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Piece-wise linear functions-based model predictive control of large-scale sewage systems

IET CONTROL THEORY AND APPLICATIONS 2010年第9期4卷 1581-1593页

作者： Ocampo-Martinez, C. Puig, V. CSIC UPC Inst Robot & Informat Ind Barcelona 08028 Spain

In this study, model predictive control (MPC) of large-scale sewage systems is addressed, considering several inherent continuous/discrete phenomena (overflows in sewers and tanks) and elements (weirs) in the system. This fact results in distinct behaviour depending on the dynamic state (flow/volume) of the network. These behaviours cannot be neglected nor can be modelled by a pure linear representation. In order to take into account these phenomena and elements in the design of the control strategy, a modelling approach based on piece-wise linear functions (PWLF) is proposed and compared against a hybrid modelling approach previously suggested by the authors. Control performance results and associated computation times of the closed-loop scheme considering both modelling approaches are compared by using a real case study based on the Barcelona sewer network. Results have shown an important reduction in the computation time when the PWLF-based model is used, with an acceptable suboptimality level in the closed-loop system performance.

关键词： PWLF modelling approach Barcelona sewer network piecewise linear functions model predictive control closed-loop scheme Control system analysis and synthesis methods sewage treatment predictive control closed loop systems piecewise linear techniques Optimal control Natural resources and environmental control large-scale sewage systems

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Structure and Weak Sharp Minimum of the Pareto Solution Set for piecewise linear Multiobjective Optimization

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2010年第1期147卷 113-124页

作者： Yang, X. Q. Yen, N. D. Hong Kong Polytech Univ Dept Appl Math Hong Kong Hong Kong Peoples R China Vietnamese Acad Sci & Technol Inst Math Hanoi Vietnam

In this paper, the Pareto solution set of a piecewise linear multiobjective optimization problem in a normed space is shown to be the union of finitely many semiclosed polyhedra. If the problem is further assumed to b... 详细信息

关键词： piecewise linear functions Multiobjective optimization problems Pareto solution sets Global weak sharp minimum Image space analysis

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An algorithm for linearizing convex extremal problems

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SBORNIK MATHEMATICS 2010年第3-4期201卷 471-492页

作者： Gorskaya, E. S. Moscow MV Lomonosov State Univ Fac Mech & Math Moscow Russia

This paper suggests a method of approximating the solution of minimization problems for convex functions of several variables under convex constraints is suggested. The main idea of this approach is the approximation of a convex function by a piecewise linear function, which results in replacing the problem of convex programming by a linear programming problem. To carry out such an approximation, the epigraph of a convex function is approximated by the projection of a polytope of greater dimension. In the first part of the paper, the problem is considered for functions of one variable. In this case, an algorithm for approximating the epigraph of a convex function by a polygon is presented, it is shown that this algorithm is optimal with respect to the number of vertices of the polygon, and exact bounds for this number are obtained. After this, using an induction procedure, the algorithm is generalized to certain classes of functions of several variables. Applying the suggested method, polynomial algorithms for an approximate calculation of the L(p)-norm of a matrix and of the minimum of the entropy function on a polytope are obtained.

关键词： convex problems piecewise linear functions approximation of functions evaluation of operator norms

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piecewise linear output measures in DEA (third revision)

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2009年第1期197卷 312-319页

作者： Cook, Wade D. Zhu, Joe York Univ Schulich Sch Business N York ON M3J 1P3 Canada Worcester Polytech Inst Dept Management Worcester MA 01609 USA

It is assumed in the standard DEA model that the aggregate output (input) is a pure linear function of each output (input). This means, for example, that if DMU j(1) generates twice as much of an output as does another DMUj(2),, then the former is credited with having created twice as much value. In many situations, however, linear pricing (mu(r)y(rj)) may not adequately reflect differences in value created from one DMU to another. In this paper, a generalization of the DEA methodology is presented that incorporates piecewise linear functions of factors. We deal specifically with those situations where for certain outputs in an input-oriented model, the weight function f(y(rj)) is described by either a non-increasing or non-decreasing set of multipliers for larger amounts of the factor. We refer to such a variable r as exhibiting diminishing marginal value (DMV) or increasing marginal value (IMV). The DMV/IMV phenomenon is common in many for-profit applications. For example, in the case that y(rj) is the amount of a consumer product r generated by DMU j, and mu(r) is the price of that product, it may well be that the market will force lower prices if greater amounts of that product are generated;discounts automatically lead to this DMV situation. Such a phenomenon can arise as well in not-for-profit settings, and we examine such a situation based on earlier work by Cook et al. [Cook, W.D., Roll, Y., Kazakov, A., 1990. A DEA model for measuring the relative efficiency of highway maintenance patrols. INFOR 28 (2), 113-124]. (C) 2008 Elsevier B.V. All rights reserved.

关键词： DEA Diminishing marginal value piecewise linear functions Assurance region

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A simplex algorithm for piecewise-linear fractional programming problems

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2007年第2期178卷 343-358页

作者： Pandey, Pooja Punnen, Abraham P. Simon Fraser Univ Dept Math Surrey BC V3T 5X3 Canada Univ New Brunswick Dept Math Sci St John NB E2L 4L5 Canada

Generalizations of the well-known simplex method for linear programming are available to solve the piecewise linear programming problem and the linear fractional programming problem. In this paper we consider a further generalization of the simplex method to solve piecewise linear fractional programming problems unifying the simplex method for linear programs, piecewise linear programs, and the linear fractional programs. Computational results are presented to obtain further insights into the behavior of the algorithm on random test problems. (c) 2006 Elsevier B.V. All rights reserved.

