithm with superlinear or even quadratic convergence is applied to solve the inner subproblems. Theoretically, the global convergence and locally asymptotically superlinear convergence of the Ssnal algorithm are established under very mild conditions. Computationally, the costs of the Ssn algorithm for solving the subproblem are significantly reduced by making full use of low-rank or high-rank property of optimal solutions of the density matrix least squares problem. In order to verify the performance of our algorithm, numerical experiments conducted on randomly generated quantum state tomography problems and density matrix least squares problems with real data demonstrate that the Ssnal algorithm is more effective and robust than the Qsdpnal solver and several state-of-the-art first-order algorithms.

关键词： Density matrix least squares problems Semismooth Newton algorithm augmented lagrangian algorithm Quadratic growth condition

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Application of an improved augmented lagrangian algorithm to the tuning of robust PID controller for hydraulic turbine governing system

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INTERNATIONAL JOURNAL OF MODELLING IDENTIFICATION AND CONTROL 2015年第2期23卷 181-191页

作者： He, Xuesong Liu, Changyu Dong, Hongkui Yan, Qiurong Huazhong Univ Sci & Technol Sch Hydropower & Informat Engn Wuhan 430074 Peoples R China Yunnan Elect Power Test & Res Inst Kunming 650217 Yunnan Peoples R China Huazhong Univ Sci & Technol Sch Elect & Elect Engn Wuhan 430074 Peoples R China

The tuning of a proportional-integral-derivative (PID) controller for hydraulic turbine governing system is a constrained engineering problem. The PID control parameters must be adjusted to capture the requirements of achieving satisfactory transient behaviour, robust stability, and disturbance rejection, which are all essential aspects for hydraulic turbine control. An optimal tuning method for robust PID controller based on the augmented lagrangian (AL) algorithm and particle swarm optimisation (PSO) is presented in this paper. The AL is extended by incorporating the PSO in a seamless manner. Moreover, it introduces a new updated strategy for the penalty factor and embeds a local search procedure within the algorithm framework to enhance the search capability and accuracy, so that the optimal control parameters can be reliably obtained from the search space. The results of the simulation experiment demonstrate the effectiveness of this improved AL algorithm in producing a robust PID controller that provides system stability over a wide range of operating conditions.

关键词： hydraulic turbine governing system PID controller robust stability augmented lagrangian algorithm particle swarm optimisation PSO

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On convergence rate of the augmented lagrangian algorithm for nonsymmetric saddle point problems

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APPLIED NUMERICAL MATHEMATICS 2005年第2期54卷 122-134页

作者： Awanou, GM Lai, MJ Univ Minnesota IMA Minneapolis MN 55455 USA Univ Georgia Dept Math Athens GA 30605 USA

We are interested in solving the system [GRAPHICS] by a variant of the augmented lagrangian algorithm. This type of problem with nonsymmetric A typically arises in certain discretizations of the Navier-Stokes equations. Here A is a (n, n) matrix, c, F is an element of R-n, L is a (m, n) matrix, and lambda G is an element of R-m. We assume that A is invertible on the kernel of L. Convergence rates of the augmented lagrangian algorithm are known in the symmetric case but the proofs in [R. Glowinski, P. LeTallec, augmented lagrangian and Operator Splitting Methods in Nonlinear Mechanics, SIAM, 19891 used spectral arguments and cannot be extended to the nonsymmetric case. The purpose of this paper is to give a rate of convergence of a variant of the algorithm in the nonsymmetric case. We illustrate the performance of this algorithm with numerical simulations of the lid-driven cavity flow problem for the 2D Navier-Stokes equations. (c) 2004 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： augmented lagrangian algorithm saddle point problems

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On convergence rate of the augmented lagrangian algorithm for nonsymmetric saddle point problems

On convergence rate of the augmented Lagrangian algorithm fo...

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6th International Symposium on Iterative Methods in Scientific Computing

作者： Awanou, GM Lai, MJ Univ Minnesota IMA Minneapolis MN 55455 USA Univ Georgia Dept Math Athens GA 30605 USA

关键词： augmented lagrangian algorithm saddle point problems

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Dynamic Optimal Power Flow with Storage Restrictions Using augmented lagrangian algorithm 20

Dynamic Optimal Power Flow with Storage Restrictions Using A...

