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Optimization over structured subsets of positive semidefinite matrices via column generation

DISCRETE OPTIMIZATION 2017年 24卷 129-151页

作者： Ahmadi, Amir Ali Dash, Sanjeeb Hall, Georgina ORFE Princeton NJ USA IBM Res Yorktown Hts NY 10598 USA

We develop algorithms to construct inner approximations of the cone of positive semidefinite matrices via linear programming and second order cone programming. Starting with an initial linear algebraic approximation suggested recently by Ahmadi and Majumdar, we describe an iterative process through which our approximation is improved at every step. This is done using ideas from column generation in largescale linear programming. We then apply these techniques to approximate the sum of squares cone in a nonconvex polynomial optimization setting, and the copositive cone for a discrete optimization problem. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Semidefinite programming Polynomial optimization Copositive programming Linear programming second order cone programming

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New Formulation and Strong MISOCP Relaxations for AC Optimal Transmission Switching Problem

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IEEE TRANSACTIONS ON POWER SYSTEMS 2017年第6期32卷 4161-4170页

作者： Kocuk, Burak Dey, Santanu S. Sun, Xu Andy Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA

As the modern transmission control and relay technologies evolve, transmission line switching has become an important option in power system operators' toolkits to reduce operational cost and improve system reliability. Most recent research has relied on the DC approximation of the power flow model in the optimal transmission switching problem. However, it is known that DC approximation may lead to inaccurate flow solutions and also overlook stability issues. In this paper, we focus on the optimal transmission switching problem with the full AC power flow model, abbreviated as AC optimal transmission switching (AC OTS). We propose a new exact formulation for ACOTS and its mixed-integer second-order cone programming relaxation. We improve this relaxation via several types of strong valid inequalities inspired by the recent development for the closely related AC optimal power flow problem [Kocuk et al., "Strong SOCP relaxations for the optimal power flow problem," Oper. Res., vol. 64, no. 6, pp. 1177-1196, 2016]. We also propose a practical algorithm to obtain high-quality feasible solutions for the AC OTS problem. Extensive computational experiments show that the proposed formulation and algorithms efficiently solve IEEE standard and congested instances and lead to significant cost benefits with provably tight bounds.

关键词： AC transmission switching cutting planes mixed-integer programming second order cone programming semidefinite programming valid inequalities

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Continuous location under the effect of 'refraction'

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MATHEMATICAL programming 2017年第1-2期161卷 33-72页

作者： Blanco, Victor Puerto, Justo Ponce, Diego Univ Granada Dept Quantitat Methods Econ & Business Granada 18011 Spain IMUS Seville Spain Univ Seville Dept Stat E-41012 Seville Spain Univ Seville OR E-41012 Seville Spain

In this paper we address the problem of locating a new facility on a d-dimensional space when the distance measure (- or polyhedral-norms) is different at each one of the sides of a given hyperplane . We relate this problem with the physical phenomenon of refraction, and extend it to any finite dimensional space and different distances at each one of the sides of any hyperplane. An application to this problem is the location of a facility within or outside an urban area where different distance measures must be used. We provide a new second order cone programming formulation, based on the -norm representation given in Blanco et al. (Comput Optim Appl 58(3):563-595, 2014) that allows to solve the problem in any finite dimensional space with second order cone or semidefinite programming tools. We also extend the problem to the case where the hyperplane is considered as a rapid transit media (a different third norm is also considered over ) that allows the demand to travel, whenever it is convenient, through to reach the new facility. Extensive computational experiments run in Gurobi are reported in order to show the effectiveness of the approach. Some extensions of these models are also presented.

关键词： Continuous location Refraction second order cone programming Norms

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Time-dependent green Weber problem

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COMPUTERS & OPERATIONS RESEARCH 2017年 88卷 316-323页

作者： Khoei, Arsham Atashi Sural, Haldun Tural, Mustafa Kemal Middle East Tech Univ Dept Ind Engn Ankara Turkey

