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A new mathematical program with complementarity constraints for optimal localization of pressure reducing valves in water distribution systems

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APPLIED WATER SCIENCE 2021年第9期11卷 152-152页

作者： Pham Duc Dai Thuyloi Univ 175 Tay Son Hanoi Vietnam

Water loss reduction in water distribution systems (WDSs) is a challenging task for water utilities worldwide. One of the most reliable and cost-effective ways to reduce water loss is to properly regulate operational pressure of the system through optimizing pressure reducing valve (PRV) placements. This well-known engineering problem can be casted into a mixed-integer nonlinear program (MINLP) where binary variables are introduced to represent positions of PRVs. Many works in the literature applied heuristic algorithms to address the optimization problem. In this paper, at first, we proposed a new optimization model and reformulated it as the mathematical program with complementarity constraints (MPCCs). It is due to the fact that the stationary point of the MPCCs is likely to be trapped into bad local solutions, a soft heuristic method is then proposed to determine the MINLP local solution in each iteration before a stationary point of the MPCCs is reached. This method not only enhances the quality of MINLP solution, but also decreases computation time for solving the MPCCs. The newly formulated MPCCs is applied to determine optimal localization of PRVs for two WDS benchmarks and a real-world WDS in Vietnam. The results are compared with others in the literature demonstrating that using our new optimization model, better and more reliable MINLP solution can be found for large scale WDSs.

关键词： Water distribution system Pressure management Leakage reduction Epanet Mixed integer nonlinear optimization mathematical program with complementarity constraints

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Partial augmented Lagrangian method for non-Lipschitz mathematical programs with complementarity constraints

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JOURNAL OF GLOBAL OPTIMIZATION 2025年第2期92卷 1-35页

作者： Li, Gao-Xi Yang, Xin-Min Long, Xian-Jun Chongqing Technol & Business Univ Sch Math & Stat Chongqing 400067 Peoples R China Chongqing Normal Univ Sch Math Sci Chongqing 401331 Peoples R China Chongqing Technol & Business Univ Chongqing Key Lab Stat Intelligent Comp & Monitori Chongqing 400067 Peoples R China

In recent years, mathematical programs with complementarity constraints (MPCC) and a non-Lipschitz objective function have been introduced and are now more prevalent than locally Lipschitz MPCC. This paper proposes a smoothing partial augmented Lagrangian (SPAL) method to tackle this problem. However, due to the disruption of the complementary structure's integrity by this method, proving its convergence becomes exceptionally challenging. We have achieved global convergence of the SPAL method. Specifically, we demonstrate that the accumulation point of the sequence generated by the SPAL method can be a strongly stationary point under the Mangasarian-Fromovitz qualification (MPCC-MFQ) and the boundedness of the multiplier corresponding to the orthogonal constraint. Moreover, if the aforementioned multiplier is unbounded, the accumulation point can be a Clarke stationary point under MPCC-MFQ and a suitable assumption. Numerical experiments indicate that the SPAL method surpasses existing methods in terms of the quality of accumulation points and running times.

关键词： mathematical program with complementarity constraints Non-Lipschitz continuity Qualification Augmented Lagrangian method

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DATA-DRIVEN DISTRIBUTIONALLY ROBUST MULTIPRODUCT PRICING PROBLEMS UNDER PURE CHARACTERISTICS DEMAND MODELS

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SIAM JOURNAL ON OPTIMIZATION 2024年第3期34卷 2917-2942页

作者： Jiang, Jie Sun, Hailin Chen, Xiaojun Chongqing Univ Coll Math & Stat Chongqing 400044 Peoples R China Nanjing Normal Univ Sch Math Sci Jiangsu Int Joint Lab BDMCA Key Lab NSLSCSMinist Educ Nanjing 210023 Peoples R China Hong Kong Polytech Univ Dept Appl Math Hong Kong Peoples R China

This paper considers a multiproduct pricing problem under pure characteristics demand models when the probability distribution of the random parameter in the problem is uncertain. We formulate this problem as a distributionally robust optimization (DRO) problem based on a constructive approach to estimating pure characteristics demand models with pricing by Pang, Su, and Lee. In this model, the consumers' purchase decision is to maximize their utility. We show that the DRO problem is well-defined, and the objective function is upper semicontinuous by using an equivalent hierarchical form. We also use the data-driven approach to analyze the DRO problem when the ambiguity set, i.e., a set of probability distributions that contains some exact information of the underlying probability distribution, is given by a general moment-based case. We give convergence results as the data size tends to infinity and analyze the quantitative statistical robustness in view of the possible contamination of driven data. Furthermore, we use the Lagrange duality to reformulate the DRO problem as a mathematical program with complementarity constraints, and give a numerical procedure for finding a global solution of the DRO problem under certain specific settings. Finally, we report numerical results that validate the effectiveness and scalability of our approach for the distributionally robust multiproduct pricing problem.

