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Correlative sparsity in primal-dual interior-point methods for LP, SDP, and SOCP

APPLIED MATHEMATICS AND OPTIMIZATION 2008年第1期58卷 69-88页

作者： Kobayashi, Kazuhiro Kim, Sunyoung Kojima, Masakazu Tokyo Inst Technol Dept Math & Comp Sci Meguro Ku Tokyo 1528552 Japan Ewha Womans Univ Dept Math Seoul 120750 South Korea

Exploiting sparsity has been a key issue in solving large-scale optimization problems. The most time-consuming part of primal-dual interior-point methods for linear programs, second-order cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by the Cholesky factorization. The computational efficiency is greatly affected by the sparsity of the coefficient matrix of the equation which is determined by the sparsity of an optimization problem (linear program, semidefinite program or second-order cone program). We show if an optimization problem is correlatively sparse, then the coefficient matrix of the Schur complement equation inherits the sparsity, and a sparse Cholesky factorization applied to the matrix results in no fill-in.

关键词： correlative sparsity primal-dual interior-point method linear program semidefinite program second-order cone program partial separability chordal graph

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Shades of grey: A critical review of grey-number optimization

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ENGINEERING OPTIMIZATION 2009年第6期41卷 573-592页

作者： Rosenberg, David E. Utah State Univ Dept Civil & Environm Engn Logan UT 84322 USA Utah State Univ Utah Water Res Lab Logan UT 84322 USA

A grey number is an uncertain number with fixed lower and upper bounds but unknown distribution. Grey numbers find use in optimization to systematically and proactively incorporate uncertainties expressed as intervals plus communicate resulting stable, feasible ranges for the objective function and decision variables. This article critically reviews their use in linear and stochastic programs with recourse. It summarizes grey model formulation and solution algorithms. It advances multiple counter-examples that yield risk-prone grey solutions that perform worse than a worst-case analysis and do not span the stable feasible range of the decision space. The article suggests reasons for the poor performance and identifies conditions for which it typically occurs. It also identifies a fundamental shortcoming of grey stochastic programming with recourse and suggests new solution algorithms that give more risk-adverse solutions. The review also helps clarify the important advantages, disadvantages, and distinctions between risk-prone and risk-adverse grey-programming and best/worst case analysis.

关键词： interval number linear program stochastic program with recourse optimization with uncertainty

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Computational analysis of storage synthesis in developing Brassica napus L. (oilseed rape) embryos: flux variability analysis in relation to ¹³C metabolic flux analysis

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PLANT JOURNAL 2011年第3期67卷 513-525页

作者： Hay, Jordan Schwender, Joerg Brookhaven Natl Lab Dept Biol Upton NY 11973 USA

Plant oils are an important renewable resource, and seed oil content is a key agronomical trait that is in part controlled by the metabolic processes within developing seeds. A large-scale model of cellular metabolism in developing embryos of Brassica napus (bna572) was used to predict biomass formation and to analyze metabolic steady states by flux variability analysis under different physiological conditions. Predicted flux patterns are highly correlated with results from prior C-13 metabolic flux analysis of B. napus developing embryos. Minor differences from the experimental results arose because bna572 always selected only one sugar and one nitrogen source from the available alternatives, and failed to predict the use of the oxidative pentose phosphate pathway. Flux variability, indicative of alternative optimal solutions, revealed alternative pathways that can provide pyruvate and NADPH to plastidic fatty acid synthesis. The nutritional values of different medium substrates were compared based on the overall carbon conversion efficiency (CCE) for the biosynthesis of biomass. Although bna572 has a functional nitrogen assimilation pathway via glutamate synthase, the simulations predict an unexpected role of glycine decarboxylase operating in the direction of NH4+ assimilation. Analysis of the light-dependent improvement of carbon economy predicted two metabolic phases. At very low light levels small reductions in CO2 efflux can be attributed to enzymes of the tricarboxylic acid cycle (oxoglutarate dehydrogenase, isocitrate dehydrogenase) and glycine decarboxylase. At higher light levels relevant to the C-13 flux studies, ribulose-1,5-bisphosphate carboxylase activity is predicted to account fully for the light-dependent changes in carbon balance.

