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Optimal Localization of Pressure Reducing Valves in Water Distribution Systems by a Reformulation Approach

WATER RESOURCES MANAGEMENT 2014年第10期28卷 3057-3074页

作者： Pham Duc Dai Li, Pu Tech Univ Ilmenau Inst Automat & Syst Engn Simulat & Optimal Proc Grp D-98693 Ilmenau Germany

Leakage reduction represents one of the most challenging tasks in managing water distribution systems (WDSs). An effective way to leakage reduction is to carry out network operational pressure management through optimizing locations and regulations for pressure reducing valves (PRVs) and system pressures. This leads to a mixed-integer nonlinear program (MINLP) with a large number of binary variables which make it difficult to solve by an available software package. In this study, instead of directly solving the MINLP problem, we reformulate it to a mathematical program with complementarity constraints which can be efficiently solved by available NLP algorithms. The binary variables are replaced by continuous ones with complementarity constraints to be satisfied by a penalization scheme. To improve the quality of the solution and also to accelerate the convergence, in each relaxed NLP the results of the binary variables are rounded to binary values with which the NLP problem is solved again to achieve a MINLP solution. The final solution will be determined by the best one among the MINLP solutions. The results from two case studies reveal new and better combinations of PRVs as compared with those given in the literature.

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

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Combining the regularization strategy and the SQP to solve MPCC - A MATLAB implementation

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2011年第18期235卷 5348-5356页

作者： Monteiro, M. Teresa T. Rodrigues, Helena Sofia Univ Minho Dept Prod & Sistemas P-4710057 Braga Portugal Inst Politecn Viana Castelo Escola Super Ciencias Empresariais P-4930678 Valenca Portugal

mathematical program with complementarity constraints (MPCC) plays a very important role in many fields such as engineering design, economic equilibrium, multilevel games, and mathematical programming theory itself. In theory its constraints fail to satisfy a standard constraint qualification such as the linear independence constraint qualification (LICQ) or the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. As a result, the developed nonlinear programming theory may not be applied to MPCC class directly. Nowadays, a natural and popular approach is trying to find some suitable approximations of an MPCC so that it can be solved by solving a sequence of nonlinear programs. This work aims to solve the MPCC using nonlinear programming techniques, namely the SQP and the regularization scheme. Some algorithms with two iterative processes, the inner and the external, were developed. A set of AMPL problems from MacMPEC database (Leyffer, 2000) [8] were tested. The comparative analysis regarding performance of algorithms was carried out. (C) 2010 Elsevier B.V. All rights reserved.

关键词： mathematical program with complementarity constraints Sequential quadratic programming Nonlinear programming Regularization scheme

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mathematical programs with complementarity constraints in the context of inverse optimal control for locomotion

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OPTIMIZATION METHODS & SOFTWARE 2017年第4期32卷 670-698页

作者： Albrecht, Sebastian Ulbrich, Michael Tech Univ Munich Dept Math Boltzmannstr 3 D-85748 Garching Germany

In this paper an inverse optimal control problem in the form of a mathematical program with complementarity constraints (MPCC) is considered and numerical experiences are discussed. The inverse optimal control problem arises in the context of human navigation where the body is modelled as a dynamical system and it is assumed that the motions are optimally controlled with respect to an unknown cost function. The goal of the inversion is now to find a cost function within a given parametrized family of candidate cost functions such that the corresponding optimal motion minimizes the deviation from given data. MPCCs are known to be a challenging class of optimization problems typically violating all standard constraint qualifications (CQs). We show that under certain assumptions the resulting MPCC fulfills CQs for MPCCs being the basis for theory on MPCC optimality conditions and consequently for numerical solution techniques. Finally, numerical results are presented for the discretized inverse optimal control problem of locomotion using different solution techniques based on relaxation and lifting.

关键词： inverse optimal control mathematical program with complementarity constraints constraint qualification constant positive-linear dependence locomotion

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OPTIMAL CONTROL OF QUASI-STATIC PLASTICITY WITH LINEAR KINEMATIC HARDENING, PART I: EXISTENCE AND DISCRETIZATION IN TIME

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2012年第5期50卷 2836-2861页

作者： Wachsmuth, Gerd Tech Univ Chemnitz Fac Math D-09107 Chemnitz Germany

In this paper we consider an optimal control problem governed by a time-dependent variational inequality arising in quasi-static plasticity with linear kinematic hardening. We address certain continuity properties of the forward operator, which imply the existence of an optimal control. Moreover, a discretization in time is derived and we show that every local minimizer of the continuous problem can be approximated by minimizers of modified, time-discrete problems.

