Mathematical Programming最新文献

As a starting point of our research, we show that, for a fixed order (gamma ge 1), each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to stationarity of order 1), satisfies stationarity conditions in terms of a coderivative construction of order (gamma ), or is asymptotically stationary with respect to a critical direction as well as order (gamma ) in a certain sense. By ruling out the latter case with a constraint qualification not stronger than directional metric subregularity, we end up with new necessary optimality conditions comprising a mixture of limiting variational tools of orders 1 and (gamma ). These abstract findings are carved out for the broad class of geometric constraints and (gamma :=2), and visualized by examples from complementarity-constrained and nonlinear semidefinite optimization. As a byproduct of the particular setting (gamma :=1), our general approach yields new so-called directional asymptotic regularity conditions which serve as constraint qualifications guaranteeing M-stationarity of local minimizers. We compare these new regularity conditions with standard constraint qualifications from nonsmooth optimization. Further, we extend directional concepts of pseudo- and quasi-normality to arbitrary set-valued mappings. It is shown that these properties provide sufficient conditions for the validity of directional asymptotic regularity. Finally, a novel coderivative-like variational tool is used to construct sufficient conditions for the presence of directional asymptotic regularity. For geometric constraints, it is illustrated that all appearing objects can be calculated in terms of initial problem data.

Given an (mtimes n) binary matrix M with (|M|=pcdot mn) (where |M| denotes the number of 1 entries), define the discrepancy of M as ({{,textrm{disc},}}(M)=displaystyle max nolimits _{Xsubset [m], Ysubset [n]}big ||M[Xtimes Y]|-p|X|cdot |Y|big |). Using semidefinite programming and spectral techniques, we prove that if ({{,textrm{rank},}}(M)le r) and (ple 1/2), then

We use this result to obtain a modest improvement of Lovett’s best known upper bound on the log-rank conjecture. We prove that any (mtimes n) binary matrix M of rank at most r contains an ((mcdot 2^{-O(sqrt{r})})times (ncdot 2^{-O(sqrt{r})})) sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank r is at most (O(sqrt{r})).

Rank regularized minimization problem is an ideal model for the low-rank matrix completion/recovery problem. The matrix factorization approach can transform the high-dimensional rank regularized problem to a low-dimensional factorized column-sparse regularized problem. The latter can greatly facilitate fast computations in applicable algorithms, but needs to overcome the simultaneous non-convexity of the loss and regularization functions. In this paper, we consider the factorized column-sparse regularized model. Firstly, we optimize this model with bound constraints, and establish a certain equivalence between the optimized factorization problem and rank regularized problem. Further, we strengthen the optimality condition for stationary points of the factorization problem and define the notion of strong stationary point. Moreover, we establish the equivalence between the factorization problem and its nonconvex relaxation in the sense of global minimizers and strong stationary points. To solve the factorization problem, we design two types of algorithms and give an adaptive method to reduce their computation. The first algorithm is from the relaxation point of view and its iterates own some properties from global minimizers of the factorization problem after finite iterations. We give some analysis on the convergence of its iterates to a strong stationary point. The second algorithm is designed for directly solving the factorization problem. We improve the PALM algorithm introduced by Bolte et al. (Math Program Ser A 146:459–494, 2014) for the factorization problem and give its improved convergence results. Finally, we conduct numerical experiments to show the promising performance of the proposed model and algorithms for low-rank matrix completion.

In this paper, we consider the problem of minimizing a general homogeneous quadratic function, subject to three real or four complex homogeneous quadratic inequality or equality constraints. For this problem, we present a sufficient and necessary test condition to detect whether its standard semi-definite programming (SDP) relaxation is tight or not. This test condition is based on only an optimal solution pair of the SDP relaxation and its dual. When the tightness is confirmed, a global optimal solution of the original problem is found simultaneously in polynomial-time. While the tightness does not hold, the SDP relaxation and its dual are proved to have the unique optimal solutions. Moreover, the Lagrangian version of such the test condition is specified for non-homogeneous cases. Based on the Lagrangian version, it is proved that several latest sufficient conditions to test the SDP tightness are contained by our test condition under the situation of two constraints. Thirdly, as an application of the test condition, S-lemma and Yuan’s lemma are generalized to three real and four complex quadratic forms first under certain exact conditions, which improves some classical results in literature. Finally, a counterexample is presented to show that the test condition cannot be simply extended to four real or five complex homogeneous quadratic constraints.

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.