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Decompositions of Semidefinite Matrices and the Perspective Reformulation of Nonseparable quadratic Programs

MATHEMATICS OF OPERATIONS RESEARCH 2020年第1期45卷 15-33页

作者： Frangioni, Antonio Gentile, Claudio Hungerford, James Univ Pisa Dipartimenlo Informat I-56127 Pisa Italy CNR Ist Anal Sistemi Informat Antonio Ruberti I-00185 Rome Italy RaceTrac Petr Inc Atlanta GA 30339 USA

We study the problem of decomposing the Hessian matrix of a mixed integer convex quadratic program (MICQP) into the sum of positive semidefinite 2 x 2 matrices. Solving this problem enables the use of perspective reformulation techniques for obtaining strong lower bounds for MICQPs with semicontinuous variables but a nonseparable objective function. An explicit formula is derived for constructing 2 x 2 decompositions when the underlying matrix is weakly scaled diagonally dominant, and necessary and sufficient conditions are given for the decomposition to be unique. For matrices lying outside this class, two exact semidefinite programming approaches and an efficient heuristic are developed for finding approximate decompositions. We present preliminary results on the bound strength of a 2 x 2 perspective reformulation for the portfolio optimization problem, showing that, for some classes of instances, the use of 2 x 2 matrices can significantly improve the quality of the bound with respect to the best previously known approach, although at a possibly high computational cost.

关键词： mixed-integer quadratic programming matrix decomposition scaled diagonal dominance semicontinuous variables portfolio optimization

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On handling indicator constraints in mixed integer programming

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2016年第3期65卷 545-566页

作者： Belotti, Pietro Bonami, Pierre Fischetti, Matteo Lodi, Andrea Monaci, Michele Nogales-Gomez, Amaya Salvagnin, Domenico FICO Birmingham W Midlands England IBM Corp Madrid Spain Univ Padua Padua Italy Univ Bologna Bologna Italy Ecole Polytech Montreal Montreal PQ Canada Huawei France R&D Math & Algorithm Sci Lab Paris France IBM Corp Milan Italy

mixed integer programming (MIP) is commonly used to model indicator constraints, i.e., constraints that either hold or are relaxed depending on the value of a binary variable. Unfortunately, those models tend to lead to weak continuous relaxations and turn out to be unsolvable in practice;this is what happens, for e.g., in the case of Classification problems with Ramp Loss functions that represent an important application in this context. In this paper we show the computational evidence that a relevant class of these Classification instances can be solved far more efficiently if a nonlinear, nonconvex reformulation of the indicator constraints is used instead of the linear one. Inspired by this empirical and surprising observation, we show that aggressive bound tightening is the crucial ingredient for solving this class of instances, and we devise a pair of computationally effective algorithmic approaches that exploit it within MIP. One of these methods is currently part of the arsenal of IBM-Cplex since version 12.6.1. More generally, we argue that aggressive bound tightening is often overlooked in MIP, while it represents a significant building block for enhancing MIP technology when indicator constraints and disjunctive terms are present.

关键词： mixed-integer linear programming mixed-integer quadratic programming Indicator constraints

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MPC-based framework incorporating pre-disaster and post-disaster actions and transportation network constraints for weather-resilient power distribution networks

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APPLIED ENERGY 2024年 362卷

作者： Souto, Laiz Parisio, Alessandra Taylor, Philip C. Univ Bristol Dept Elect & Elect Engn Queens Bldg Bristol BS8 1TR England Univ Manchester Dept Elect & Elect Engn Manchester M13 9PL England