关键词： flux balance analysis carbon partitioning constraint-based model linear program stoichiometric model

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Market clearing engine with unbundled reserves

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INTERNATIONAL TRANSACTIONS ON ELECTRICAL ENERGY SYSTEMS 2015年第2期25卷 288-303页

作者： Chattopadhyay, Deb Gan, Deqiang Univ Queensland Brisbane Qld Australia Zhejiang Univ Sch Elect Engn Hangzhou 310003 Zhejiang Peoples R China

Electricity markets have evolved over the years leading to sophisticated spot markets for contingency reserve in operation in a number of countries. This paper proposes a security-constrained optimal power flow based co-optimization of energy and reserve services. The proposed paradigm improves on the existing market clearing engines because it enables an endogenous determination of reserve requirements, allocation of reserve to generators and unbundled energy/reserve prices, to accurately reflect the impact of underlying line/generator contingencies. We highlight the salient aspects of the proposed formulation using a set of numerical examples. Copyright (c) 2013 John Wiley & Sons, Ltd.

关键词： dispatch linear program ancillary services pricing security analysis

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A type of biased consensus-based distributed neural network for path planning

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NONlinear DYNAMICS 2017年第3期89卷 1803-1815页

作者： Zhang, Yinyan Li, Shuai Guo, Hongliang Hong Kong Polytech Univ Dept Comp Kowloon Hong Kong Peoples R China Univ Elect Sci & Technol China Sch Automat Engn Ctr Robot Chengdu Sichuan Peoples R China

In this paper, a unified scheme is proposed for solving the classical shortest path problem and the generalized shortest path problem, which are highly nonlinear. Particularly, the generalized shortest path problem is more complex than the classical shortest path problem since it requires finding a shortest path among the paths from a vertex to all the feasible destination vertices. Different from existing results, inspired by the optimality principle of Bellman's dynamic programming, we formulate the two types of shortest path problems as linear programs with the decision variables denoting the lengths of possible paths. Then, biased consensus neural networks are adopted to solve the corresponding linear programs in an efficient and distributed manner. Theoretical analysis guarantees the performance of the proposed scheme. In addition, two illustrative examples are presented to validate the efficacy of the proposed scheme and the theoretical results. Moreover, an application to mobile robot navigation in a maze further substantiates the efficacy of the proposed scheme.

关键词： Biased consensus neural network Nonlinear equation linear program Path planning Shortest path

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EXACT REGULARIZATION OF CONVEX programS

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SIAM JOURNAL ON OPTIMIZATION 2008年第4期18卷 1326-1350页

作者： Friedlander, Michael P. Tseng, Paul Univ British Columbia Dept Comp Sci Vancouver BC V6T 1Z4 Canada Univ Washington Dept Math Seattle WA 98195 USA

The regularization of a convex program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. For a general convex program, we show that the regularization is exact if and only if a certain selection problem has a Lagrange multiplier. Moreover, the regularization parameter threshold is inversely related to the Lagrange multiplier. We use this result to generalize an exact regularization result of Ferris and Mangasarian [Appl. Math. Optim., 23 (1991), pp. 266-273] involving a linearized selection problem. We also use it to derive necessary and sufficient conditions for exact penalization, similar to those obtained by Bertsekas [Math. programming, 9 (1975), pp. 87-99] and by Bertsekas, Nedic, and Ozdaglar [Convex Analysis and Optimization, Athena Scientific, Belmont, MA, 2003]. When the regularization is not exact, we derive error bounds on the distance from the regularized solution to the original solution set. We also show that existence of a "weak sharp minimum" is in some sense close to being necessary for exact regularization. We illustrate the main result with numerical experiments on the l(1) regularization of benchmark (degenerate) linear programs and semidefinite/second-order cone programs. The experiments demonstrate the usefulness of l(1) regularization in finding sparse solutions.

关键词： convex program conic program linear program regularization exact penalization Lagrange multiplier degeneracy sparse solutions interior-point algorithms

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A SYMMETRIC REDUCTION OF THE NT DIRECTION

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SIAM JOURNAL ON OPTIMIZATION 2014年第2期24卷 714-732页

作者： Jarre, Florian Hergenroeder, Chantal Univ Dusseldorf Math Inst D-40225 Dusseldorf Germany

A stable symmetrization of the linear systems arising in interior-point methods for solving linear programs is introduced. A comparison of the condition numbers of the resulting interior-point linear systems with other commonly used approaches indicates that the new approach may be best suitable for an iterative solution. It is shown that there is a natural generalization of this symmetrization to the Nesterov-Todd (NT) search direction for solving semidefinite programs. The generalization includes a novel pivoting strategy to minimize the norm of the right and side and heavily relies on the symmetry properties of the NT direction. The search directions generated by iterative solvers typically have fairly low relative accuracy. Nevertheless, in some preliminary numerical examples, a suitably adapted interior-point approach results in a rather small number of outer iterations.

