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An Accelerated Proximal Alternating Direction Method of Multipliers for Optimal Decentralized Control of Uncertain Systems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2025年第1期204卷 1-37页

作者： Yang, Bo Zhao, Xinyuan Li, Xudong Sun, Defeng Beijing Univ Technol Beijing 100124 Peoples R China Fudan Univ Shanghai 200433 Peoples R China Hong Kong Polytech Univ Hong Kong Peoples R China

To ensure the system stability of the H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}_{2}$$\end{document}-guaranteed cost optimal decentralized control (ODC) problem, we formulate an approximate semidefinite programming (SDP) problem that leverages the block diagonal structure of the decentralized controller's gain matrix. To minimize data storage requirements and enhance computational efficiency, we employ the Kronecker product to vectorize the SDP problem into a conic programming (CP) problem. We then propose a proximal alternating direction method of multipliers (PADMM) to solve the dual of the resulting CP problem. By using the equivalence between the semi-proximal ADMM and the (partial) proximal point algorithm, we identify the non-expansive operator of PADMM, enabling the use of Halpern fixed-point iteration to accelerate the algorithm. Finally, we demonstrate that the sequence generated by the proposed accelerated PADMM exhibits a fast convergence rate for the Karush-Kuhn-Tucker residual. Numerical experiments confirm that the accelerated algorithm outperforms the well-known COSMO, MOSEK, and SCS solvers in efficiently solving large-scale CP problems, particularly those arising from H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}_{2}$$\end{document}-guaranteed cost ODC problems.

关键词： Semidefinite programming conic programming Acceleration Halpern iteration Decentralized control

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Facial Approach for Constructing Stationary Points for Mathematical Programs with Cone Complementarity Constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2025年第1期204卷 1-27页

作者： Madariaga, Javier I. Ramirez, C. Hector North Carolina State Univ Dept Math Raleigh NC 27695 USA Univ Chile Dept Ingn Matemat Santiago Chile Univ Chile Ctr Modelamiento Matemat FB210005 CNRS IRL 2807 Santiago Chile

This paper studies stationary points in mathematical programs with cone complementarity constraints (CMPCC). We begin by reviewing various formulations of CMPCC and revisiting definitions for Bouligand, proximal strong, regular strong, Wachsmuth's strong, L-strong, weak, as well as Mordukhovich and Clarke stationary points, establishing a comprehensive framework for CMPCC. Building on key principles related to cone faces and their properties, we introduce a novel stationarity concept, facial stationarity, which naturally extends the weak stationarity condition in the CMPCC context. Finally, we analyze the hierarchical relations between these different types of stationary points.

关键词： conic programming Complementarity constraints Stationary point Facial stationarity

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Energy-constrained orienteering problem for green tourist trip design: Mathematical formulation and heuristic solution approaches

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COMPUTERS & INDUSTRIAL ENGINEERING 2025年 200卷

作者： Karabas, Tolga Tural, Mustafa Kemal Middle East Tech Univ Dept Ind Engn Ankara Turkiye

Planning a touristic itinerary is a challenging task that requires personalized tour generation for tourists interested in visiting available points of interest (POIs). The negative externalities of using vehicles (e.g., emissions) can be reduced by considering such aspects in itinerary planning. To address this need, we propose the Energy-Constrained Tourist Trip Design Problem to plan environmentally friendly touristic itineraries. We model this problem as an Energy-Constrained Orienteering Problem (ECOP), where we consider continuous vehicle speed, speed-dependent travel time, and fuel consumption -accordingly emissions- which have not been considered in the scope of the OP before. First, the ECOP is formulated as a mixed-integer second-order cone programming (MISOCP) model which is able to solve only small-size instances to optimality within a reasonable time. Second, for large-size instances, we develop two heuristic methods, namely Greedy Insertion (GI) and Iterated Local Search (ILS) algorithms. In the latter, several improvement heuristics are coupled with an SOCP-based speed optimization procedure. Based on our computational experiments, the GI heuristic is the fastest among the proposed methods, while generally yielding suboptimal solutions compared to the ILS algorithm, but for larger instances, these solutions are better than the ones obtained via the exact method. The ILS algorithm outperforms the exact method by reaching good-quality solutions in a relatively short computing time. The ECOP has prospective applications on selective routing problems involving other transportation modes (e.g., cruise, aerial) and has a significant potential on reducing transportation-related GHG emissions.