Enhancing power system resilience to extreme weather requires an effective coordination of pre-disaster preventive response actions and post-disaster repair and restoration services. In this context, this article presents a two -stage framework to enhance resilience of power distribution networks to severe weather events. It considers operating and resource constraints in interdependent power and transportation networks based on official guidance from transport appraisal studies. It also incorporates sources of information available in a smart city context, such as energy measurements, traffic flows, and geographical information systems. In the first stage, a pre -disaster operational planning strategy is defined to prepare the grid for a high -risk outage scenario with the objective of minimizing the value of lost loads, formulated as a mixed -integer quadratic programming model. In the second stage, a post -disaster corrective strategy is implemented over a receding time horizon to compensate for deviations between the actions computed in the first stage and the actions required to minimize power outages, embedded into a model predictive control scheme. The effectiveness of the proposed framework is demonstrated on a real -world large-scale distribution network in the United Kingdom over a range of outage scenarios. Simulation results show that the proposed framework is capable of effectively minimizing first -stage operational costs and second -stage deviations between the projected and actual load demand supply. Numerical results obtained with the proposed framework indicate that the load energy unserved is at least twice smaller than with typical practices adopted by distribution network operators and computational times take two minutes or less. Therefore, it can be effectively used by distribution network operators to ensure an appropriate level of preparedness to power outages caused by extreme weather along with prompt restoration and repair services.

关键词： Extreme weather mixed-integer quadratic programming Model predictive control Power distribution networks Power system resilience Power system restoration Smart grids

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Linear approximations to improve lower bounds of a physician scheduling problem in emergency rooms

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JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 2023年第3期74卷 888-904页

作者： Devesse, Valdemar Abrao P. A. Akartunali, Kerem Arantes, Marcio da S. Toledo, Claudio F. M. Univ Sao Paulo BR-13566590 Sao Carlos SP Brazil Univ Strathclyde Glasgow Lanark Scotland SENAI Innovat Inst Embedded Syst Florianopolis SC Brazil

The physician assignment process consists of coverage of shifts and duties allocated to physicians in a planning period, taking into account work regulations, individual preferences, and organizational rules, which mostly conflict with each other. In this work, we propose a reformulated mixed-integer programming model based on the literature to tackle fairness in physician scheduling in Emergency Rooms (ERs). In particular, we propose two mixed-integer quadratic programming formulations that consider quadratic costs and two models with linear costs. Our approaches provide balanced schedules concerning target hours and weekends in terms of fairness. Our models also provide a high degree of demand coverage, providing decision-makers a significant advantage.

关键词： Physician scheduling problem mixed-integer quadratic programming fairness healthcare management

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A branch-and-cut algorithm for mixed-integer Bilinear programming

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2020年第2期282卷 506-514页

作者： Fischetti, Matteo Monaci, Michele Univ Padua DEI Via Gradenigo 6-A I-35100 Padua Italy Univ Bologna DEI Guglielmo Marconi Viale Risorgimento 2 I-40136 Bologna Italy

In this paper, we consider the mixed-integer Bilinear programming problem, a widely-used reformulation of the classical mixed-integer quadratic programming problem. For this problem we describe a branch and -cut algorithm for its exact solution, based on a new family of intersection cuts derived from bilinearspecific disjunctions. We also introduce a new branching rule that is specifically designed for bilinear problems. We computationally analyze the behavior of the proposed algorithm on a large set of mixed-integer quadratic instances from the MINLPIib problem library. Our results show that our method, even without intersection cuts, is competitive with a state-of-the-art mixed-integer nonlinear solver. As to intersection cuts, their extensive use at each branching node tends to slow down the solver for most problems in our test bed, but they are extremely effective for some specific instances. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Combinatorial optimization mixed-integer quadratic programming Bilinear programming Branch-and-cut algorithms Intersection cuts

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Compute-and-Forward on a Multiaccess Relay Channel: Coding and Symmetric-Rate Optimization

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2014年第4期13卷 1932-1947页

作者： El Soussi, Mohieddine Zaidi, Abdellatif Vandendorpe, Luc Catholic Univ Louvain ICTEAM B-1348 Louvain Belgium Univ Paris Est Marne La Vallee F-77454 Marne La Vallee 2 France

