Bilevel linear optimization problems are the linear optimization problems with two sequential decision steps of the leader and the follower. In this paper, we focus on the ambiguity of coefficients of the follower in ...
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Bilevel linear optimization problems are the linear optimization problems with two sequential decision steps of the leader and the follower. In this paper, we focus on the ambiguity of coefficients of the follower in his objective function that hinder the leader from exactly calculating the rational response of the follower. Under the assumption that the follower's possible range of the ambiguous coefficient vector is known as a certain convex polytope, the leader can deduce the possible set of rational responses of the follower. The leader further assumes that the follower's response is the worst-case scenario to his objective function, and then makes a decision according to the maximin criteria. We thus formulate the bilevel linear optimization problem with ambiguous objective function of the follower as a special kind of three-level programming problem. In our formulation, we show that the optimal solution locates on the extreme point and propose a solution method based on the enumeration of possible rational responses of the follower. A numerical example is used to illustrate our proposed computational method.
The power transfer capability (PTC) of AC transmission lines is increased by using series capacitor and shunt reactor compensation. This paper presents a method to calculate the degree of series capacitor and shunt re...
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The power transfer capability (PTC) of AC transmission lines is increased by using series capacitor and shunt reactor compensation. This paper presents a method to calculate the degree of series capacitor and shunt reactor compensation in order to increase the power transfer capability of overhead transmission lines. This method is based on linear programming which takes into consideration line modelling, series and shunt compensation, voltage and reactive power constraints, and power factor before and after compensation. The proposed method has been applied to the existing 500 kV transmission line interconnected between the High-Dam generation station and Cairo load center in Egypt. Copyright (c) 2005 John Wiley & Sons, Ltd.
In many practical linear programming problems, it is often important to know how different optimality criteria (optimal solution, optimal basis, optimal partition, etc.) change under input data perturbations. Our aim ...
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In many practical linear programming problems, it is often important to know how different optimality criteria (optimal solution, optimal basis, optimal partition, etc.) change under input data perturbations. Our aim is to compute tolerances (intervals) for the objective function and the right-hand side coefficients such that these coefficients can independently and simultaneously vary inside their tolerances while preserving the corresponding optimality criterion. We put tolerance analysis in a unified framework that is convenient for algorithmic processing and that is applicable not only in linear programming but for other linear systems as well. We survey the known results (pioneered by R.E. Wendell) and propose an improvement that is optimal in some sense (the resulting tolerances are maximal and they take into account proportionality). We apply our approach to several optimality invariancies: optimal basis, support set and optimal partition invariancy. Thus, the approach is useful not only for simplex method solvers, but for the interior points methods, too. We also discuss time complexity and show that it is NP-hard to determine the maximal tolerances.
It has been shown [8] that numerous interior-point algorithms for linear programming (LP) generate solution sequences that converge to strict complementarity solutions, or interior solutions on the optimal face. In th...
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It has been shown [8] that numerous interior-point algorithms for linear programming (LP) generate solution sequences that converge to strict complementarity solutions, or interior solutions on the optimal face. In this note we further establish a theoretical base for Gay's test (Gay, 1989) to identify the optimal face, and develop a new termination procedure to obtain an exact solution on the optimal face. We also report some numerical results for solving a set of LP test problems, each of which has a highly degenerate and unbounded optimal face.
Basing on the algebraic representation of Petri net models of Ada tasking programs, we have defined a set of linear programming problems to support the static analysis of various queries about tasking behavior, such a...
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ISBN:
(纸本)0818628308
Basing on the algebraic representation of Petri net models of Ada tasking programs, we have defined a set of linear programming problems to support the static analysis of various queries about tasking behavior, such as queries about deadlock existence, possible parallelism, possible occurrence of an event, reachable states, definite occurrence of events, and quantitative questions. The linear programming problems are defined in such a way that their solutions can be used to guide net simulation to confirm the analysis conclusion. The stress of this method is that the enumeration of the entire state space of a Petri net is avoided. The defined problems are valid as long as the underlying net model is a safe net.
Elevator group control dispatches elevators to passengers' calls in a dynamic environment where new calls constantly emerge. At the moment of making a dispatching decision, it is not known when and at which floors...
