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second-order cone programming models for the unitary weighted Weber problem and for the minimum sum of the squares clustering problem

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INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH 2025年第2期32卷 961-972页

作者： Linhares, Marcella Braga de Assis Pinto, Renan Vicente Maculan, Nelson Negreiros, Marcos Univ Fed Rio De Janeiro PESC COPPE Rio de Janeiro Brazil Fed Rural Univ Rio Janeiro Dept Math Rio De Janeiro Brazil Univ Estadual Ceara Sci Comp Lab Fortaleza CE Brazil

In this work, new mixed integer nonlinear optimization models are proposed for two clustering problems: the unitary weighted Weber problem and the minimum sum of squares clustering. The proposed formulations are convex quadratic models with linear and second-order cone constraints that can be efficiently solved by interior point algorithms. Their continuous relaxation is convex and differentiable. The numerical experiments show the proposed models are more efficient than some classical models for these problems known in the literature.

关键词： combinatorial optimization nonlinear programming clustering problem second-order cone programming

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A new formulation for second-order cone programming support vector machine

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INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS 2024年第3期15卷 1101-1111页

作者： Zong, Zemin Mu, Xuewen Xidian Univ Sch Math & Stat Xian 710071 Shanxi Peoples R China

A new second-order cone programming (SOCP) formulation inspired by the soft-margin linear programming support vector machine (LP-SVM) formulation and cost-sensitive framework is proposed. Our proposed method maximizes the slack variables related to each class by appropriately relaxing the bounds on the VC dimension using the l(infinity)-norm, and penalizes them using the corresponding regularization parametrization to control the trade-off between margin and slack variables. The proposed method has two main advantages: firstly, a flexible classifier is constructed that extends the advantages of the soft-margin LP-SVM problem to the second-order cone;secondly, due to the elimination of a conic restriction, only two SOCP problems containing second-order cone constraints need to be solved. Thus similar results to the SOCP-SVM problem are obtained with less calculational effort. Numerical experiments show that our method achieves the better classification performance than the conventional SOCP-SVM formulations and standard linear SVM formulations.

关键词： second-order cone programming Support vector machine Linear programming SVM

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Operational Coordination Optimization of Electricity and Natural Gas Networks Based on Sequential Symmetrical second-order cone programming

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JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY 2025年第2期13卷 488-499页

作者： Min, Liang Lou, Chengwei Yang, Jin Yu, James Yu, Zhibin Power Grid Dispatching & Control Branch Co NARI Te Nanjing Peoples R China China Agr Univ Coll Informat & Elect Engn Beijing Peoples R China Univ Glasgow Sch Engn Glasgow G12 8QQ Scotland Scottish Power Energy Network Glasgow G2 5AD Scotland Univ Liverpool Dept Mech & Aerosp Engn Liverpool England

The variable and unpredictable nature of renewable energy generation (REG) presents challenges to its large-scale integration and the efficient and economic operation of the electricity network, particularly at the distribution level. In this paper, an operational coordination optimization method is proposed for the electricity and natural gas networks, aiming to overcome the identified negative impacts. The method involves the implementation of bi-directional energy flows through power-to-gas units and gas-fired power plants. A detailed model of the three-phase power distribution system up to each phase is employed to improve the representation of multi-energy systems to consider real-world end-user consumption. This method allows for the full consideration of unbalanced operational scenarios. Meanwhile, the natural gas network is modelled and analyzed with steady-state gas flows and the dynamics of the line pack in pipelines. The sequential symmetrical second-order cone programming (SS-SOCP) method is employed to facilitate the simultaneous analysis of three-phase imbalance and line pack while accelerating the solution process. The efficacy of the operational coordination optimization method is demonstrated in case studies comprising a modified IEEE 123-node power distribution system with a 20-node natural gas network. The studies show that the operational coordination optimization method can simultaneously minimize the total operational cost, the curtailment of installed REG, the voltage imbalance of three-phase power system, and the overall carbon emissions.

关键词： Natural gas Power systems Pipelines Electricity Optimization Mathematical models Load flow programming Numerical models Power system dynamics Operational coordination multi-energy system power-to-gas electricity network natural gas network second-order cone programming

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A second-order cone programming-based microgrid bidding strategy considering real-time market price correlation

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ELECTRICAL ENGINEERING 2025年 1-14页

作者： El-Meligy, Mohammed A. Sharaf, Mohamed Alnowibet, Khalid A. Abdelgawad, Abdelatty E. King Saud Univ Adv Mfg Inst POB 800 Riyadh 11421 Saudi Arabia King Saud Univ Coll Sci Stat & Operat Res Dept Riyadh 11451 Saudi Arabia King Saud Univ Coll Engn Ind Engn Dept POB 800 Riyadh 11421 Saudi Arabia