关键词： pure characteristics demand model stochastic optimization distributional robustness data-driven mathematical program with complementarity constraints

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A New Augmented Lagrangian Method for MPCCs-Theoretical and Numerical Comparison with Existing Augmented Lagrangian Methods

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MATHEMATICS OF OPERATIONS RESEARCH 2022年第2期47卷 1229-1246页

作者： Guo, Lei Deng, Zhibin East China Univ Sci & Technol Sch Business Shanghai 200237 Peoples R China Univ Chinese Acad Sci Chinese Acad Sci Sch Econ & Management Key Lab Big Data Min & Knowledge Management Beijing 100190 Peoples R China

We propose a new augmented Lagrangian (AL) method for solving the mathematical program with complementarity constraints (MPCC), where the complementarity constraints are left out of the AL function and treated directly. Two observations motivate us to propose this method: The AL subproblems are closer to the original problem in terms of the constraint structure;and the AL subproblems can be solved efficiently by a nonmonotone projected gradient method, in which we have closed-form solutions at each iteration. The former property helps us show that the proposed method converges globally to an M-stationary (better than C-stationary) point under MPCC relaxed constant positive linear dependence condition. Theoretical comparison with existing AL methods demonstrates that the proposed method is superior in terms of the quality of accumulation points and the strength of assumptions. Numerical comparison, based on problems in MacMPEC, validates the theoretical results.

关键词： mathematical program with complementarity constraints augmented Lagrangian method nonmonotone projected gradient method M-stationarity

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mathematical programs with complementarity constraints and a non-Lipschitz objective: optimality and approximation

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mathematical programMING 2021年第1-2期185卷 455-485页

作者： Guo, Lei Chen, Xiaojun East China Univ Sci & Technol Sch Business Shanghai 200237 Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China

We consider a class of mathematical programs with complementarity constraints (MPCC) where the objective function involves a non-Lipschitz sparsity-inducing term. Due to the existence of the non-Lipschitz term, existing constraint qualifications for locally Lipschitz MPCC cannot ensure that necessary optimality conditions hold at a local minimizer. In this paper, we present necessary optimality conditions and MPCC-tailored qualifications for the non-Lipschitz MPCC. The proposed qualifications are related to the constraints and the non-Lipschitz term, which ensure that local minimizers satisfy these necessary optimality conditions. Moreover, we present an approximation method for solving the non-Lipschitz MPCC and establish its convergence. Finally, we use numerical examples of sparse solutions of linear complementarity problems and the second-best road pricing problem in transportation science to illustrate the effectiveness of our approximation method for solving the non-Lipschitz MPCC.

关键词： mathematical program with complementarity constraints Non-Lipschitz continuity Sparse solution Optimality condition Approximation method

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REFORMULATION OF THE M-STATIONARITY CONDITIONS AS A SYSTEM OF DISCONTINUOUS EQUATIONS AND ITS SOLUTION BY A SEMISMOOTH NEWTON METHOD

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SIAM JOURNAL ON OPTIMIZATION 2021年第2期31卷 1459-1488页

作者： Harder, Felix Mehlitz, Patrick Wachsmuth, Gerd Brandenburg Tech Univ Cottbus Senftenberg Inst Math D-03046 Cottbus Germany

We show that the Mordukhovich-stationarity system associated with a mathematical program with complementarity constraints (MPCC) can be equivalently written as a system of discontinuous equations which can be tackled with a semismooth Newton method. It will be demonstrated that the resulting algorithm can be interpreted as an active set strategy for MPCCs. Local fast convergence of the method is guaranteed under validity of an MPCC-tailored version of LICQ and a suitable strong second-order condition. In case of linear-quadratic MPCCs, the LICQ-type constraint qualification can be replaced by a weaker condition which depends on the underlying multipliers. We discuss a suitable globalization strategy for our method. Some numerical results are presented in order to illustrate our theoretical findings.

