A classical result of variational analysis, known as Attouch theorem, establishes an equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding subdifferential maps up to a normalization condition which fixes the integration constant. In this work, we show that in finite dimensions and under a mild boundedness assumption, we can replace subdifferentials (sets of vectors) by slopes (scalars, corresponding to the distance of the subdifferentials to zero) and still obtain the same characterization: namely, the epigraphical convergence of functions is equivalent to the epigraphical convergence of their slopes. This surprising result goes in line with recent developments on slope determination (Boulmezaoud et al. in SIAM J Optim 28(3):2049–2066, 2018; Pérez-Aros et al. in Math Program 190(1–2):561-583, 2021) and slope sensitivity (Daniilidis and Drusvyatskiy in Proc Am Math Soc 151(11):4751-4756, 2023) for convex functions.

We consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido–Isoda function. To any given instance of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.

We show how to round any half-integral solution to the subtour-elimination relaxation for the TSP, while losing a less-than-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-$$\\end{document} 1.5 factor. Such a rounding algorithm was recently given by Karlin, Klein, and Oveis Gharan based on sampling from max-entropy distributions. We build on an approach of Haddadan and Newman to show how sampling from the matroid intersection polytope, combined with a novel use of max-entropy sampling, can give better guarantees.

Appendix E contained response of the referees, but it was inadvertently published in the original publication of the article. Now, it has been removed and original article has been corrected. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We introduce a Bi-level OPTimization (BiOPT) framework for minimizing the sum of two convex functions, where one of them is smooth enough. The BiOPT framework offers three levels of freedom: (i) choosing the order p of the proximal term; (ii) designing an inexact pth-order proximal-point method in the upper level; (iii) solving the auxiliary problem with a lower-level non-Euclidean method in the lower level. We here regularize the objective by a (p+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p+1)$$\\end{document}th-order proximal term (for arbitrary integer p≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 1$$\\end{document}) and then develop the generic inexact high-order proximal-point scheme and its acceleration using the standard estimating sequence technique at the upper level. This follows at the lower level with solving the corresponding pth-order proximal auxiliary problem inexactly either by one iteration of the pth-order tensor method or by a lower-order non-Euclidean composite gradient scheme. Ultimately, it is shown that applying the accelerated inexact pth-order proximal-point method at the upper level and handling the auxiliary problem by the non-Euclidean composite gradient scheme lead to a 2q-order method with the convergence rate O(k-(p+1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(k^{-(p+1)})$$\\end{document} (for q=⌊p/2⌋\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=\\lfloor p/2\\rfloor $$\\end{document} and the iteration counter k), which can result to a superfast method for some specific class of problems.

The minimum sum-of-squares clustering (MSSC), or k-means type clustering, has been recently extended to exploit prior knowledge on the cardinality of each cluster. Such knowledge is used to increase performance as well as solution quality. In this paper, we propose a global optimization approach based on the branch-and-cut technique to solve the cardinality-constrained MSSC. For the lower bound routine, we use the semidefinite programming (SDP) relaxation recently proposed by Rujeerapaiboon et al. (SIAM J Optim 29(2):1211–1239, 2019). However, this relaxation can be used in a branch-and-cut method only for small-size instances. Therefore, we derive a new SDP relaxation that scales better with the instance size and the number of clusters. In both cases, we strengthen the bound by adding polyhedral cuts. Benefiting from a tailored branching strategy which enforces pairwise constraints, we reduce the complexity of the problems arising in the children nodes. For the upper bound, instead, we present a local search procedure that exploits the solution of the SDP relaxation solved at each node. Computational results show that the proposed algorithm globally solves, for the first time, real-world instances of size 10 times larger than those solved by state-of-the-art exact methods.

We consider first-order methods with constant step size for minimizing locally Lipschitz coercive functions that are tame in an o-minimal structure on the real field. We prove that if the method is approximated by subgradient trajectories, then the iterates eventually remain in a neighborhood of a connected component of the set of critical points. Under suitable method-dependent regularity assumptions, this result applies to the subgradient method with momentum, the stochastic subgradient method with random reshuffling and momentum, and the random-permutations cyclic coordinate descent method.

Given a matrix X ∈ R m×n + with non-negative entries, the cone factorization problem over a cone K ⊆ R k concerns computing {a 1 , . . . , a m } ⊆ K and {b 1 , . . . , b n } ⊆ K * belonging to its dual so that . Cone factorizations are fundamental to mathematical optimization as they allow us to express convex bodies as feasible regions of linear conic programs. In this paper, we introduce and analyze the symmetric-cone multiplicative update (SCMU) algorithm for computing cone factorizations when K is symmetric; i.e., it is selfdual and homogeneous. Symmetric cones are of central interest in mathematical optimization as they provide a common language for studying linear optimization over the nonnegative orthant (linear programs), over the second-order cone (second order cone programs), and over the cone of positive semidefinite matrices (semidefinite programs). The SCMU algorithm is multiplicative in the sense that the iterates are updated by applying a meticulously chosen automorphism of the cone computed using a generalization of the geometric mean to symmetric cones. Using an extension of Lieb's concavity theorem and von Neumann's trace inequality to symmetric cones, we show that the squared loss objective is non-decreasing along the trajectories of the SCMU algorithm. Specialized to the nonnegative orthant, the SCMU algorithm corresponds to the seminal algorithm by Lee and Seung for computing Nonnegative Matrix Factorizations.