This study establishes a non-deterministic microgrid bidding strategy methodology participating in a day-ahead energy market. In this regard, a stochastic programming-based model is mathematically constructed, fully considering the uncertainty of day-ahead market prices, electricity demand, and renewable generation by creating several scenarios. The real-time electricity market is also considered, where the real-time price uncertainty is modeled using robust optimization. Even though robust optimization has already been employed to model real-time price uncertainty, the correlation among real-time prices at different hours is neglected. Thus, how real-time price correlation affects the optimal microgrid bidding strategy remains unclear. This study develops a novel mathematical model that captures real-time price correlations by constructing an ellipsoidal uncertainty set. In this context, unlike the current approaches in which the real-time price deviation ranges are modeled as constants at each hour, they are assumed to lie within a prespecified ellipsoidal uncertainty set. In doing so, a mixed-integer second-order cone programming problem is created, which existing solving methods can efficiently handle. The presented scheme is applied to a typical microgrid, and the efficiency and excellence of the presented framework are validated by comparing numerical results with traditional models.

关键词： Correlation Ellipsoidal uncertainty set Microgrid bidding strategy Robust optimization second-order cone programming Stochastic programming

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A second-order sequential optimality condition for nonlinear second-order cone programming problems

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2025年第3期90卷 911-939页

作者： Fukuda, Ellen H. Okabe, Kosuke Kyoto Univ Grad Sch Informat Sakyo KuYoshida Honmachi Kyoto Kyoto 6068501 Japan

In the last two decades, the sequential optimality conditions, which do not require constraint qualifications and allow improvement on the convergence assumptions of algorithms, had been considered in the literature. It includes the work by Andreani et al. (IMA J Numer Anal 37:1902-1929, 2017), with a sequential optimality condition for nonlinear programming, that uses the second-order information of the problem. More recently, Fukuda et al. (Set-Valued Var Anal 31:15, 2023) analyzed the conditions that use second-order information, in particular for nonlinear second-order cone programming problems (SOCP). However, such optimality conditions were not defined explicitly. In this paper, we propose an explicit definition of approximate-Karush-Kuhn-Tucker 2 (AKKT2) and complementary-AKKT2 (CAKKT2) conditions for SOCPs. We prove that the proposed AKKT2/CAKKT2 conditions are satisfied at local optimal points of the SOCP without any constraint qualification. We also present two algorithms that are based on augmented Lagrangian and sequential quadratic programming methods and show their global convergence to points satisfying the proposed conditions.

关键词： Optimality conditions second-order optimality second-order cone programming Conic optimization

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Distributed Slack-Bus Based DC Optimal Power Flow With Transmission Loss: A second-order cone programming Approach and Sufficient Conditions

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IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING 2024年第3期21卷 3873-3885页

作者： Huang, Yuhan Ding, Tao Mu, Chenggang Zhang, Xiaosheng He, Yuankang Shahidehpour, Mohammad Xi An Jiao Tong Univ Sch Elect Engn Xian 710049 Peoples R China Northwest Branch State Grid Corp China Xian 710048 Shaanxi Peoples R China IIT ECE Dept Chicago IL 60616 USA

This paper proposed sufficient conditions of the zero duality gap for the second-order conic programming approach of the DC optimal power flow that quadratically embeds the transmission loss. The mechanism of the changes for locational marginal price with differently selecting the slack bus is revisited, based on which a load-weighted distributed slack bus method is introduced. Furthermore, a second-order conic programming-based convexification method is proposed by employing the duality analysis. A favorable property of the proposed method is that mild sufficient conditions of the zero duality gap can be derived by the analysis of Karush-Kuhn-Tucker conditions. Numerical results illustrate the effectiveness of the proposed method and physically prove the proposed sufficient conditions. Compared with traditional and similar market-clearing methods, the proposed method has better performance on convergence, robustness, and accuracy. Note to Practitioners-This paper was motivated by the problem of the pricing mechanism in the deregulated electricity market but it is also applicable to rapidly solving the power flow of power systems with sufficient reactive power. Existing approaches use the B-coefficient to estimate the transmission loss in the model non-linearly. In this paper, an improved approach considering transmission loss is proposed, which further introduces the distributed slack bus method and subsequently employs the second-order cone programming algorithm. The proposed method hedges the risk of appearing multiple solutions or lack of accuracy. Rigorous mathematical proofs are derived to support the method. For industrial practice, three sufficient conditions suitable for different kinds of power systems are given, indicating when the proposed approach can be efficiently applied The proposed method can be integrated into the existing electricity market trading platform for independent system operators and facilitates real-time market clearing and electricity

关键词： DC optimal power flow distributed slack bus duality analysis Karush-Kuhn-Tucker condition locational marginal price second-order cone programming

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AUBIN PROPERTY AND STRONG REGULARITY ARE EQUIVALENT FOR NONLINEAR second-order cone programming

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SIAM Journal on Optimization 2025年第2期35卷 712-738页

