Convergence Properties of a Regularization Scheme for Mathematical Programs with Complementarity Constraints

By Stefan Scholtes. SIAM Journal on Optimization, 2000

453 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Management Science and Operations Research, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/2469e7fcef3ff263e5452daa7c9c96d8950c9b2b

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Citations in the corpus

Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints (2006, background): “Note that (P ) satisfies an MPEC-MFCQ [20] but violates an MPEC-LICQ (as can be seen easily by observing that four constraints are active at the solution).”
On the Guignard constraint qualification for mathematical programs with equilibrium constraints (2005, unknown): “For a more detailed analysis, the reader is referred to the recent papers [3,7,10,19], for example.”
Convergence of a Penalty Method for Mathematical Programming with Complementarity Constraints (2004, unknown): “The expression on the left above is actually the gradient of the so-called MPCC Lagrangian L(z;λ, µ, ξ, η) at (z̄, λ̄, µ̄, ξ̄, η̄) with respect to z, where L(z;λ, µ, ξ, η) = f(z) + g(z)Tλ+ h(z)Tµ−G(z)T ξ −H(z)Tµ.”
Using EPECs to Model Bilevel Games in Restructured Electricity Markets with Locational Prices (2007, methodology): “An alternative is to use an iterative scheme like the regularization method, see Scholtes (2001).”
Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications (2008, background): “, l, see, for example, the two books [20, 27] for a general treatment and many applications of MPECs, the more recent works [10, 11, 12, 25, 26, 28, 34, 36, 37] for some more refined theoretical results, or [4, 7, 8, 13, 14, 18, 19, 30, 35] for a number of suitable methods.”
Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation (2008, methodology): “For instance, we may use Scholtes’ regularization method [22] to relax the j-th complementarity constraint by replacing it with a perturbed system of inequalities”
Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints (2003, methodology): “for MPECs in [364], a penalty method in [185] and piece-wise SQP methods [201,268,343,344].”
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).”
On M-stationary points for mathematical programs with equilibrium constraints (2005, unknown): “For example, Scholtes [19] has introduced an algorithm which, under certain assumptions to the MPEC (1), converges to an M-stationary point, but not in general to a strongly stationary point.”
Optimality Conditions for Disjunctive Programs with Application to Mathematical Programs with Equilibrium Constraints (2007, background): “It was coined in the context of MPECs by Scholtes [23].”
Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games (2009, background): “Taking this implied bound into consideration (which is redundant in the full formulation below), firm I's problem is then stated as the following MPEC: For fixed but arbitrary ρ II ∈ [0, 1], determine ρ I , q I,I , q I,II , γ I,I , and γ I,II to maximize ρ I q I,I subject to 0 ≤ ρ I ≤ 1 0 ≤ q I,I ⊥ ”
First-and second-order optimality conditions for mathematical programs with vanishing constraints (2007, methodology): “In this case, a weaker stationary concept was introduced in [6], with the terminology coming from a similar concept for MPECs, see [9, 12, 3].”