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A Neurodynamic Approach for Solving robust linear programming under the Polyhedral Uncertainty Set

NEURAL PROCESSING LETTERS 2023年第8期55卷 10669-10683页

作者： Hu, Jin He, Li Peng, Yiheng Zeng, Chunna Chongqing Jiaotong Univ Sch Math & Stat Chongqing 400074 Peoples R China Chongqing Normal Univ Coll Math Sci Chongqing 401331 Peoples R China

This paper converts the robust linear programming under the polyhedral uncertainty set to standard linear programming. After eliminating the parameter uncertainty, we design a projection neural network to address the problem. The equilibrium point of the neural network model is theoretically proved to be stable in the Lyapunov sense and globally convergent to the optimal solution to robust linear programming. Two numerical examples are provided to illustrate the validity and performance of the proposed approach.

关键词： robust linear programming Polyhedral uncertainty set Neurodynamic approach Lyapunov function Global convergence

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Randomized Constraints Consensus for Distributed robust linear programming

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IFAC-PapersOnLine 2017年第1期50卷 4973-4978页

作者： Chamanbaz M. Notarstefano G. Bouffanais R. Arak University of Technology Arak Iran Singapore University of Technology and Design Singapore Department of Engineering Università del Salento Lecce Italy

In this paper we consider a network of processors aiming at cooperatively solving linear programming problems subject to uncertainty. Each node only knows a common cost function and its local uncertain constraint set. We propose a randomized, distributed algorithm working under time-varying, asynchronous and directed communication topology. The algorithm is based on a local computation and communication paradigm. At each communication round, nodes perform two updates: (i) a verification in which they check—in a randomized setup—the robust feasibility (and hence optimality) of the candidate optimal point, and (ii) an optimization step in which they exchange their candidate bases (minimal sets of active constraints) with neighbors and locally solve an optimization problem whose constraint set includes: a sampled constraint violating the candidate optimal point (if it exists), agent's current basis and the collection of neighbor's basis. As main result, we show that if a processor successfully performs the verification step for a sufficient number of communication rounds, it can stop the algorithm since a consensus has been reached. The common solution is—with high confidence—feasible (and hence optimal) for the entire set of uncertainty except a subset having arbitrary small probability measure. We show the effectiveness of the proposed distributed algorithm on a multi-core platform in which the nodes communicate asynchronously. © 2017

关键词： control of large-scale network systems Distributed Optimization Large scale optimization problems Optimization Randomized Algorithms robust linear programming

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Randomized Constraints Consensus for Distributed robust linear programming

Randomized Constraints Consensus for Distributed Robust Line...

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20th World Congress of the International-Federation-of-Automatic-Control (IFAC)

作者： Chamanbaz, Mohammadreza Notarstefano, Giuseppe Bouffanais, Roland Arak Univ Technol Arak Iran Univ Salento Dept Engn Lecce Italy Singapore Univ Technol & Design Singapore Singapore

In this paper we consider a network of processors aiming at cooperatively solving linear programming problems subject to uncertainty. Each node only knows a common cost function and its local uncertain constraint set. We propose a randomized, distributed algorithm working under time-varying, asynchronous and directed communication topology. The algorithm is based on a local computation and communication paradigm. At each communication round, nodes perform two updates: (i) a verification in which they check in a randomized setup the robust feasibility (and hence optimality) of the candidate optimal point, and (ii) an optimization step in which they exchange their candidate bases (minimal sets of active constraints) with neighbors and locally solve an optimization problem whose constraint set includes: a sampled constraint violating the candidate optimal point (if it exists), agent's current basis and the collection of neighbor's basis. As main result, we show that if a processor successfully performs the verification step for a sufficient number of communication rounds, it can stop the algorithm since a consensus has been reached. The common solution is with high confidence feasible (and hence optimal) for the entire set of uncertainty except a subset having arbitrary small probability measure. We show the effectiveness of the proposed distributed algorithm on a multi-core platform in which the nodes communicate asynchronously. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