关键词： active set method mathematical program with complementarity constraints M-stationarity nonlinear M-stationarity function semismooth Newton method

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Integration of optimal cleaning scheduling and control of heat exchanger networks under fouling: MPCC solution

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COMPUTERS & CHEMICAL ENGINEERING 2019年 126卷 128-146页

作者： Santamaria, Federico Lozano Macchietto, Sandro Imperial Coll London Dept Chem Engn South Kensington Campus London SW7 2AZ England

Fouling is a major source of energy inefficiencies that decreases the performance of process units. Fouling mitigation alternatives are required to ensure a sustainable, profitable, and safe operation. In heat exchanger networks, two mitigation alternatives are: flow distribution control, and periodical cleanings. This paper addresses the optimal control and optimal cleaning scheduling, individually and simultaneously, for heat exchangers in refining operations. The problem is formulated as a MINLP, which uses an accurate model of the process, logic disjunctions, and a continuous time representation. To solve it efficiently, it is reformulated as a mathematical program with complementarity constraints (MPCC). The modelling and solution strategies are demonstrated in several case studies, from small to moderately large networks for realistic applications. The formulation is versatile, and large network problems are solved in a reasonable computational time. The integration of optimal scheduling and control decreases the operational cost substantially relative to independent mitigation alternatives. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Heat exchanger networks Energy integration Optimal operation Optimal control Optimal cleaning scheduling Crude oil fouling mathematical program with complementarity constraints

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THE STANDARD PESSIMISTIC BILEVEL PROBLEM

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SIAM JOURNAL ON OPTIMIZATION 2019年第2期29卷 1634-1656页

作者： Lampariello, Lorenzo Sagratella, Simone Stein, Oliver Roma Tre Univ Dept Business Studies Rome Italy Sapienza Univ Rome Dept Comp Control & Management Engn Antonio Ruber Rome Italy Karlsruhe Inst Technol Inst Operat Res Karlsruhe Germany

Pessimistic bilevel optimization problems, as do optimistic ones, possess a structure involving three interrelated optimization problems. Moreover, their finite infima are only attained under strong conditions. We address these difficulties within a framework of moderate assumptions and a perturbation approach which allows us to approximate such finite infima arbitrarily well by minimal values of a sequence of solvable single-level problems. To this end, as has already been done for optimistic problems, for the first time in the literature we introduce the standard version of the pessimistic bilevel problem. For its algorithmic treatment, we reformulate it as a standard optimistic bilevel program with a two follower Nash game in the lower level. The latter lower-level game, in turn, is replaced by its Karush-Kuhn-Tucker conditions, resulting in a single-level mathematical program with complementarity constraints. We prove that the perturbed pessimistic bilevel problem, its standard version, the two follower game, as well as the mathematical program with complementarity constraints are equivalent with respect to their global minimal points. We study the more intricate connections between their local minimal points in detail. As an illustration, we numerically solve a regulator problem from economics for different values of the perturbation parameters.

关键词： pessimistic bilevel programming standard optimistic bilevel problem generalized Nash equilibrium problem mathematical program with complementarity constraints

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Production scheduling and linear MPC: Complete integration via complementarity conditions

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COMPUTERS & CHEMICAL ENGINEERING 2019年 125卷 287-305页

作者： Simkoff, Jodie M. Baldea, Michael Univ Texas Austin McKetta Dept Chem Engn 200 East Dean Keeton StStop C0400 Austin TX 78712 USA

Competitive global market conditions and the availability of real-time pricing data call for agile and flexible operations in the chemical process industries. As the operating paradigm shifts, process scheduling decisions may consider a shorter time scale, and interactions with the process control layer become significant. The integration of these two decision-making layers is an active area of research. In this work, we propose a novel framework for the integration of production scheduling and linear model predictive control. Our approach explicitly represents the closed-loop response of the system by embedding the MKT conditions of the controller in a nonlinear programming formulation of the optimal scheduling problem. We present two case studies which demonstrate that control-informed scheduling leads to superior realized performance compared with a hierarchical, sequential decision-making structure. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Optimal scheduling Process control MPC Integrated scheduling and control mathematical program with complementarity constraints

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Solution of bilevel optimization problems using the KKT approach

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OPTIMIZATION 2019年第8期68卷 1471-1489页

作者： Dempe, S. Franke, S. TU Bergakad Freiberg Freiberg Germany

Using the Karush-Kuhn-Tucker conditions for the convex lower level problem, the bilevel optimization problem is transformed into a single-level optimization problem (a mathematical program with complementarity constraints). A regularization approach for the latter problem is formulated which can be used to solve the bilevel optimization problem. This is verified if global or local optimal solutions of the auxiliary problems are computed. Stationary solutions of the auxiliary problems converge to C-stationary solutions of the mathematical program with complementarity constraints.

关键词： Bilevel optimization Karush-Kuhn-Tucker transformation mathematical program with complementarity constraints

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