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ISBN:
(纸本)9789898565402
Elevator group control dispatches elevators to passengers' calls in a dynamic environment where new calls constantly emerge. At the moment of making a dispatching decision, it is not known when and at which floors new passengers will register new calls, what is the number of passengers waiting behind these and existing calls, and what are their destinations. Robust dispatching decisions require that future passenger traffic is forecast based on the realized passenger flow in a building. The problem is that this flow cannot be directly measured. It can, however, be estimated by finding the passenger counts for the origins and destinations of every elevator trip occurring in a building. An elevator trip consists of successive stops in one direction of travel with passengers inside the elevator. We formulate the elevator trip origin-destination matrix estimation problem as a minimum cost network flow problem. We also present a branch-and-bound algorithm for finding all solutions to the problem and study its performance based on numerical experiments.
We provide the first meaningful documentation and analysis of the 'Idiot' crash implemented by Forrest in Clp that aims to obtain an approximate solution to linear programming (LP) problems for warm-starting t...
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We provide the first meaningful documentation and analysis of the 'Idiot' crash implemented by Forrest in Clp that aims to obtain an approximate solution to linear programming (LP) problems for warm-starting the primal simplex method. The underlying algorithm is a penalty method with naive approximate minimization in each iteration. During initial iterations an approach similar to augmented Lagrangian is used. Later the technique corresponds closely to a classical quadratic penalty method. We discuss the extent to which it can be used to obtain fast approximate solutions of LP problems, in particular when applied to linearizations of quadratic assignment problems.
The unicellular microalga Phaeodactylum tricornutum exhibits the ability to accumulate triacylglycerols to a high specific content when nutrients are limited in the culture medium. Therefore, the organism is a promisi...
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The unicellular microalga Phaeodactylum tricornutum exhibits the ability to accumulate triacylglycerols to a high specific content when nutrients are limited in the culture medium. Therefore, the organism is a promising candidate for biodiesel production. Mathematical modeling can substantially contribute to process development and optimization of algae cultivation on different levels. In our work we describe a linear programming approach to model and simulate the growth and storage molecule accumulation of P. tricornutum. The model is based on mass and energy balances and shows that the organism realizes the inherent drive for maximization of energy to biomass conversion and growth. The model predicts that under nutrient limiting conditions both storage carbohydrates and lipids are synthesized simultaneously but at different rates. The model was validated with data gained from batch growth experiments.
The success of deep neural networks and their potential use in many safety-critical applications has motivated research on formal verification of deep neural networks. A fundamental primitive enabling the formal analy...
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The success of deep neural networks and their potential use in many safety-critical applications has motivated research on formal verification of deep neural networks. A fundamental primitive enabling the formal analysis of neural networks is the output range analysis. Existing approaches on output range analysis either focus on some simple activation functions, such as relu,or compute a relaxed result for some activation functions, such as exponential linear unit elu. In this article, we propose an approach to compute the output range for feed-forward deep neural networks via linear programming. The key idea is to encode the activation functions, such as elu and sigmoid, as linear constraints in term of the line between the left and right end-points of the input range and the tangent lines on some special points in the input range. A strategy to partition the network to get a tighter range is presented. The experimental results show that our approach gets a tighter result than RobustVerifier on elu networks and sigmoid networks. Moreover, our approach performs better than (the linear encodings implemented in) Crown on elu networks with alpha =0.5, 1.0$ and sigmoid networks, and better than CNN-Cert and DeepCert on elu networks with alpha = 0.5 or 1.0. For elu networks with alpha = 2.0, our approach can achieve results that are closed to Crown, CNN-Cert, and DeepCert. Finally, we also found that the network partition helps to achieve a tighter result as well as to improve the efficiency for elu networks.
With recent advances in computer hardware and software, more process simulation strategies are now being considered in an equation-oriented mode. This simulation mode relies on an equation solving engine which must de...
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With recent advances in computer hardware and software, more process simulation strategies are now being considered in an equation-oriented mode. This simulation mode relies on an equation solving engine which must deal effectively with nonsmooth and constrained relations which are often enforced as procedures in the sequential modular mode. Also, with complex flowsheet topologies and output specifications due to design or optimization calculations, conditional and nonsmooth relations introduce derivative discontinuities into the problem which frequently cause these solvers to fail. In an earlier paper (Bullard and Biegler, Computers chem Engng 15, 4, 1991) we introduced an iterated linear programming (LP) based approach for solving the constrained simulation problem. The present article proposes two related approaches to handle a general class of nonsmooth problems. The first uses continuous variables while the second also includes binary variables. These approaches are demonstrated on a number of examples, including pipeline network flow problems containing check valves which may either be on or off, flow transitions from laminar to turbulent and equidistribution problems which minimize the absolute value of the approximation error.
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