Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2501.13835

A Globally Convergent Method for Computing B-stationary Points of Mathematical Programs with Equilibrium Constraints

Armin Nurkanović·Sven Leyffer
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper introduces a computationally efficient method that converges globally to B-stationary points of mathematical programs with equilibrium constraints (MPECs). B-stationarity is necessary for optimality and means that no feasible first-order direction can improve the objective. It can be certified by solving a linear program with equilibrium constraints (LPEC) constructed at a given feasible point. The proposed method solves a finite sequence of LPECs, which either certify B-stationarity or provide an active set estimate for the complementarity constraints, along with branch nonlinear programs (BNLPs) obtained by fixing the complementarity active set in the MPEC. In particular, the method proceeds in two phases: the first identifies a feasible BNLP or a stationary point of a constraint infeasibility minimization problem, and the second solves a sequence of BNLPs until a B-stationary point of the MPEC is found. We prove that under the MPEC-MFCQ, the method requires solving only a finite number of BNLPs and LPECs for convergence. Moreover, we show that, unless the current iterate is B-stationary, the combinatorial LPECs need not be solved to optimality. For convergence, it suffices to compute a nonzero feasible point, which in practice often requires solving a single linear program, yielding significant computational savings. Numerical experiments show that the proposed method is more robust and faster than relaxation-based methods and mixed-integer NLP reformulations (which, in contrast to the proposed approach, do not provide a certificate of B-stationarity), even on medium- to large-scale instances.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Armin Nurkanović, Sven Leyffer. 2026-09-08. A Globally Convergent Method for Computing B-stationary Points of Mathematical Programs with Equilibrium Constraints. https://arxiv.org/abs/2501.13835

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Genericity of Polyak-Lojasiewicz Inequalities for Entropic Mean-Field Neural ODEs

We address the behavior of idealized deep residual neural networks (ResNets), modeled via an optimal control problem set over continuity (or adjoint transport) equations. The continuity equations describe the statistical evolution of the features in the asymptotic regime where the layers of the network form a continuum. The velocity field is expressed through the network activation function, which is itself viewed as a function of the statistical distribution of the network parameters (weights and biases). From a mathematical standpoint, the control is interpreted in a relaxed sense, taking values in the space of probability measures over the set of parameters. We investigate the optimal behavior of the network when the cost functional arises from a regression problem and includes an additional entropic regularization term on the distribution of the parameters. In this framework, we focus in particular on the existence of stable optimizers --that is, optimizers at which the Hessian of the cost is non-degenerate. We show that, for an open and dense set of initial data, understood here as probability distributions over features and associated labels, there exists a unique stable global minimizer of the control problem. Moreover, we show that such minimizers satisfy a local Polyak--Lojasiewicz inequality, which can lead to exponential convergence of the corresponding gradient descent when the initialization lies sufficiently close to the optimal parameters. This result thus demonstrates the genericity (with respect to the distribution of features and labels) of the Polyak--Lojasiewicz condition in ResNets with a continuum of layers and under entropic penalization.

math.OC↗
arXiv2026

A regret minimization approach to fixed-point iterations

We propose a conversion scheme that turns regret minimizing algorithms into fixed point iterations, with convergence guarantees following from regret bounds. The resulting iterations can be seen as a grand extension of the classical Krasnoselskii--Mann iterations, as the latter are recovered by converting the Online Gradient Descent algorithm. This approach yields new simple iterations for finding fixed points of non-self operators. We also focus on converting algorithms from the AdaGrad family of regret minimizers, and thus obtain fixed point iterations with adaptive guarantees of a new kind. Numerical experiments on various problems demonstrate faster convergence of AdaGrad-based fixed point iterations over Krasnoselskii--Mann iterations.

math.OC↗
arXiv2026

Variational Analysis in Spectral Decomposition Systems

This work is concerned with the variational analysis of functions defined on Euclidean spaces whose values depend solely on certain invariants (``spectrum'') of their arguments, a class we term ``spectral functions.'' Building on our previous work \cite{PartI} on the convex analysis of such functions, we work in the abstract framework of spectral decomposition systems, which covers a wide range of previously studied settings, including eigenvalue decomposition of Hermitian matrices and singular value decomposition of rectangular matrices, and allows the derivation of new results in more general settings such as normal decomposition systems. The main results of this work provide constructive formulae for computing the regular, limiting, and Clarke subdifferentials of a spectral function in terms of the corresponding objects of the associated invariant function. Finally, we obtain a generalization of Lidski\uı's theorem on the spectrum of additive perturbations of Hermitian matrices to arbitrary spectral decomposition systems.

math.OC↗