Mathematical Programming最新文献

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.

The best practical techniques for exact solution of instances of the constrained maximum-entropy sampling problem, a discrete-optimization problem arising in the design of experiments, are via a branch-and-bound framework, working with a variety of concave continuous relaxations of the objective function. A standard and computationally-important bound-enhancement technique in this context is (ordinary) scaling, via a single positive parameter. Scaling adjusts the shape of continuous relaxations to reduce the gaps between the upper bounds and the optimal value. We extend this technique to generalized scaling, employing a positive vector of parameters, which allows much more flexibility and thus potentially reduces the gaps further. We give mathematical results aimed at supporting algorithmic methods for computing optimal generalized scalings, and we give computational results demonstrating the performance of generalized scaling on benchmark problem instances.

We study rectangle stabbing problems in which we are given n axis-aligned rectangles in the plane that we want to stab, that is, we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (Stabbing), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In the horizontal–vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (that is, either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan et al. (ISAAC, 2018) initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand et al. (A QPTAS for stabbing rectangles, 2021) have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing, but it was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for Stabbing, settling this question. For HV-Stabbing, we obtain a ((2+varepsilon ))-approximation. We also obtain PTASs for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle’s width is at most its height, and (iii) when all rectangles are (delta )-large, that is, have at least one edge whose length is at least (delta ), while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as generalized minimum Manhattan network.

We develop a fully asynchronous proximal bundle method for solving non-smooth, convex optimization problems. The algorithm can be used as a drop-in replacement for classic bundle methods, i.e., the function must be given by a first-order oracle for computing function values and subgradients. The algorithm allows for an arbitrary number of master problem processes computing new candidate points and oracle processes evaluating functions at those candidate points. These processes share information by communication with a single supervisor process that resembles the main loop of a classic bundle method. All processes run in parallel and no explicit synchronization step is required. Instead, the asynchronous and possibly outdated results of the oracle computations can be seen as an inexact function oracle. Hence, we show the convergence of our method under weak assumptions very similar to inexact and incremental bundle methods. In particular, we show how the algorithm learns important structural properties of the functions to control the inaccuracy induced by the asynchronicity automatically such that overall convergence can be guaranteed.

Handling symmetries in optimization problems is essential for devising efficient solution methods. In this article, we present a general framework that captures many of the already existing symmetry handling methods. While these methods are mostly discussed independently from each other, our framework allows to apply different methods simultaneously and thus outperforming their individual effect. Moreover, most existing symmetry handling methods only apply to binary variables. Our framework allows to easily generalize these methods to general variable types. Numerical experiments confirm that our novel framework is superior to the state-of-the-art symmetry handling methods as implemented in the solver SCIP on a broad set of instances.

We propose a new first-order method for minimizing nonconvex functions with Lipschitz continuous gradients and Hölder continuous Hessians. The proposed algorithm is a heavy-ball method equipped with two particular restart mechanisms. It finds a solution where the gradient norm is less than (varepsilon ) in (O(H_{nu }^{frac{1}{2 + 2 nu }} varepsilon ^{- frac{4 + 3 nu }{2 + 2 nu }})) function and gradient evaluations, where (nu in [0, 1]) and (H_{nu }) are the Hölder exponent and constant, respectively. This complexity result covers the classical bound of (O(varepsilon ^{-2})) for (nu = 0) and the state-of-the-art bound of (O(varepsilon ^{-7/4})) for (nu = 1). Our algorithm is (nu )-independent and thus universal; it automatically achieves the above complexity bound with the optimal (nu in [0, 1]) without knowledge of (H_{nu }). In addition, the algorithm does not require other problem-dependent parameters as input, including the gradient’s Lipschitz constant or the target accuracy (varepsilon ). Numerical results illustrate that the proposed method is promising.

A central tool for understanding first-order optimization algorithms is the Kurdyka–Łojasiewicz inequality. Standard approaches to such methods rely crucially on this inequality to leverage sufficient decrease conditions involving gradients or subgradients. However, the KL property fundamentally concerns not subgradients but rather “slope”, a purely metric notion. By highlighting this view, and avoiding any use of subgradients, we present a simple and concise complexity analysis for first-order optimization algorithms on metric spaces. This subgradient-free perspective also frames a short and focused proof of the KL property for nonsmooth semi-algebraic functions.

In this paper we propose polarized consensus-based dynamics in order to make consensus-based optimization (CBO) and sampling (CBS) applicable for objective functions with several global minima or distributions with many modes, respectively. For this, we “polarize” the dynamics with a localizing kernel and the resulting model can be viewed as a bounded confidence model for opinion formation in the presence of common objective. Instead of being attracted to a common weighted mean as in the original consensus-based methods, which prevents the detection of more than one minimum or mode, in our method every particle is attracted to a weighted mean which gives more weight to nearby particles. We prove that in the mean-field regime the polarized CBS dynamics are unbiased for Gaussian targets. We also prove that in the zero temperature limit and for sufficiently well-behaved strongly convex objectives the solution of the Fokker–Planck equation converges in the Wasserstein-2 distance to a Dirac measure at the minimizer. Finally, we propose a computationally more efficient generalization which works with a predefined number of clusters and improves upon our polarized baseline method for high-dimensional optimization.

We study linear bilevel programming problems whose lower-level objective is given by a random cost vector with known distribution. We consider the case where this distribution is nonatomic, allowing to reformulate the problem of the leader using the Bayesian approach in the sense of Salas and Svensson (SIAM J Optim 33(3):2311–2340, 2023), with a decision-dependent distribution that concentrates on the vertices of the feasible set of the follower’s problem. We call this a vertex-supported belief. We prove that this formulation is piecewise affine over the so-called chamber complex of the feasible set of the high-point relaxation. We propose two algorithmic approaches to solve general problems enjoying this last property. The first one is based on enumerating the vertices of the chamber complex. This approach is not scalable, but we present it as a computational baseline and for its theoretical interest. The second one is a Monte-Carlo approximation scheme based on the fact that randomly drawn points of the domain lie, with probability 1, in the interior of full-dimensional chambers, where the problem (restricted to this chamber) can be reduced to a linear program. Finally, we evaluate these methods through computational experiments showing both approaches’ advantages and challenges.

We investigate the information complexity of mixed-integer convex optimization under different types of oracles. We establish new lower bounds for the standard first-order oracle, improving upon the previous best known lower bound. This leaves only a lower order linear term (in the dimension) as the gap between the lower and upper bounds. This is derived as a corollary of a more fundamental “transfer” result that shows how lower bounds on information complexity of continuous convex optimization under different oracles can be transferred to the mixed-integer setting in a black-box manner. Further, we (to the best of our knowledge) initiate the study of, and obtain the first set of results on, information complexity under oracles that only reveal partial first-order information, e.g., where one can only make a binary query over the function value or subgradient at a given point. We give algorithms for (mixed-integer) convex optimization that work under these less informative oracles. We also give lower bounds showing that, for some of these oracles, every algorithm requires more iterations to achieve a target error compared to when complete first-order information is available. That is, these oracles are provably less informative than full first-order oracles for the purpose of optimization.