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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 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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Constraint qualifications for constrained Lipschitz optimization problems and applications to a MPCC

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2012年第2期75卷 526-542页

作者： Zhang, Shaowu Zhang, Jie Zhang, Liwei Wang, Wei Dalian Univ Technol Inst ORCT Sch Math Sci Dalian 116024 Peoples R China Dalian Univ Technol Fac Elect Informat & Elect Engn Dalian 116024 Peoples R China Liaoning Normal Univ Sch Math Dalian 116029 Peoples R China

Three constraint qualifications (the weak generalized Robinson constraint qualification, the bounded constraint qualification, and the generalized Abadie constraint qualification), which are weaker than the generalized Robinson constraint qualification (GRCQ) given by Yen (1997) [1], are introduced for constrained Lipschitz optimization problems. Relationships between those constraint qualifications and the calmness of the solution mapping are investigated. It is demonstrated that the weak generalized Robinson constraint qualification and the bounded constraint qualification are easily verifiable sufficient conditions for the calmness of the solution mapping, whereas the proposed generalized Abadie constraint qualification, described in terms of graphical derivatives in variational analysis, is weaker than the calmness of the solution mapping. Finally, those constraint qualifications are written for a mathematical program with complementarity constraints (MPCC), and new constraint qualifications ensuring the C-stationary point condition of a MPCC are obtained. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Constrained Lipschitz optimization Constraint qualifications Calmness mathematical program with complementarity constraints

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Arc-length method for frictional contact problems using mathematical programming with complementarity constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2006年第1期131卷 89-113页

作者： Kanno, Y. Martins, J. A. C. Univ Tokyo Dept Math Informat Tokyo Japan Inst Super Tecn Dept Civil & Architectural Engn Lisbon Portugal ICIST Lisbon Portugal

A new formulation as well as a new solution technique is proposed for an equilibrium path-following method in two-dimensional quasistatic frictional contact problems. We consider the Coulomb friction law as well as a geometrical nonlinearity explicitly. Based on a criterion of maximum dissipation of energy, we propose a formulation as a mathematical program with complementarity constraints (MPEC) in order to avoid unloading solutions in which most contact candidate nodes become stuck. A regularization scheme for the MPEC is proposed, which can be solved by using a conventional nonlinear programming approach. The equilibrium paths of various structures are computed in cases such that there exist some limit points and/or infinite number of successive bifurcation points.

关键词： contact problems Coulomb's friction arc-length method mathematical program with complementarity constraints maximum dissipation

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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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Sensitivity analysis for cone-constrained optimization problems under the relaxed constraint qualifications

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MATHEMATICS OF OPERATIONS RESEARCH 2005年第2期30卷 333-353页

作者： Arutyunov, AV Izmailov, AF Peoples Friendship Univ Moscow 117198 Russia Moscow MV Lomonosov State Univ Fac Computat Math & Cybernet Dept Operat Res Moscow 119992 Russia

We present the local sensitivity analysis for cone-constrained optimization problems under the CQ-type conditions significantly weaker than those traditionally used in this context. Our basic sensitivity results are established under the first or second-order sufficient optimality conditions combined with the estimate of the distance to the feasible set of the perturbed problem. We demonstrate how such an estimate can be obtained under the assumptions weaker than Robinson's CQ, and establish the corresponding sensitivity results. Finally, we apply our results to sensitivity analysis and relaxation schemes for mathematical programs with complementarity constraints.

关键词： sensitivity analysis parametric optimization cone-constrained problem constraint qualification abnormal point sufficient optimality condition mathematical program with complementarity constraints

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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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Lower-order penalty methods for mathematical programs with complementarity constraints

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OPTIMIZATION METHODS & SOFTWARE 2004年第6期19卷 693-720页

作者： Yang, XQ Huang, XX Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China Chongqing Normal Univ Dept Math & Comp Sci Chongqing 400047 Peoples R China

In this article, a smooth mathematical program with complementarity constraints (MPCC) is reformulated as a non-smooth constrained optimization problem by using the Fischer-Burmeister function. A lower-order penalty method is applied to transform the resulted constrained optimization problem into unconstrained optimization problems. Lower-order penalty functions may not be locally Lipschitz. However, they require weaker conditions to guarantee an exact penalization property than the classical l(1) penalty functions. We derive optimality conditions for the penalty problems using a smooth approximate variational principle, and establish that the limit point of a sequence of points that satisfy the second-order necessary optimality conditions of penalty problems is a strongly stationary point (hence a B-stationary point) of the original MPCC if the limit point is feasible to MPCC, and a linear independence constraint qualification for MPCC and an upper level strict complementarity condition hold at the limit point. Furthermore, the limit point also satisfies a second-order necessary condition of MPCC. Numerical examples are presented to demonstrate and compare the effectiveness of the proposed methods.

关键词： mathematical program with complementarity constraints penalty function optimality condition B-stationary point convergence

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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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