关键词： linear program semidefinite program condition number quasi-minimal residual iteration interior-point algorithm

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Network duality for multiuser MIMO beamforming networks and applications

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IEEE TRANSACTIONS ON COMMUNICATIONS 2007年第3期55卷 618-630页

作者： Song, Bongyong Cruz, Rene L. Rao, Bhaskar D. Univ Calif San Diego Dept Elect & Comp Engn La Jolla CA 92093 USA

For any multiple-input multiple-output (MIMO) network with linear beamformers, it is shown that there exists a dual network that attains the same signal-to-interference-plus-noise ratio. (SINR) performance. This network duality concept is a generalization of the virtual uplink concept investigated by Rashid-Farrokhi et al.. We first develop the dual relation using a generic duality theory in linear programming. This approach naturally leads to the construction of a dual network, and provides machinery for handling a generalized cost function. We then consider the joint MIMO beamforming and power control problem with individual SINR constraints. We apply the network duality to this problem and propose a high-performance iterative algorithm which, compared with past centralized approaches, has improved convergence behavior. Finally, we propose a simpler distributed version of the algorithm tailored for a cellular downlink. This algorithm can be readily implemented in a distributed fashion, since it does not require information exchange between base stations. The algorithm achieves performance close to that of centralized ones, and outperforms other decentralized approaches available in the literature.

关键词： beamforming duality linear program multiple-input multiple-output (MIMO) power control

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Algorithms for art gallery illumination

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JOURNAL OF GLOBAL OPTIMIZATION 2017年第1期68卷 23-45页

作者： Ernestus, Maximilian Friedrichs, Stephan Hemmer, Michael Kokemueller, Jan Kroeller, Alexander Moeini, Mahdi Schmidt, Christiane TU Braunschweig Algorithms Grp IBR Muhlenpfordtstr 23 D-38106 Braunschweig Germany Max Planck Inst Informat Saarbrucken Germany Saarbrucken Grad Sch Comp Sci Saarbrucken Germany Tech Univ Kaiserslautern Chair Business Informat Syst & Operat Res BISOR Postfach 3049Erwin Schrodinger Str D-67653 Kaiserslautern Germany Linkoping Univ Commun & Transport Syst ITN Linkoping Sweden

The art gallery problem (AGP) is one of the classical problems in computational geometry. It asks for the minimum number of guards required to achieve visibility coverage of a given polygon. The AGP is well-known to be NP-hard even in restricted cases. In this paper, we consider the AGP with fading (AGPF): A polygonal region is to be illuminated with light sources such that every point is illuminated with at least a global threshold, light intensity decreases over distance, and we seek to minimize the total energy consumption. Choosing fading exponents of zero, one, and two are equivalent to the AGP, laser scanner applications, and natural light, respectively. We present complexity results as well as a negative solvability result. Still, we propose two practical algorithms for AGPF with fixed light positions (e.g. vertex guards) independent of the fading exponent, which we demonstrate to work well in practice. One is based on a discrete approximation, the other on non-linear programming by means of simplex-partitioning strategies. The former approach yields a fully polynomial-time approximation scheme for the AGPF with fixed light positions. The latter approach obtains better results in our experimental evaluation.

关键词： Art gallery problem Fading Computational geometry linear program Non-linear program Lipschitz function Algorithm engineering

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Preference relation approach for obtaining OWA operators weights

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INTERNATIONAL JOURNAL OF APPROXIMATE REASONING 2008年第2期47卷 166-178页

作者： Ahn, Byeong Seok Chung Ang Univ Coll Business Adm Seoul 156756 South Korea

Actual result of aggregation performed by an ordered weighted averaging (OWA) operator heavily depends upon the weighting vector used. A number of approaches for obtaining the associated weights have been suggested in the academic literature. In this paper, we present a method for determining the OWA weights when (1) the preferences of some subset of alternatives over other subset of alternatives are specified in a holistic manner across all the criteria, and (2) the consequences (criteria values) are specified in one of three different formats: precise numerical values, intervals and fuzzy numbers. The OWA weights are to be estimated in the direction of minimizing deviations from the OWA weights implied by the preference relations, thus as consistent as possible with a priori preference relations. (c) 2007 Elsevier Inc. All rights reserved.

关键词： OWA weights preference relations linear program

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