Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity

By H. Scheel, Stefan Scholtes. Mathematics of Operations Research, 2000

776 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Economics and Econometrics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/c1174bd9f50f0bb96f1cba1b2d98b10d3bf61fe2

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Citations in the corpus

Convergence of a Penalty Method for Mathematical Programming with Complementarity Constraints (2004, neutral): “The first equation in (8) is 0 = ∇f(z) + ρ∇ ( ψ ◦ (G,H) ) (z) +∇g(z)λ +∇h(z)μ − ∇G(z) ξ −∇H(z)η.”
Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications (2008, methodology): “(Another tightened nonlinear program in the context of MPECs was also used in [34] in order to define MPEC-tailored constraint qualifications.”
Using EPECs to Model Bilevel Games in Restructured Electricity Markets with Locational Prices (2007, result): “This result is essentially a corollary of the regularization convergence analysis of Scholtes (2001) which requires that the “solution” of (18), in (ii-2), is a stationary point that satisfies a second-order necessary condition as described in Scholtes (2001).”
Stochastic Equilibrium Problems and Stochastic Mathematical Programs with Equilibrium Constraints: A Survey 1 (2009, background): “is solved to get a weak stationary point; see [53] for various stationarity concepts for MPEC.”
On M-stationary points for mathematical programs with equilibrium constraints (2005, methodology): “The other stationarity concepts differ from M-stationarity only in the restriction imposed upon λi and λ H i for i ∈ β, as detailed in the following list: • strong stationarity [18, 15]: λi ≥ 0 ∧ λi ≥ 0 ∀i ∈ β; • M-stationarity [13]: (λi > 0 ∧ λi > 0) ∨ λi λi = 0 ∀i ∈ β; • C-stationarity [18]: λi λi”
On the Coderivative of the Projection Operator onto the Second-order Cone (2008, background): “In compliance with [19], we could thus call the point (0, 0) strongly stationary.”
Stochastic Programming with Equilibrium Constraints (2006, background): “In that respect we may refer to [13] and [19], and references therein.”
Sustainability SI: Optimal Prices of Electricity at Public Charging Stations for Plug-in Electric Vehicles (2016, background): “Note that the manifold suboptimization algorithm yields a strongly stationary solution, whose quality may not be good because strongly stationary conditions are one of the weaker optimality conditions (e.g., Scheel and Scholtes 2000; Fletcher and Leyffer 2004; Lou et al. 2010).”
Equilibria in an Oligopolistic Electricity Pool With Stepwise Offer Curves (2012, background): “However, the nature of the MPECs entails that the Mangasarian-Fromovitz constraint qualification (MFCQ) fails at every feasible point [6] which makes difficult the definition of second-order sufficient conditions for optimality.”
Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints (2006, background): “In [19], four stationarity concepts are introduced for MPEC (1.”
Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints (2005, background): “This improves the result of [18] in whic was shown that MPEC–MFCQ is a constraint qualification for C-stationarity.”
An overview of bilevel optimization (2007, methodology): “Among them are Dempe (1992a, 1992b), Yezza (1996), Outrata (1993), Ye and Zhu (1995) and Scheel and Scholtes (2000), using tools from nonsmooth analysis (see e.g. Clarke 1990).”
A Fritz John Approach to First Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints (2003, methodology): “A slightly different theorem under MPEC-MFCQ is stated in [16], where the existence of Lagrange multipliers (λG)∗ and (λH)∗ satisfying (λi ) (λi ) ∗ ≥ 0 ∀i ∈ β (18) is shown.”
An SOS1-Based Approach for Solving MPECs with a Natural Gas Market Application (2013, background): “It has been noted (Scheel and Scholtes 2000) that different constraint qualifications in fact result in solvers converging to different local solutions.”