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Power Systems Computation Conference (PSCC)

作者： Ruppert, Manuel Slednev, Viktor Ardone, Armin Fichtner, Wolf Karlsruhe Inst Technol Chair Energy Econ Karlsruhe Germany

ISBN: (纸本)9781910963104

Incorporating time-coupled storage restrictions in the nonlinear dynamic optimal power flow poses a major challenge in terms of performance and solvability. As a consequence, the application of existing works that combine AC optimal power flow with integration of storage are usually restricted to problems of small size or short periods of time coupling. In this work, an approach using an augmented lagrangian algorithm for the decoupling of the storage component has been developed, which is suitable for solving large-scale problems. The iterative methodology, which is highly scalable and allows for parallelization up to the total amount of time steps, takes multiple storage units into account. The approach has been tested using a data set of the German transmission network consisting of 950 buses and 26 hydro storage units. The proposed method shows robust results in terms of convergence as well as a promising performance characteristic for an increasing number of simulated time steps.

关键词： augmented lagrangian algorithm Dynamic Optimal Power Flow Energy Storage Power System Management Progressive Hedging

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An augmented lagrangian algorithm for decomposition of symmetric tensors of order-4

An augmented Lagrangian algorithm for decomposition of symme...

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2017 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2017

作者： Phan, Anh-Huy Yamagishi, Masao Cichocki, Andrzej Brain Science Institute RIKEN Wakoshi Japan Tokyo Institute of Technology Japan

ISBN: (纸本)9781509041176

Decomposition of symmetric tensors has found numerous applications in blind sources separation, blind identification, clustering, and analysis of social interactions. In this paper, we consider fourth order symmetric tensors, and its symmetric tensor decomposition. By imposing unit-length constraints on components, we resort the optimisation problem to the constrained eigenvalue decomposition in which eigenvectors are represented in form of rank-1 matrices. To this end, we develop an augmented lagrangian algorithm with simple update rules. The proposed algorithm has been compared with the Trust-Region solver over manifold, and achieved higher success rates. The algorithm is also validated for blind identification, and achieves more stable results than the ALSCAF algorithm. © 2017 IEEE.

关键词： augmented lagrangian algorithm spherical quadratic programming symmetric tensor decomposition

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An Inexact Semismooth Newton-Based augmented lagrangian algorithm for Multi-Task Lasso Problems

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ASIA-PACIFIC JOURNAL OF OPERATIONAL RESEARCH 2024年第3期41卷 2350027-2350027页

作者： Lin, Lanyu Liu, Yong-Jin Fuzhou Univ Sch Math & Stat Fuzhou 350108 Peoples R China Fuzhou Univ Ctr Appl Math Fujian Prov Sch Math & Stat Fuzhou 350108 Peoples R China

This paper is concerned with the l(1),infinity-norm ball constrained multi-task learning problem, which has received extensive attention in many research areas such as machine learning, cognitive neuroscience, and signal processing. To address the challenges of solving large-scale multi-task Lasso problems, this paper develops an inexact semismooth Newton-based augmented lagrangian (Ssnal) algorithm. When solving the inner problems in the Ssnal algorithm, the semismooth Newton (Ssn) algorithm with superlinear or even quadratic convergence is applied. Theoretically, this paper presents the global and asymptotically superlinear local convergence of the Ssnal algorithm under standard conditions. Computationally, we derive an efficient procedure to construct the generalized Jacobian of the projector onto l(1),infinity-norm ball, which is an important component of the Ssnal algorithm, making the computational cost in the Ssn algorithm very cheap. Comprehensive numerical experiments on the multi-task Lasso problems demonstrate that the Ssnal algorithm is more efficient and robust than several existing state-of-the-art first-order algorithms.

关键词： Multi-task Lasso problem augmented lagrangian algorithm semismooth Newton algorithm generalized Jacobian

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An augmented lagrangian algorithm for decomposition of symmetric tensors of order-4

An augmented Lagrangian algorithm for decomposition of symme...

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IEEE International Conference on Acoustics, Speech and Signal Processing

作者： Anh-Huy Phan Masao Yamagishi Andrzej Cichocki Brain Science Institute RIKEN Wakoshi Japan Tokyo Institute of Technology Japan

ISBN: (纸本)9781509041183