关键词： conic programming Green logistics Iterated local search Orienteering problem Speed optimization Tourist trip planning

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Risk assessment of integrated transmission and distribution systems considering load transfer

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ELECTRIC POWER SYSTEMS RESEARCH 2025年 239卷

作者： Lu, Jiangang Zhao, Ruifeng Hou, Zufeng Lin, Guihui Zhang, Yong Wang, Chao Pan, Kaiyan Liu, Haixin Guangdong Power Grid Co Ltd Elect Power Dispatching & Control Ctr Guangzhou 518000 Peoples R China Guangdong Power Co Ltd Zhuhai Power Supply Bur Zhuhai 519000 Peoples R China Dongfang Elect Cooperat Yantai 264010 Peoples R China

Integrated transmission and distribution system (ITDS) analysis is gaining increasing attention from academics and industry. Load transfer, a critical factor in maintaining system reliability and reducing system risk during outages, is often overlooked in conventional system risk assessment. This paper proposes a framework that incorporates load transfer capabilities into ITDS modeling, enabling a comprehensive evaluation of its impact on risk mitigation. The framework integrates the uncertainty of electricity demand, initial outages suggested by contingency motifs, and detailed ITDS models. A centralized model using mixed-integer conic programming is formulated to identify optimal load transfer strategies. Recognizing the typical operational separation of transmission and distribution systems, a practical two-stage optimization method with limited information exchange is also proposed to determine degraded system states. Case studies demonstrate that incorporating load transfer can significantly reduce system risk, underscoring the importance of integrating this capability into risk assessments and operational strategies.

关键词： Risk assessment Integrated transmission and distribution system Load transfers conic programming

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Cuts for conic mixed-integer programming

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12th International Integer programming and Combinatorial Optimization Conference

作者： Atamturk, Alper Narayanan, Vishnu Univ Calif Berkeley Dept Ind Engn & Operat Res Berkeley CA 94720 USA

ISBN: (纸本)9783540727910

A conic integer program is an integer programming problem with conic constraints. conic integer programming has important applications in finance, engineering, statistical learning, and probabilistic integer programming. Here we study mixed-integer sets defined by second-order conic constraints. We describe general-purpose conic mixed-integer rounding cuts based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve continuous conic programming relaxations at the nodes of the search tree. Our preliminary computational experiments with the new cuts show that they are quite effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs.

关键词： integer programming conic programming branch-and-cut

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Interior point methods in the year 2025

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EURO JOURNAL ON COMPUTATIONAL OPTIMIZATION 2025年 13卷

作者： Gondzio, Jacek Univ Edinburgh Sch Math Peter Guthrie Tait Rd Edinburgh EH9 3FD Scotland Univ Edinburgh Maxwell Inst Math Sci Peter Guthrie Tait Rd Edinburgh EH9 3FD Scotland

Interior point methods (IPMs) have hugely influenced the field of optimization. Their fast development has been triggered by the seminal paper of Narendra Karmarkar published in 1984 which delivered a polynomial algorithm for linear programming and suggested that it might be implemented into a very efficient method in practice. Indeed, this has been demonstrated within a few years after 1984 and has gained IPMs a status of exceptionally powerful optimization tool. Linear programming (LP) is at the centre of many operational research techniques including mixed-integer programming, network optimization and various decomposition techniques. Therefore, any progress in LP has far-reaching consequences. IPMs certainly did not disappoint in this context: they have become a heavily used methodology in modern optimization and operational research. Their accuracy, efficiency and reliability have been particularly appreciated when IPMs are applied to truly large scale problems which challenge any alternative approaches. In this survey we will discuss several issues related to interior point methods. We will recall techniques which provide the building blocks of IPMs, and observe that actually at least some of them have been developed before 1984. We will briefly comment on the worst-case complexity results for different variants of IPMs and then focus on key aspects of their implementation. We will also address some of the most spectacular features of IPMs and discuss their potential advantages when applied in decomposition algorithms, cutting planes scheme and column generation technique.

