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Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints

COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2012年第1期51卷 199-221页

作者： Izmailov, A. F. Pogosyan, A. L. Solodov, M. V. IMPA BR-22460320 Rio De Janeiro RJ Brazil Moscow MV Lomonosov State Univ VMK Fac OR Dept Moscow 119991 Russia

We consider a reformulation of mathematical programs with complementarity constraints, where by introducing an artificial variable the constraints are converted into equalities which are once but not twice differentiable. We show that the Lagrange optimality system of such a reformulation is semismooth and BD-regular at the solution under reasonable assumptions. Thus, fast local convergence can be obtained by applying the semismooth Newton method. Moreover, it turns out that the squared residual of the Lagrange system is continuously differentiable (even though the system itself is not), which opens the way for a natural globalization of the local algorithm. Preliminary numerical results are also reported.

关键词： mathematical program with complementarity constraints Semismooth Newton method BD-regularity Second-order sufficiency Merit function

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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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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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A semismooth sequential quadratic programming method for lifted mathematical programs with vanishing constraints

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COMPUTATIONAL MATHEMATICS AND mathematical PHYSICS 2011年第6期51卷 919-941页

作者： Izmailov, A. F. Pogosyan, A. L. Moscow MV Lomonosov State Univ Fac Computat Math & Cybernet Moscow 119991 Russia

mathematical programs with vanishing constraints are a difficult class of optimization problems with important applications to optimal topology design problems of mechanical structures. Recently, they have attracted increasingly more attention of experts. The basic difficulty in the analysis and numerical solution of such problems is that their constraints are usually nonregular at the solution. In this paper, a new approach to the numerical solution of these problems is proposed. It is based on their reduction to the so-called lifted mathematical programs with conventional equality and inequality constraints. Special versions of the sequential quadratic programming method are proposed for solving lifted problems. Preliminary numerical results indicate the competitiveness of this approach.

关键词： mathematical program with vanishing constraints lifted problem mathematical program with complementarity constraints constraint qualifications optimality conditions sequential quadratic programming

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The C-Index: A New Stability Concept for Quadratic programs with complementarity constraints

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MATHEMATICS OF OPERATIONS RESEARCH 2011年第3期36卷 504-526页

作者： Ralph, Daniel Stein, Oliver Univ Cambridge Judge Business Sch Cambridge CB2 1AG England Karlsruhe Inst Technol D-76128 Karlsruhe Germany

We introduce nondegeneracy and the C-index for C-stationary points of a QPCC, that is, for a mathematical program with a quadratic objective function and linear complementarity constraints. The C-index characterizes the qualitative local behavior of a QPCC around a nondegenerate C-stationary point. The article focuses on the structure of the C-stationary set of QPCCs depending on a real parameter. We show that, for generic QPCC data, the C-index changes exactly at turning points of the C-stationary set, and that it changes by exactly one. To illustrate this concept, we introduce and analyze two homotopy methods for finding C-stationary points. Numerical results illustrate that, for randomly generated test problems, the two homotopy methods very often identify B-stationary points.

关键词： mathematical program with complementarity constraints C-stationarity C-index MPEC MPCC QPCC homotopy

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On the convergence properties of modified augmented Lagrangian methods for mathematical programming with complementarity constraints

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JOURNAL OF GLOBAL OPTIMIZATION 2010年第2期46卷 217-232页

作者： Luo, H. Z. Sun, X. L. Xu, Y. F. Wu, H. X. Fudan Univ Sch Management Dept Management Sci Shanghai 200433 Peoples R China Zhejiang Univ Technol Dept Appl Math Hangzhou 310032 Zhejiang Peoples R China Hangzhou Dianzi Univ Dept Math Hangzhou 310018 Zhejiang Peoples R China

In this paper, we present new convergence results of augmented Lagrangian methods for mathematical programs with complementarity constraints (MPCC). Modified augmented Lagrangian methods based on four different algorithmic strategies are considered for the constrained nonconvex optimization reformulation of MPCC. We show that the convergence to a global optimal solution of the problem can be ensured without requiring the boundedness condition of the multipliers.

关键词： mathematical program with complementarity constraints Modified augmented Lagrangian methods Nonconvex constrained optimization Convergence to global solution

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Convergence Properties of Modified and Partially-Augmented Lagrangian Methods for mathematical programs with complementarity constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2010年第3期145卷 489-506页

作者： Luo, H. Z. Sun, X. L. Xu, Y. F. Fudan Univ Sch Management Dept Management Sci Shanghai 200433 Peoples R China Zhejiang Univ Technol Dept Appl Math Hangzhou 310032 Zhejiang Peoples R China

We present new convergence properties of partially augmented Lagrangian methods for mathematical programs with complementarity constraints (MPCC). Four modified partially augmented Lagrangian methods for MPCC based on different algorithmic strategies are proposed and analyzed. We show that the convergence of the proposed methods to a B-stationary point of MPCC can be ensured without requiring the boundedness of the multipliers.

关键词： mathematical program with complementarity constraints Modified partially augmented Lagrangian methods Convergence to B-stationary point Constraint qualifications Boundedness of the multipliers

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Optimality conditions and newton-type methods for mathematical programs with vanishing constraints

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COMPUTATIONAL MATHEMATICS AND mathematical PHYSICS 2009年第7期49卷 1128-1140页

作者： Izmailov, A. F. Pogosyan, A. L. Moscow MV Lomonosov State Univ Fac Computat Math & Cybernet Moscow 119992 Russia

A new class of optimization problems is discussed in which some constraints must hold in certain regions of the corresponding space rather than everywhere. In particular, the optimal design of topologies for mechanical structures can be reduced to problems of this kind. Problems in this class are difficult to analyze and solve numerically because their constraints are usually irregular. Some known first- and second-order necessary conditions for local optimality are refined for problems with vanishing constraints, and special Newton-type methods are developed for solving such problems.

关键词： mathematical program with vanishing constraints mathematical program with complementarity constraints constraint qualification optimality conditions sequential quadratic programming active-set method

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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 ACTIVE-SET NEWTON METHOD FOR mathematical programS WITH complementarity constraints

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SIAM JOURNAL ON OPTIMIZATION 2008年第3期19卷 1003-1027页

作者： Izmailov, A. F. Solodov, M. V. Moscow MV Lomonosov State Univ Fac Computat Math & Cybernet Dept Operat Res Moscow 119899 Russia IMPA BR-22460320 Rio De Janeiro Brazil

For a mathematical program with complementarity constraints (MPCC), we propose an active-set Newton method, which has the property of local quadratic convergence under the MPCC linear independence constraint qualification (MPCC-LICQ) and the standard second-order sufficient condition (SOSC) for optimality. Under MPCC-LICQ, this SOSC is equivalent to the piecewise SOSC on branches of MPCC, which is weaker than the special MPCC-SOSC often employed in the literature. The piecewise SOSC is also more natural than MPCC-SOSC because, unlike the latter, it has an appropriate second-order necessary condition as its counterpart. In particular, our assumptions for local quadratic convergence are weaker than those required by standard SQP when applied to MPCC and are equivalent to assumptions required by piecewise SQP for MPCC. Moreover, each iteration of our method consists of solving a linear system of equations instead of a quadratic program. Some globalization issues of the local scheme are also discussed, and illustrative examples and numerical experiments are presented.

关键词： mathematical program with complementarity constraints active set method Newton method SQP second-order sufficiency quadratic convergence

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