关键词： Distributed Optimization Randomized Algorithms robust linear programming Optimization and control of large-scale network systems Large scale optimization problems

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Tractable approximate robust geometric programming

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OPTIMIZATION AND ENGINEERING 2008年第2期9卷 95-118页

作者： Hsiung, Kan-Lin Kim, Seung-Jean Boyd, Stephen Stanford Univ Dept Elect Engn Informat Syst Lab Stanford CA 94305 USA

The optimal solution of a geometric program (GP) can be sensitive to variations in the problem data. robust geometric programming can systematically alleviate the sensitivity problem by explicitly incorporating a model of data uncertainty in a GP and optimizing for the worst-case scenario under this model. However, it is not known whether a general robust GP can be reformulated as a tractable optimization problem that interior-point or other algorithms can efficiently solve. In this paper we propose an approximation method that seeks a compromise between solution accuracy and computational efficiency. The method is based on approximating the robust GP as a robust linear program (LP), by replacing each nonlinear constraint function with a piecewise-linear (PWL) convex approximation. With a polyhedral or ellipsoidal description of the uncertain data, the resulting robust LP can be formulated as a standard convex optimization problem that interior-point methods can solve. The drawback of this basic method is that the number of terms in the PWL approximations required to obtain an acceptable approximation error can be very large. To overcome the "curse of dimensionality" that arises in directly approximating the nonlinear constraint functions in the original robust GP, we form a conservative approximation of the original robust GP, which contains only bivariate constraint functions. We show how to find globally optimal PWL approximations of these bivariate constraint functions.

关键词： geometric programming linear programming piecewise-linear function robust geometric programming robust linear programming robust optimization

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A robust optimization methodology for preliminary aircraft design

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ENGINEERING OPTIMIZATION 2016年第5期48卷 883-899页

作者： Prigent, S. Marechal, P. Rondepierre, A. Druot, T. Belleville, M. AIRBUS Future Project Off Toulouse France Univ Toulouse Inst Math Toulouse Toulouse France INSA Toulouse France

This article focuses on a robust optimization of an aircraft preliminary design under operational constraints. According to engineers' know-how, the aircraft preliminary design problem can be modelled as an uncertain optimization problem whose objective (the cost or the fuel consumption) is almost affine, and whose constraints are convex. It is shown that this uncertain optimization problem can be approximated in a conservative manner by an uncertain linear optimization program, which enables the use of the techniques of robust linear programming of Ben-Tal, ElGhaoui, and Nemirovski [robust Optimization, Princeton University Press, 2009]. This methodology is then applied to two real cases of aircraft design and numerical results are presented.

关键词： robust linear programming optimization under uncertainty application to aircraft design

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A unifying framework for duality and modeling in robust linear programs

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OMEGA-INTERNATIONAL JOURNAL OF MANAGEMENT SCIENCE 2013年第6期41卷 984-997页

作者： Soyster, A. L. Murphy, F. H. Northeastern Univ Boston MA 02115 USA Temple Univ Fox Sch Business Philadelphia PA 19122 USA

In this paper, our major theme is a unifying framework for duality in robust linear programming. We show that there are two pair of dual programs allied with a robust linear program;one in which the primal is constructed to be "ultra-conservative" and one in which the primal is constructed to be "ultra-optimistic." Furthermore, as one would expect, if the uncertainly in the primal is row-based, the corresponding uncertainty in the dual is column-based, and vice-versa. Several examples are provided that illustrate the properties of these primal and dual models. A second theme of the paper is about modeling in robust linear programming. We replace the ordinary activity vectors (points) and right-hand sides with well-known geometric objects such as hyper-rectangles, parallel line segments and hyper-spheres. In this manner, imprecision and uncertainty can be explicitly modeled as an inherent characteristic of the model. This is in contrast to the usual approach of using vectors to model activities and/or constraints and then, subsequently, imposing some further constraints in the model to accommodate imprecision and uncertainties. The unifying duality structure is then applied to these models to understand and interpret the marginal prices. The key observation is that the optimal solutions to these dual problems are comprised of two parts: a traditional "centrality" component along with a "robustness" component. (C) 2012 Elsevier Ltd, All rights reserved.