We consider an extension of the classical Weber problem, named as the green Weber problem (GWP), in which the customers have one-sided time windows. The GWP decides on the location of the single facility in the plane and the speeds of the vehicles serving the customers from the facility within the one-sided time windows so as to minimize the total amount of carbon dioxide emitted in the whole distribution system. We also introduce time-dependent congestion which limits the vehicle speeds in different time periods and call the resulting problem as the time-dependent green Weber problem (TD-GWP). In the TD-GWP, the vehicles are allowed to wait during more congested time periods. We formulate the GWP and TD-GWP as second order cone programming problems both of which can be efficiently solved to optimality. We show that if the traffic congestion is non-increasing, then there exists an optimal solution in which the vehicles do not wait at all. Computational results are provided comparing the locations of the facility and the resulting carbon dioxide emissions of the classical Weber problem with those of the GWP and comparing the GWP with the TD-GWP in terms of carbon dioxide emissions in different traffic congestion patterns. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Weber problem Carbon dioxide emission Facility location Green facility location Time-dependent congestion second order cone programming

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SCA Power Optimization in Cell-Free Massive MIMO with Short-Term Power Constraints 10

SCA Power Optimization in Cell-Free Massive MIMO with Short-...

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10th International Conference on Wireless Communications and Signal Processing (WCSP)

作者： Zhang, Yao Cao, Haotong Guo, Yongan Yang, Longxiang Nanjing Univ Posts & Telecommun Coll Commun & Informat Engn Nanjing 210003 Jiangsu Peoples R China

ISBN: (纸本)9781538661192

In this paper, the performance of downlink cell-free massive multiple-input multiple-output (mMIMO) system with short-term power constraints is investigated. To maximize the sum downlink user signal-interference-noise-ratio (SINR), this paper proposes a sequential convex approximation (SCA) power optimization algorithm for cell-free mMIMO with normalized conjugate beamforming (CB) precoder. This proposed SCA algorithm is proved to be a second order cone programming (SOCP). Then, under the condition that the achievable data SINR is greater than or equal to a certain quality of service (QoS) threshold, a new sum power minimization algorithm is presented. It also can be reformulated as a SOCP. Numerical results indicate that the achievable sum downlink SINR monotonically increases at each iteration and the sum SINR tends to a stable value after 7 iterations. In addition, the power minimization algorithm works well in satisfying target users' SINR constraints and minimizing the sum downlink transmission power.

关键词： cell-free massive MIMO system short-term average power constraint sequential convex approximation QoS threshold second order cone programming

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Learning the optimal kernel for Fisher discriminant analysis via second order cone programming

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2010年第3期203卷 692-697页

作者： Khemchandani, Reshma Jayadeva Chandra, Suresh Indian Inst Technol Dept Math New Delhi 110016 India IBM India Res Lab Inst Area Vasant Kunj New Delhi 110070 India

Kernel Fisher discriminant analysis (KFDA) is a popular classification technique which requires the user to predefine an appropriate kernel. Since the performance of KFDA depends on the choice of the kernel, the problem of kernel selection becomes very important. In this paper we treat the kernel selection problem as an optimization problem over the convex set of finitely many basic kernels, and formulate it as a second order cone programming (SOCP) problem. This formulation seems to be promising because the resulting SOCP can be efficiently solved by employing interior point methods. The efficacy of the optimal kernel, selected from a given convex set of basic kernels, is demonstrated on LICI machine learning benchmark datasets. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Fisher discriminant analysis Kernel methods Machine learning Kernel optimization Support vector machines Convex optimization second order cone programming Semidefinite programming

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Monocular 3D tracking of deformable surfaces using sequential second order cone programming

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PATTERN RECOGNITION 2010年第1期43卷 244-254页

作者： Shen, Shuhan Liu, Yuncai Lu, Wu-Sheng Shanghai Jiao Tong Univ Inst Image Proc & Pattern Recognit Shanghai 200030 Peoples R China Univ Victoria Dept Elect & Comp Engn Victoria BC Canada

Reconstructing structures of deformable objects from monocular image sequences is important for applications like visual servoing and augmented reality. In this paper, we propose a method to recover 3D shapes of deformable surfaces using sequential second order cone programming (SOCP). The key of our approach is to represent the surface as a triangulated mesh and introduce two sets of constraints, one for model-to-image keypoint correspondences which are SOCP constraints, another for retaining the original lengths of the mesh edges which are non-convex constraints. In the process of tracking, the surface structure is iteratively updated by solving sequential SOCP feasibility problems in which the non-convex constraints are replaced by a set of convex constraints over a local convex region. The shape constraints used in our approach is more generic than previous methods, that enables us to reliably recover surface shapes with smooth, sharp and other complex deformations. The capability and efficiency of our approach are evaluated quantitatively with synthetic image sequences and qualitatively with real image sequences. (C) 2009 Elsevier Ltd. All rights reserved.