We consider a system in which two users communicate with a destination with the help of a half-duplex relay. Based on the compute-and-forward scheme, we develop and evaluate the performance of coding strategies that are of network coding spirit. In this framework, instead of decoding the users' information messages, the destination decodes two integer-valued linear combinations that relate the transmitted codewords. Two decoding schemes are considered. In the first one, the relay computes one of the linear combinations and then forwards it to the destination. The destination computes the other linear combination based on the direct transmissions. In the second one, accounting for the side information available at the destination through the direct links, the relay compresses what it gets using lattice-based Wyner-Ziv compression and conveys it to the destination. The destination then computes the two linear combinations, locally. For both coding schemes, we discuss the design criteria, and derive the allowed symmetric-rate. Next, we address the power allocation and the selection of the integer-valued coefficients to maximize the offered symmetric-rate;an iterative coordinate descent method is proposed. The analysis shows that the first scheme can outperform standard relaying techniques in certain regimes, and the second scheme, while relying on feasible structured lattice codes, can at best achieve the same performance as regular compress-and-forward for the multiaccess relay network model that we study. The results are illustrated through some numerical examples.

关键词： Compute-and-forward network coding lattice codes relay channel geometric programming mixed-integer quadratic programming

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A two-stage approach to the UCITS-constrained index-tracking problem

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COMPUTERS & OPERATIONS RESEARCH 2019年 103卷 167-183页

作者： Strub, O. Trautmann, N. Univ Bern Dept Business Adm Schutzenmattstr 14 CH-3012 Bern Switzerland

Undertakings for Collective Investments in Transferable Securities (UCITS) are investment funds that are regulated by the European Union. UCITS have become increasingly popular, resulting in a total corresponding amount of assets under management of (sic) 8.5 trillion by the end of 2016. We present a two-stage approach to the problem of how to construct a portfolio of assets for a UCITS that aims to replicate the returns of a financial index subject to the constraints imposed by the UCITS regulations. In the first stage, we apply a genetic algorithm that treats subsets of the index constituents as individuals to construct a good feasible solution in a short CPU time. In this genetic algorithm, we use a new representation of subsets, which is the first to exhibit all of the following four desirable properties: feasibility, efficiency, locality, and heritability. In the second stage, we apply local branching based on a new mixed-integer quadratic programming formulation to improve the best solution obtained in the first stage. In a numerical experiment on real-world data, the approach yields very good feasible solutions in a short CPU time. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： Portfolio management Index tracking mixed-integer quadratic programming Heuristics

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Online Voltage Regulation of Active Distribution Networks: A Deep Neural Encoding-Decoding Approach

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IEEE TRANSACTIONS ON POWER SYSTEMS 2024年第2期39卷 4574-4586页

作者： Sang, Linwei Xu, Yinliang Wu, Wenchuan Long, Huan Tsinghua Univ Tsinghua Shenzhen Int Grad Sch Tsinghua Berkeley Shenzhen Inst Shenzhen 518055 Peoples R China Tsinghua Univ Dept Elect Engn Beijing 100084 Peoples R China Southeast Univ Sch Elect Engn Nanjing 210096 Peoples R China

Voltage regulation plays a vital role in active distribution networks, which can be posed as mixed-integer quadratic programming (MIQP). To alleviate the real-time MIQP solution burden, this article proposes a deep neural encoding-decoding approach for online voltage regulation, including offline encoding and online decoding processes. Offline encoding extracts the integer variables and active constraints of the MIQP regulation problem and reduces the MIQP into quadratic programming (QP). The KKT-based solution matrices derived from the reduced QP are stored via the compact triangular matrix factorization way. Deep neural network (DNN) is trained to encode the optimal voltage regulation solution into the regulation strategy. Online decoding utilizes the trainedDNNto identify the optimal regulation strategies, which selects the top-k most likely strategies under various scenarios. Then it decodes the k strategies into precomputed KKT-based linear solution matrices for real-time regulation. The above online voltage regulation can be calculated efficiently without any solver. Case studies verify the proposed approach's voltage regulation effectiveness and calculation efficiency.