关键词： complementarity condition quasistatic plasticity time-dependent variational inequality mathematical program with complementarity constraints evolution variational inequality rate-dependent

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A lifting method for generalized semi-infinite programs based on lower level Wolfe duality

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2013年第1期54卷 189-210页

作者： Diehl, M. Houska, B. Stein, O. Steuermann, P. Karlsruhe Inst Technol KIT Inst Operat Res Karlsruhe Germany Katholieke Univ Leuven Optimizat Engn Ctr OPTEC Louvain Belgium Katholieke Univ Leuven Elect Engn Dept ESAT Louvain Belgium

This paper introduces novel numerical solution strategies for generalized semi-infinite optimization problems (GSIP), a class of mathematical optimization problems which occur naturally in the context of design centering problems, robust optimization problems, and many fields of engineering science. GSIPs can be regarded as bilevel optimization problems, where a parametric lower-level maximization problem has to be solved in order to check feasibility of the upper level minimization problem. The current paper discusses several strategies to reformulate this class of problems into equivalent finite minimization problems by exploiting the concept of Wolfe duality for convex lower level problems. Here, the main contribution is the discussion of the non-degeneracy of the corresponding formulations under various assumptions. Finally, these non-degenerate reformulations of the original GSIP allow us to apply standard nonlinear optimization algorithms.

关键词： Semi-infinite optimization Lower level duality Lifting approach Adaptive convexification 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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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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mathematical programs with complementarity constraints in traffic and telecommunications networks

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PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-mathematical PHYSICAL AND ENGINEERING SCIENCES 2008年第1872期366卷 1973-1987页

作者： Ralph, Daniel Univ Cambridge Judge Business Sch Cambridge CB2 1AG England

Given a suitably parametrized family of equilibrium models and a higher level criterion by which to measure an equilibrium state, mathematical programs with equilibrium constraints (MPECs) provide a framework for improving or optimizing the equilibrium state. An example is toll design in traffic networks, which attempts to reduce total travel time by choosing which arcs to toll and what toll levels to impose. Here, a Wardrop equilibrium describes the traffic response to each toll design. Communication networks also have a deep literature on equilibrium flows that suggest some MPECs. We focus on mathematical programs with complementarity constraints ( MPCCs), a subclass of MPECs for which the lower level equilibrium system can be formulated as a complementarity problem and therefore, importantly, as a nonlinear program (NLP). Although MPECs and MPCCs are typically non-convex, which is a consequence of the upper level objective clashing with the users' objectives in the lower level equilibrium program, the last decade of research has paved the way for finding local solutions of MPCCs via standard NLP techniques.

关键词： mathematical program with equilibrium constraints mathematical program with complementarity constraints bi-level optimization system design traffic networks telecommunications networks

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An algorithm for global solution to bi-parametric linear complementarity constrained linear programs

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JOURNAL OF GLOBAL OPTIMIZATION 2015年第2期62卷 263-297页

作者： Lee, Yu-Ching Pang, Jong-Shi Mitchell, John E. Univ Illinois Dept Ind & Enterprise Syst Engn Urbana IL 61801 USA Univ So Calif Daniel J Epstein Dept Ind & Syst Engn Los Angeles CA 90089 USA Rensselaer Polytech Inst Dept Math Sci Troy NY 12180 USA

A linear program with linear complementarity constraints (LPCC) is among the simplest mathematical programs with complementarity constraints. Yet the global solution of the LPCC remains difficult to find and/or verify. In this work we study a specific type of the LPCC which we term a bi-parametric LPCC. Reformulating the bi-parametric LPCC as a non-convex quadratically constrained program, we develop a domain-partitioning algorithm that solves a series of the linear subproblems and/or convex quadratically constrained subprograms obtained by the relaxations of the complementarity constraint. The choice of an artificial constants-pair allows us to control the domain on which the partitioning is done. Numerical results of robustly solving 105 randomly generated bi-parametric LPCC instances of different structures associated with different numbers of complementarity constraints by the algorithm are presented.

关键词： mathematical program with complementarity constraints Bi-parametric program Domain partitioning Global optimization algorithm

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Capacity Expansion in the Integrated Supply Network for an Electricity Market

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IEEE TRANSACTIONS ON POWER SYSTEMS 2011年第4期26卷 2275-2284页

作者： Jin, Shan Ryan, Sarah M. Iowa State Univ Dept Ind & Mfg Syst Engn Ames IA 50011 USA

constraints in fuel supply, electricity generation, and transmission interact to affect the welfare of strategic generators and price-sensitive consumers. We consider a mixed integer bilevel programming model in which the leader makes capacity expansion decisions in the fuel transportation, generation, and transmission infrastructure of the electricity supply network to maximize social welfare less investment cost. Based on the leader's expansion decisions, the multiple followers including the fuel suppliers, ISO, and generation companies simultaneously optimize their respective objectives of cost, social welfare, and profit. The bilevel program is formulated as a mathematical program with complementarity constraints. The computational challenge posed by the discrete character of transmission expansions has been managed by multiple model reformulations. A lower bound provided by a nonlinear programming reformulation increases the efficiency of solving a binary variable reformulation to global optimality. A single-level optimization relaxation serves as a competitive benchmark to assess the effect of generator strategic operational behavior on the optimal capacity configuration.

关键词： Capacity expansion electricity market model mathematical program with complementarity constraints mixed integer bilevel program

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