The main aim of this work is to provide some basis for the development of interior point algorithms to minimize piecewiselinear objective functions. Specifically, we study a piecewiselinear separable and convex obje...
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The main aim of this work is to provide some basis for the development of interior point algorithms to minimize piecewiselinear objective functions. Specifically, we study a piecewiselinear separable and convex objective function, subject to linear constraints. The available methods in the literature for this class of problem are of the Simplex type, except for specific cases, such as linear fitting in the sense of L-1-norm. A common practice for the resolution of piecewiselinear programs consists of transforming them into equivalent linear programs and exploring their structure. This strategy is suitable for simplex-type methods, but inadequate for interior point methods. We show how to extend known interior point methods devised for linearprogramming to piecewise linear programming without resorting to equivalent linear programs. We also show that the generated interior points for the original piecewiselinear program are not interior points for the equivalent linear program. Finally, some computational experiments are presented.
Network piecewise linear programming is a useful model in operations research. It could be solved as network linearprogramming through a reformulation approach which greatly increases the number of variables. This pa...
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Network piecewise linear programming is a useful model in operations research. It could be solved as network linearprogramming through a reformulation approach which greatly increases the number of variables. This paper describes a direct simplex algorithm and its implementation for network piecewise linear programming. By allowing nonbasic, variables to take breakpoint values and using nominal costs at breakpoints of the objective function, this method keeps the number of variables in the original level and simplifies the computation of the reduced cost. This is important for applications in which the number of breakpoints is large;for instance, the stochastic network flow problem and the piecewiselinearized convex network program. The method takes good advantage of tree data structures and combines ratio test with the computation of reduced costs. Computational experiment is conduced on the 40 benchmark network programs of Klingman et al. by replacing the linear costs with piecewiselinear costs. The result of the test shows that solving a network problem with piecewiselinear cost is not much harder than solving a linear problem on the same network.
The paper is a manifestation of the fundamental importance of the linear program with linear complementarity constraints (LPCC) in disjunctive and hierarchical programming as well as in some novel paradigms of mathema...
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The paper is a manifestation of the fundamental importance of the linear program with linear complementarity constraints (LPCC) in disjunctive and hierarchical programming as well as in some novel paradigms of mathematical programming. In addition to providing a unified framework for bilevel and inverse linear optimization, nonconvex piecewise linear programming, indefinite quadratic programs, quantile minimization, and a"" (0) minimization, the LPCC provides a gateway to a mathematical program with equilibrium constraints, which itself is an important class of constrained optimization problems that has broad applications. We describe several approaches for the global resolution of the LPCC, including a logical Benders approach that can be applied to problems that may be infeasible or unbounded.
Economic model predictive control (EMPC) is an effective control strategy for wastewater treatment plants (WWTPs), which are crucial for preventing water pollution and improving the water quality. However, industrial ...
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Economic model predictive control (EMPC) is an effective control strategy for wastewater treatment plants (WWTPs), which are crucial for preventing water pollution and improving the water quality. However, industrial processes typically involve large-scale nonlinear and strongly coupled systems, and it is computationally expensive to solve nonlinear optimization problems based on nonlinear prediction models such as EMPC. In this study, to facilitate the application of EMPC to large-scale nonlinear systems such as WWTPs, the lattice trajectory piecewiselinear (PWL) model is used to approximate the nonlinear system with a predefined error bound. The resulting optimization problem can be expressed as a continuous PWL programming problem if the cost function of the EMPC is linear. Therefore, an iterative descent algorithm is proposed to transform the PWL programming problem into a series of linearprogramming problems. The stability of the nonlinear system operating under EMPC is analyzed. The EMPC scheme based on the lattice trajectory PWL model is applied to a WWTP benchmark problem, in which the state is 78-dimensional and the EMPC cost is linear. The proposed strategy outperforms the EMPC schemes based on the nonlinear prediction model and trajectory PWL prediction model. Overall, the EMPC scheme based on the lattice trajectory PWL prediction model can improve the computational efficiency of optimization problems while ensuring control performance.(c) 2023 Elsevier Ltd. All rights reserved.
Optimizing scheduling is an effective way to improve the profit of refineries;it usually requires accurate models to describe the complex and nonlinear refining processes. However, conventional nonlinear models will r...
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Optimizing scheduling is an effective way to improve the profit of refineries;it usually requires accurate models to describe the complex and nonlinear refining processes. However, conventional nonlinear models will result in a complex mixed integer nonlinearprogramming (MINLP) problem for scheduling. This paper presents a piecewiselinear (PWL) modeling approach, which can describe global nonlinearity with locally linear functions, to refinery scheduling. Specifically, a high level canonical PWL representation is adopted to give a simple yet effective partition of the domain of decision variables. Furthermore, a unified partitioning strategy is proposed to model multiple response functions defined on the same domain. Based on the proposed PWL partitioning and modeling strategy, the original MINLP can be replaced by mixed integer linearprogramming (MILP), which can be readily solved using standard optimization algorithms. The effectiveness of the proposed strategy is demonstrated by a case study originated from a refinery in China. (C) 2015 Elsevier Ltd. All rights reserved.
The dual of an optimization problem with piecewiselinear, convex, and continous objective function and linear restrictions is investigated. It is shown that the dual problem is of the same type as the primal and can ...
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The dual of an optimization problem with piecewiselinear, convex, and continous objective function and linear restrictions is investigated. It is shown that the dual problem is of the same type as the primal and can be solved by the same methods as were proposed in a previous work of the author. Experiences arising from the application of different variants of these methods to solve a number of examples, where the primal as well as the dual version was employed, are reported.
Minimum cost network flow problems with a piecewiselinear convex cost function are used to model various optimization problems. They are also used extensively to approximate nonlinear cost functions which may otherwi...
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