关键词： robust linear programming Duality

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Two-stage robust LP with ellipsoidal right-hand side uncertainty is NP-hard

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OPTIMIZATION LETTERS 2012年第7期6卷 1463-1475页

作者： Minoux, Michel Univ Paris 06 F-75005 Paris France

As shown in previous work, robust linear programming problems featuring polyhedral right-hand side (RHS) uncertainty (a) arise in many practical applications;(b) frequently lead to robust equivalents belonging to the class of strongly NP-hard problems. In the present paper the case of ellipsoidal RHS uncertainty is investigated and similar complexity results are shown to hold even when restricting to simplified specially structured problems related to robust production planning under uncertain customer requirements. The proof is based on a reduction which significantly differs from the one used in the case of polyhedral RHS uncertainty.

关键词： robust optimization robust linear programming Polyhedral uncertainty Ellipsoidal uncertainty Complexity of robust linear programming

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On 2-stage robust LP with RHS uncertainty: complexity results and applications

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JOURNAL OF GLOBAL OPTIMIZATION 2011年第3期49卷 521-537页

作者： Minoux, Michel Univ Paris 06 Paris France

We investigate here the class-denoted R-LP-RHSU-of two-stage robust linear programming problems with right-hand-side uncertainty. Such problems arise in many applications e.g: robust PERT scheduling (with uncertain task durations);robust maximum flow (with uncertain arc capacities);robust network capacity expansion problems;robust inventory management;some robust production planning problems in the context of power production/distribution systems. It is shown that such problems can be formulated as large scale linear programs with associated nonconvex separation subproblem. A formal proof of strong NP-hardness for the general case is then provided, and polynomially solvable subclasses are exhibited. Differences with other previously described robust LP problems (featuring row-wise uncertainty instead of column wise uncertainty) are highlighted.

关键词： robust linear programming Right-hand side uncertainty Complexity of robust linear programming robust network optimization robust production planning robust PERT scheduling robust inventory management

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Farkas' lemma for separable sublinear inequalities without qualifications

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OPTIMIZATION LETTERS 2009年第4期3卷 537-545页

作者： Jeyakumar, V. Li, G. Y. Univ New S Wales Dept Appl Math Sydney NSW 2052 Australia

We show that the celebrated Farkas lemma for linear inequality systems continues to hold for separable sublinear inequality systems. As a consequence, we establish a qualification-free characterization of optimality for separable sublinear programming problems which include classes of robust linear programming problems. We also deduce that the Lagrangian duality always holds for these programming problems without qualifications.

关键词： Generalized Farkas' lemma Separable sublinear inequalities robust linear programming Optimality conditions Duality

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robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L₁-gain and L_∞-gain characterization

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INTERNATIONAL JOURNAL OF robust AND NONlinear CONTROL 2013年第17期23卷 1932-1954页

作者： Briat, Corentin Swiss Fed Inst Technol Zurich ETH Z Dept Biosyst Sci & Engn D BSSE CH-4058 Basel Switzerland KTH ACCESS Linnaeus Ctr Stockholm Sweden

Copositive linear Lyapunov functions are used along with dissipativity theory for stability analysis and control of uncertain linear positive systems. Unlike usual results on linear systems, linear supply rates are employed here for robustness and performance analysis using L-1-gain and L-gain. robust stability analysis is performed using integral linear constraints for which several classes of uncertainties are discussed. The approach is then extended to robust stabilization and performance optimization. The obtained results are expressed in terms of robust linear programming problems that are equivalently turned into finite dimensional ones using Handelman's theorem. Several examples are provided for illustration. Copyright (c) 2012 John Wiley & Sons, Ltd.

关键词： positive linear systems robustness integral linear constraints robust control robust linear programming relaxation

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