关键词： Interior point methods Linear programming conic programming Theory and computational perspective History and future developments

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Advances in the simulation of viscoplastic fluid flows using interior-point methods

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2018年 330卷 368-394页

作者： Bleyer, Jeremy Univ Paris Est Ecole Ponts ParisTech ENPC Lab Navier UMR 8205CNRSIFSTTAR 6-8 Av Blaise Pascal F-77455 Champs Sur Marne France

We present a primal-dual interior point algorithm for the resolution of steady-state viscoplastic fluid flows formulated as a conic optimization problem. We give a complete description of the algorithm including some advanced aspects such as a predictor-corrector and scaling scheme to improve its efficiency. Our interior-point approach is shown to be largely more efficient than Augmented Lagrangian (AL) approaches which are traditionally used to solve such problems. In particular, the interior-point approach is roughly 5 times faster than the modern accelerated version of AL algorithms. The yield surfaces are shown to be accurately predicted and various examples ranging from channel flows to three-dimensional flows through a porous medium demonstrate its efficiency. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Yield stress fluids Viscoplasticity Bingham model Interior point method conic programming

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Central Swaths

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FOUNDATIONS OF COMPUTATIONAL MATHEMATICS 2013年第3期13卷 405-454页

作者： Renegar, James Cornell Univ Sch Operat Res & Informat Engn Ithaca NY 14850 USA

We develop a natural generalization to the notion of the central path, a concept that lies at the heart of interior-point methods for convex optimization. The generalization is accomplished via the "derivative cones" of a "hyperbolicity cone", the derivatives being direct and mathematically appealing relaxations of the underlying (hyperbolic) conic constraint, be it the non-negative orthant, the cone of positive semidefinite matrices, or other. We prove that a dynamics inherent to the derivative cones generates paths always leading to optimality, the central path arising from a special case in which the derivative cones are quadratic. Derivative cones of higher degree better fit the underlying conic constraint, raising the prospect that the paths they generate lead to optimality quicker than the central path.

关键词： Hyperbolicity cone Hyperbolic polynomial Hyperbolic programming Central path conic programming Convex optimization

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Polynomial convergence of infeasible-interior-point methods over symmetric cones

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SIAM JOURNAL ON OPTIMIZATION 2006年第4期16卷 1211-1229页

作者： Rangarajan, BK Cornell Univ Sch Operat Res & Ind Engn Ithaca NY 14853 USA

We establish polynomial-time convergence of infeasible-interior-point methods for conic programs over symmetric cones using a wide neighborhood of the central path. The convergence is shown for a commutative family of search directions used in Schmieta and Alizadeh [ Math. Program. 96 ( 2003), pp. 409 - 438]. Monteiro and Zhang [ Math. Program., 81 ( 1998), pp. 281 - 299] introduced this family of directions when analyzing semidefinite programs. These conic programs include linear and semidefinite programs. This extends the work of Rangarajan and Todd [ Tech. rep. 1388, School of OR & IE, Cornell University, Ithaca, NY, 2003], which established convergence of infeasible-interior-point methods for self-scaled conic programs using the NT direction. Our work is built on earlier analyses by Faybusovich [ J. Comput. Appl. Math., 86 ( 1997), pp. 149 - 175] and Schmieta and Alizadeh [ Math. Program. 96 ( 2003), pp. 409 - 438]. Of independent interest, we provide a constructive proof of Lyapunov lemma in the Jordan algebraic setting.

关键词： conic programming infeasible-interior-point methods symmetric cones Euclidean Jordan algebras semidefinite programming

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Lower bound limit analysis of cohesive-frictional materials using second-order cone programming

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2006年第4期66卷 604-634页

作者： Makrodimopoulos, A Martin, CM Univ Oxford Dept Engn Sci Oxford OX1 3PJ England

The formulation of limit analysis by means of the finite element method leads to an optimization problem with a large number of variables and constraints. Here we present a method for obtaining strict lower bound solutions using second-order cone programming (SOCP), for which efficient primaldual interior-point algorithms have recently been developed. Following a review of previous work, we provide a brief introduction to SOCP and describe how lower bound limit analysis can be formulated in this way. Some methods for exploiting the data structure of the problem are also described, including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage. The benefits of employing SOCP are then illustrated with numerical examples. Through the use of an effective algorithm/software, very large optimization problems with up to 700000 variables are solved in minutes on a desktop machine. The numerical examples concern plane strain conditions and the Mohr-Coulomb criterion, however we show that SOCP can also be applied to any other problem of lower bound limit analysis involving a yield function with a conic quadratic form (notable examples being the Drucker-Prager criterion in 2D or 3D, and Nielsen's criterion for plates). Copyright (c) 2005 John Wiley & Sons, Ltd.

关键词： limit analysis lower bound cohesive-frictional finite element optimization conic programming

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