A log-exponential smoothing method for mathematical programs with complementarity constraints

Li, Yanyan; Tan, Ting; Li, Xingsi

doi:10.1016/j.amc.2011.11.046

Cited by 12 publications

(12 citation statements)

References 13 publications

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“…Microscopic and mesoscopic vehicular emission and noise models, particularly with an explicit consideration of multiple vehicle types, need to be integrated into the EC-TEP in order to enhance the modeling realism of environmental constraints. 2 In the literature, recent studies proposed different types of gap functions, for example, perturbed Fischer-Burmeister gap functions and log-exponential smoothing function (Li, Tan, and Li 2012). We plan to compare their theoretical and computational performances in the context of EC-TEP reformulation.…”

Section: Conclusion and Future Researchmentioning

confidence: 99%

Reformulating Environmentally Constrained Traffic Equilibrium via a Smooth Gap Function

Chen

Cheng

2015

International Journal of Sustainable Transportation

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Various government laws have recently been enacted to alleviate the environmental deterioration of transportation systems. Environmental constraint is a valid means to explicitly reflect various environmental protection requirements imposed by the government. In this paper, we examine the environmentally constrained traffic equilibrium problem (EC-TEP), which is a fundamental tool for modeling and evaluating environmental protection requirements. Specifically, we provide an equivalent reformulation for the EC-TEP. The proposed reformulation adapts the concept of gap function to simultaneously reformulate the nonlinear complementarity conditions associated with the generalized user equilibrium conditions, environmental constraints, and conservation constraints as an equivalent unconstrained optimization problem. This gap function reformulation has two desirable features: (1) it can handle a general environmental constraint structure (linear or nonlinear; link-based or area-based) and a general link and route cost structure, enhancing the modeling adaptability and flexibility; (2) it is smooth and unconstrained, permitting a number of existing efficient algorithms for its solution. A gradient-based solution algorithm with a self-regulated averaging stepsize scheme is customized to solve the reformulated unconstrained optimization problem. Numerical examples are also provided to demonstrate the modeling flexibility of the proposed EC-TEP reformulation.

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Section: Conclusion and Future Researchmentioning

confidence: 99%

Reformulating Environmentally Constrained Traffic Equilibrium via a Smooth Gap Function

Chen

Cheng

2015

International Journal of Sustainable Transportation

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“…Unfortunately, the relation between equilibrium flow and congestion pricing variables may not always be differentiable, which has been theoretically investigated in [21] and [22] and numerically illustrated in [23] and [24]. The combined problem of congestion pricing and signal settings for equilibrium flow can be formulated as a case of mathematical program with equilibrium constraints (MPEC) or MPCC (mathematical program with complementarity constraints) [25][26][27]. Lawphongpanich and Hearn [18] proposed a cutting constraint algorithm (CCA) approach to solving a second-best toll pricing problem.…”

Section: Introductionmentioning

confidence: 99%

A cutting plane projection method for bi-level area traffic control optimization with uncertain travel demand

Chiou

2015

Applied Mathematics and Computation

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“…Thus, many fundamental theoretical results and powerful algorithms for an ordinary smooth optimization problem cannot be directly employed to solve (1). Actually, some specific approaches have been proposed for solving MPCC (1), such as the sequential quadratic programming approach in [3][4][5][6][7][8], the interior point methods in [9,10], the penalty approach in [11][12][13], the lifting method in [14], the relaxation approach in [15][16][17][18][19][20][21][22], and the smoothing methods in [23][24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

A New Smoothing Method for Mathematical Programs with Complementarity Constraints Based on Logarithm-Exponential Function

Wan

2018

Mathematical Problems in Engineering

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We present a new smoothing method based on a logarithm-exponential function for mathematical program with complementarity constraints (MPCC). Different from the existing smoothing methods available in the literature, we construct an approximate smooth problem of MPCC by partly smoothing the complementarity constraints. With this new method, it is proved that the Mangasarian-Fromovitz constraint qualification holds for the approximate smooth problem. Convergence of the approximate solution sequence, generated by solving a series of smooth perturbed subproblems, is investigated. Under the weaker constraint qualification MPCC-Cone-Continuity Property, it is proved that any accumulation point of the approximate solution sequence is a M-stationary point of the original MPCC. Preliminary numerical results indicate that the developed algorithm based on the partly smoothing method is efficient, particularly in comparison with the other similar ones.

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A log-exponential smoothing method for mathematical programs with complementarity constraints

Cited by 12 publications

References 13 publications

Reformulating Environmentally Constrained Traffic Equilibrium via a Smooth Gap Function

Reformulating Environmentally Constrained Traffic Equilibrium via a Smooth Gap Function

A cutting plane projection method for bi-level area traffic control optimization with uncertain travel demand

A New Smoothing Method for Mathematical Programs with Complementarity Constraints Based on Logarithm-Exponential Function

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