作者： Chen, Liang Chen, Ruoning Sun, Defeng Zhu, Junyuan School of Mathematics Hunan University Changsha China Department of Mathematical Sciences Tsinghua University Beijing China Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Hong Kong

This paper solves a fundamental open problem in variational analysis on the equivalence between the Aubin property and the strong regularity for nonlinear second-order cone programming (SOCP) at a locally optimal solution. We achieve this by introducing a reduction approach to the Aubin property characterized by the Mordukhovich criterion and a lemma of alternative choices on cones to replace the S-lemma used in Outrata and Ramírez [SIAM J. Optim., 21 (2011), pp. 789-823] and Opazo, Outrata, and Ramírez [SIAM J. Optim., 27 (2017), pp. 2143-2151], where the same SOCP was considered under the strict complementarity condition except for possibly only one block of constraints. As a byproduct, we also offer a new approach to the well-known result of Dontchev and Rockafellar [SIAM J. Optim., 6 (1996), pp. 1087-1105] on the equivalence of the two concepts in conventional nonlinear programming. © 2025 Society for Industrial and Applied Mathematics.

关键词： second-order cone programming

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IPRSOCP: A Primal-Dual Interior-Point Relaxation Algorithm for second-order cone programming

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JOURNAL OF THE OPERATIONS RESEARCH SOCIETY OF CHINA 2024年 1-31页

作者： Zhang, Rui-Jin Wang, Zhao-Wei Liu, Xin-Wei Dai, Yu-Hong Chinese Acad Sci Acad Math & Syst Sci Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Hebei Univ Technol Inst Math Tianjin 300401 Peoples R China

Inspired by the smoothing barrier augmented Lagrangian function in Liu et al. (Math Methods Oper Res 96(3):351-382, 2022), we propose a primal-dual interior-point relaxation algorithm for second-order cone programming, called IPRSOCP. Two features of the IPRSOCP algorithm are as follows. One is that the iterative points of the proposed algorithm need not lie inside the interior region, convening the use of warm-start. The other is that an explicit form of the Schur complement matrix is explored such that the low-rank structure of the Schur complement matrix can be used to improve numerical stability and efficiency. Under suitable assumptions, it is shown that the barrier parameter in the IPRSOCP algorithm tends to zero and the generated sequence of iterations is globally convergent to the solution. Numerical results demonstrate that the IPRSOCP algorithm is competitive with the open-source benchmark solvers, SeDuMi, SDPT3, and ECOS, in terms of robustness and efficiency.

关键词： second-order cone programming Interior-point relaxation method Smoothing barrier augmented Lagrangian function Schur complement matrix Global convergence

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Path Deformation Method with Constraints on Normal Curvature for Wheeled Robots in Precision Agriculture Based on second-order cone programming

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AUTOMATION AND REMOTE CONTROL 2024年第2期85卷 123-131页

作者： Tormagov, T. A. Moscow Inst Phys & Technol Moscow Russia

In precision agriculture, path planning for agricultural robots with complete covering a three-dimensional landscape is an essential task. For robots with front wheels steering the normal curvature of the trajectories should be limited to some value determined by the characteristics of the vehicle. The paper considers a method of deformation of these paths to account for obstacles for trajectories described by homogeneous cubic B-splines. We propose an optimization problem that allows calculating paths with minimizing skips in the coverage. The considered problem is convex and belongs to the class of second-order cone programming, which entails the possibility of its computationally efficient solution. The computational examples are presented.

关键词： precision farming precision agriculture obstacle avoidance complete coverage path planning second-order cone programming SOCP

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Large deformation of cable networks with fiber sliding as a second-order cone programming

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INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES 2024年 298卷

作者： Tkachuk, Mykola Tkachuk, Anton Natl Tech Univ Inst Educ & Sci Mech Engn & Transport Kharkiv Polytech Inst UA-61002 Kharkiv Ukraine Karlstad Univ Dept Engn & Phys SE-65188 Karlstad Sweden

A new model for irreversible large deformations of fiber networks is developed. The fibers are considered as inextensible cables that slide relative to each other in the frictional junctions. This sliding is constituted by a rate-independent flow rule. The nonsmooth dissipation potential for each sliding system is defined as a product of the yield strength and the absolute value of the fiber sliding. The response of the cable segments is nonsmooth as well, since it shows asymmetry with respect to tension and compression. A principle of minimum incremental potential and a pure complementary energy principle are derived for the equilibrium incremental loading of the network at large deformations. They form a pair of primal and dual second- order cone programming problems with matching sets of displacement-based and force-based variables. These problems can be effectively solved by interior point methods that have many advantages compared to the gradient-based methods or dynamic relaxation. The model is extended by a simple mechanism of fiber pull-out resulting from fiber sliding at the free unconstrained ends. This can be used for the microstructural analysis of failure of needle-punched nonwoven materials.

关键词： Cable networks Fiber sliding second-order cone programming Complementary energy principle Nonwoven fabrics

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