关键词： 3D tracking Deformable tracking second order cone programming Convex optimization

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Support vector machine classification with noisy data: a second order cone programming approach

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INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 2010年第7期39卷 757-781页

作者： Trafalis, Theodore B. Alwazzi, Samir A. Univ Oklahoma Lab Optimizat & Intelligent Syst Sch Ind Engn Norman OK 73019 USA

In this paper, we investigate the stability of linear and quadratic programming support vector machines (SVMs) with bounded noise in the input data using a robust optimisation model. For a linear discriminant function, this model is expressed as a second order cone optimisation problem. Using the concept of the kernel function, we generalise for nonlinear discriminant functions. Intuitively, it looks quite clear that large margin classifiers are robust in terms of bounded input noise. However, there is no theoretical analysis investigating this behaviour. We show that the SVM solution is stable under bounded perturbations of the data both in the linear programming and quadratic programming. Computational results are also presented for toy and real-world data.

关键词： robust optimisation kernel methods support vector machines semi definite programming second order cone programming

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Achievable Rate of Beamforming Dual-hop Multi-antenna Relay Network in the Presence of a Jammer

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KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS 2017年第8期11卷 3789-3808页

作者： Feng, Guiguo Guo, Wangmei Gao, Jingliang Xidian Univ State Key Lab Integrated Serv Networks ISN Xian 710071 Shaanxi Peoples R China

This paper studies a multi-antenna wireless relay network in the presence of a jammer. In this network, the source node transmits signals to the destination node through a multi-antenna relay node which adopts the amplify-and-forward scheme, and the jammer attempts to inject additive signals on all antennas of the relay node. With the linear beamforming scheme at the relay node, this network can be modeled as an equivalent Gaussian arbitrarily varying channel (GAVC). Based on this observation, we deduce the mathematical closed-forms of the capacities for two special cases and the suboptimal achievable rate for the general case, respectively. To reduce complexity, we further propose an optimal structure of the beamforming matrix. In addition, we present a second order cone programming (SOCP)-based algorithm to efficiently compute the optimal beamforming matrix so as to maximize the transmission rate between the source and the destination when the perfect channel state information (CSI) is available. Our numerical simulations show significant improvements of our propose scheme over other baseline ones.

关键词： wireless relay network linear beamforming Gaussian arbitrarily varying channel second order cone programming

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DSO and Aggregator Sharing Concept for Distributed Battery Storage System

DSO and Aggregator Sharing Concept for Distributed Battery S...

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IEEE International Conference on Environment and Electrical Engineering

作者： Mirna Grzanic Tomislav Capuder Slavko Krajcar Faculty of Electrical Engineering and Computing University of Zagreb Zagreb Croatia

Increase in electricity demand, mostly due to integration of new technologies electrifying heat and transport, as well as increasing share of distributed generation, create new challenges for distribution system operator (DSO) in terms of reliability and quality of power supply. This is particularly manifested during daily extremes, suggesting there is insufficient capacity of the distribution grid. To avoid expensive and unnecessary investments in new cables and transformers (since events that require network reinforcement are short and rare), the DSO can define a methodology for implementation of so called non-grid solutions. This paper analyses a concept in which the DSO signs a contract with the aggregator of flexible resources, offering incentives, such as reduced network fee, for using battery storage when necessary. Since the aggregator is looking for a feasible business case due to high investment cost of storage, the incentives given by the DSO provide a cost-effective investment for the aggregator. The aggregator uses battery storage to minimize the cost of purchasing electricity on the market while the DSO is utilizing it to postpone network reinforcement. The problem is cast as a bilevel problem where the operation of the distribution grid is modelled by second-order-cone-programming (SOCP) relaxation of optimal power flows bidding for the right to use aggregator's battery for preventing violating networks technical constraints. Over a set of scenarios, we demonstrate how coordinated usage of battery storage can postpone network reinforcement while ensuring secure power supply, as well as bring additional cost benefits for the aggregator.

关键词： Aggregator Battery Storage Distribution System Operator Reduced Network Fee second order cone programming

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