关键词： Deep learning voltage regulation online optimization mixed-integer quadratic programming

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quadratic CONVEX REFORMULATIONS FOR SEMICONTINUOUS quadratic programming

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SIAM JOURNAL ON OPTIMIZATION 2017年第3期27卷 1531-1553页

作者： Wu, Baiyi Sun, Xiaoling Li, Duan Zheng, Xiaojin Guangdong Univ Foreign Studies Sch Finance Guangzhou Guangdong Peoples R China Fudan Univ Sch Management Dept Management Sci Shanghai 200433 Peoples R China Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China Tongji Univ Sch Econ & Management Shanghai 200092 Peoples R China

We consider in this paper a class of semicontinuous quadratic programming problems, which arises in many real-world applications such as production planning, portfolio selection, and subset selection in regression. We build upon the idea of the quadratic convex reformulation approach, i.e., adding to the original objective function an additional convex term that is zero at all points of the feasible region. Exploiting the structure of semicontinuous variables, we derive the most general set of quadratic functions that can be added. We also incorporate the state-of-the-art perspective reformulation into our new framework. Within this framework, the set of optimal reformulations with the tightest continuous relaxation can be found by solving a semidefinite programming problem. We also explore the relationship among different reformulations within our framework and simplify the search for optimal ones in certain cases with the help of second-order cone programming. Among the proposed set of optimal reformulations, we can choose one with or without the perspective component. If the perspective component is present, the new reformulation can be solved by the perspective cut algorithm in the literature. If the perspective component is not present, the new reformulation is a standard mixed-integer quadratic programming (MIQP) problem and can be plugged into and solved by any MIQP solvers. Our preliminary numerical tests in both portfolio selection and subset selection problems have demonstrated the effectiveness of our new reformulations, and the comparison results indicate that the performance of the new reformulation without the perspective component, when solved in general MIQP solvers, is competitive to the perspective cut approach.

关键词： mixed-integer quadratic programming semicontinuous quadratic program perspective cut reformulation quadratic convex reformulation semidefinite program portfolio selection subset selection

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A deterministic method for the unit commitment problem in power systems

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COMPUTERS & OPERATIONS RESEARCH 2016年 66卷 241-247页

作者： Zheng, Haiyan Jian, Jinbao Yang, Linfeng Loan, Ran Guangxi Univ Coll Elect Engn Nanning 530004 Peoples R China Yulin Normal Univ Guangxi Coll & Univ Key Lab Complex Syst Optimiza Coll Math & Informat Sci Yulin 537000 Peoples R China Guangxi Univ Coll Comp Elect & Informat Nanning 530004 Peoples R China Henan Univ Technol Coll Sci Zhengzhou 450001 Peoples R China

The unit commitment (UC) problem in power systems is generally formulated as a large-scale nonlinear mixed-integer combinatorial optimization problem, which is difficult to solve. This paper presents a deterministic method named cut-and-branch for solving UC, which is based on cuts and branch-and-bound search as well as heuristic rounding technique. First, a suitable mixed-integer quadratic programming (MIQP) model of UC is presented by some linearization technique, then the MIQP is solved by the proposed cut-and-branch method. In the proposed method, two classes of cuts are introduced to give a stronger representation of the corresponding continuous relaxed problem: one is the approximate integer cut derived from a natural understanding of the problem which is simple but highly efficient, and the other is the generalized flow cover inequality. Furthermore, the continuous relaxed problem incorporating the proposed cuts can obtain some better initial feasible solutions and reduce the numbers of nodes during the branch-and-bound search. The simulation results for six systems with up to 100 units and 24 h show that the proposed method has nice convergence, which can find global optimal solution in theory. (c) 2015 Elsevier Ltd. All rights reserved.

关键词： Unit commitment mixed-integer quadratic programming Approximate integer cut Generalized flow cover inequality Cut-and-branch method

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