关键词： fractional programming simplex method piecewise linear functions

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Nonlinear uncertainty model of a magnetic suspension system

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MATHEMATICAL AND COMPUTER MODELLING 2004年第9-10期40卷 1075-1087页

作者： Agamennoni, O Skrjanc, I Lepetic, M Chiacchiarini, H Matko, D Univ Nacl Sur Dept Ingn Elect & Comp RA-8000 Bahia Blanca Buenos Aires Argentina CIC Buenos Aires DF Argentina Univ Ljubljana Fac Elect Engn Ljubljana Slovenia Consejo Nacl Invest Cient & Tecn RA-1033 Buenos Aires DF Argentina

The nonlinear identification of a nominal model as well as the uncertainty bounds of a magnetic suspension system is developed. This system has a nonsymmetric dynamic behavior;it has an undershoot but not an overshoot. The proposed model structure is a cascade of a global linear fuzzy dynamic block followed by a piecewise linear function. This model structure allows a proper identification of the system dynamic and a tight description of the uncertainties. (C) 2004 Elsevier Ltd. All rights reserved.

关键词： nonlinear systems identification uncertain systems fuzzy logic piecewise linear functions

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A piecewise linear dual phase-1 algorithm for the simplex method

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2003年第1期26卷 63-81页

作者： Maros, I Univ London Imperial Coll Sci Technol & Med Dept Comp London SW7 2AZ England

A dual phase-1 algorithm for the simplex method that handles all types of variables is presented. In each iteration it maximizes a piecewise linear function of dual infeasibilities in order to make the largest possible step towards dual feasibility with a selected outgoing variable. The algorithm can be viewed as a generalization of traditional phase-1 procedures. It is based on the multiple use of the expensively computed pivot row. By small amount of extra work per iteration, the progress it can make is equivalent to many iterations of the traditional method. While this is its most important feature, it possesses some additional favorable properties, namely, it can be efficient in coping with degeneracy and numerical difficulties. Both theoretical and computational issues are addressed. Some computational experience is also reported which shows that the potentials of the method can materialize on real world problems.

关键词： linear programming dual simplex method phase-1 piecewise linear functions

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Lebesgue constant for the Stromberg wavelet

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JOURNAL OF APPROXIMATION THEORY 2003年第1期122卷 13-23页

作者： Bechler, P Polish Acad Sci Inst Math PL-00950 Warsaw Poland

Let S denote the Stromberg wavelet in L-2(R) and P-s,P-n (s is an element of Z, n is an element of Z boolean OR {infinity}), the orthogonal projection onto the space spanned by the functions 2(r/2) S(2(r)t - m), where r less than or equal to s, m < n + 1 (i.e. P-s,P-n are partial sums for the orthonormal wavelet basis generated by S). We show that the maximum of the norms of the extensions of the operators P-s,P-n onto L-infinity (R) is equal to 2 + (2 - root3)(2). (C) 2003 Elsevier Science (USA). All rights reserved.

关键词： Stromberg wavelet Lebesgue constant best constant piecewise linear functions orthogonal projections

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Multiple objective programming with piecewise linear functions

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Journal of Multi-Criteria Decision Analysis 2000年第6期8卷

作者： Stefan Nickel Margaret M. Wiecek Fachbereich Mathematik Universität Kaiserslautern Germany On sabbatical leave from the Department of Mathematical Sciences Clemson University South Carolina USA.

An approach to generating all efficient solutions for multiple objective programs with piecewise linear objective functions and linear constraints is presented. The approach is based on the decomposition of the feasible set into subsets, referred to as cells, so that the original problem reduces to a series of single objective linear programs and feasibility tests over the cells. The concepts of cell-efficiency and complex-efficiency are introduced and their relationship with efficiency is examined. A generic algorithm for finding efficient solutions for bi-objective piecewise linear programs is proposed. Applications in location theory as well as in worst case analysis are highlighted. Copyright © 1999 John Wiley & Sons, Ltd.

关键词： multiple objective programming piecewise linear functions cell-efficiency complex-efficiency

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On a dual method for a specially structured linear programming problem with application to stochastic programming

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OPTIMIZATION METHODS & SOFTWARE 2002年第3期17卷 445-492页

作者： Fábián, CI Prékopa, A Ruf-Fiedler, O Rutgers State Univ Rutgers Ctr Operat Res RUTCOR Piscataway NJ 08854 USA Eotvos Lorand Univ Dept OR H-1518 Budapest Hungary Swiss Reinsurance Co Life Dept Mythenquai 50 60 CH-8022 Zurich Switzerland

This article revises and improves on a Dual Type Method (DTM), developed by Prekopa. (Prekopa, A. (1990). Dual method for the solution of a one-stage stochastic programming problem with random RHS obeying a discrete probability distribution, ZOR-Methods and Models of Operations Research , 34 , 441-461), in two ways. The first one allows us, in each iteration, to perform the largest step toward the optimum. The second one consists of exploiting the structure of the working basis , which has to be inverted in each iteration, and updating its inverse in product form, as it is usual in case of the standard dual method. The improved method has been implemented. A report on its performance on the solution of some stochastic programming problems is also presented.

关键词： linear programming piecewise linear functions stochastic programming

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