Mathematical Programming最新文献

In the non-uniform sparsest cut problem, we are given a supply graph G and a demand graph D, both with the same set of nodes V. The goal is to find a cut of V that minimizes the ratio of the total capacity on the edges of G crossing the cut over the total demand of the crossing edges of D. In this work, we study the non-uniform sparsest cut problem for supply graphs with bounded treewidth k. For this case, Gupta et al. (ACM STOC, 2013) obtained a 2-approximation with polynomial running time for fixed k, and it remained open the question of whether there exists a c-approximation algorithm for a constant c independent of k, that runs in (textsf{FPT}) time. We answer this question in the affirmative. We design a 2-approximation algorithm for the non-uniform sparsest cut with bounded treewidth supply graphs that runs in (textsf{FPT}) time, when parameterized by the treewidth. Our algorithm is based on rounding the optimal solution of a linear programming relaxation inspired by the Sherali-Adams hierarchy. In contrast to the classic Sherali-Adams approach, we construct a relaxation driven by a tree decomposition of the supply graph by including a carefully chosen set of lifting variables and constraints to encode information of subsets of nodes with super-constant size, and at the same time we have a sufficiently small linear program that can be solved in (textsf{FPT}) time.

Abstract

In this paper, a special case of the generalized 4-block n-fold IPs is investigated, where (B_i=B) and B has a rank at most 1. Such IPs, called almost combinatorial 4-block n-fold IPs, include the generalized n-fold IPs as a subcase. We are interested in fixed parameter tractable (FPT) algorithms by taking as parameters the dimensions of the blocks and the largest coefficient. For almost combinatorial 4-block n-fold IPs, we first show that there exists some (lambda le g(gamma )) such that for any nonzero kernel element ({textbf{g}}) , (lambda {textbf{g}}) can always be decomposed into kernel elements in the same orthant whose (ell _{infty }) -norm is bounded by (g(gamma )) (while ({textbf{g}}) itself might not admit such a decomposition), where g is a computable function and (gamma ) is an upper bound on the dimensions of the blocks and the largest coefficient. Based on this, we are able to bound the (ell _{infty }) -norm of Graver basis elements by ({mathcal {O}}(g(gamma )n)) and develop an ({mathcal {O}}(g(gamma )n^{3+o(1)}hat{L}^2)) -time algorithm (here (hat{L}) denotes the logarithm of the largest absolute value occurring in the input). Additionally, we show that the (ell _{infty }) -norm of Graver basis elements is (varOmega (n)) . As applications, almost combinatorial 4-block n-fold IPs can be used to model generalizations of classical problems, including scheduling with rejection, bi-criteria scheduling, and a generalized delivery problem. Therefore, our FPT algorithm establishes a general framework to settle these problems.

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))th-order proximal term (for arbitrary integer (pge 1)) 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 ({mathcal {O}}(k^{-(p+1)})) (for (q=lfloor p/2rfloor ) and the iteration counter k), which can result to a superfast method for some specific class of problems.

The network pricing problem (NPP) is a bilevel problem, where the leader optimizes its revenue by deciding on the prices of certain arcs in a graph, while expecting the followers (also known as the commodities) to choose a shortest path based on those prices. In this paper, we investigate the complexity of the NPP with respect to two parameters: the number of tolled arcs, and the number of commodities. We devise a simple algorithm showing that if the number of tolled arcs is fixed, then the problem can be solved in polynomial time with respect to the number of commodities. In contrast, even if there is only one commodity, once the number of tolled arcs is not fixed, the problem becomes NP-hard. We characterize this asymmetry in the complexity with a novel property named strong bilevel feasibility. Finally, we describe an algorithm to generate valid inequalities to the NPP based on this property, whose numerical results illustrate its potential for effectively solving the NPP with a high number of commodities.

In this paper, we propose a first second-order scheme based on arbitrary non-Euclidean norms, incorporated by Bregman distances. They are introduced directly in the Newton iterate with regularization parameter proportional to the square root of the norm of the current gradient. For the basic scheme, as applied to the composite convex optimization problem, we establish the global convergence rate of the order $O\left(k^{-2}\right)$ both in terms of the functional residual and in the norm of subgradients. Our main assumption on the smooth part of the objective is Lipschitz continuity of its Hessian. For uniformly convex functions of degree three, we justify global linear rate, and for strongly convex function we prove the local superlinear rate of convergence. Our approach can be seen as a relaxation of the Cubic Regularization of the Newton method (Nesterov and Polyak in Math Program 108(1):177-205, 2006) for convex minimization problems. This relaxation preserves the convergence properties and global complexities of the Cubic Newton in convex case, while the auxiliary subproblem at each iteration is simpler. We equip our method with adaptive search procedure for choosing the regularization parameter. We propose also an accelerated scheme with convergence rate $O\left(k^{-3}\right)$, where k is the iteration counter.

We investigate statistical properties of the optimal value of the Sample Average Approximation of stochastic programs, continuing the study (Krätschmer in Nonasymptotic upper estimates for errors of the sample average approximation method to solve risk averse stochastic programs, 2023. Forthcoming in SIAM J. Optim.). Central Limit Theorem type results are derived for the optimal value. As a crucial point the investigations are based on a new type of conditions from the theory of empirical processes which do not rely on pathwise analytical properties of the goal functions. In particular, continuity or convexity in the parameter is not imposed in advance as usual in the literature on the Sample Average Approximation method. It is also shown that the new condition is satisfied if the paths of the goal functions are Hölder continuous so that the main results carry over in this case. Moreover, the main results are applied to goal functions whose paths are piecewise Hölder continuous as e.g. in two stage mixed-integer programs. The main results are shown for classical risk neutral stochastic programs, but we also demonstrate how to apply them to the Sample Average Approximation of risk averse stochastic programs. In this respect we consider stochastic programs expressed in terms of absolute semideviations and divergence risk measures.

Recently, semidefinite programming performance estimation has been employed as a strong tool for the worst-case performance analysis of first order methods. In this paper, we derive new non-ergodic convergence rates for the alternating direction method of multipliers (ADMM) by using performance estimation. We give some examples which show the exactness of the given bounds. We also study the linear and R-linear convergence of ADMM in terms of dual objective. We establish that ADMM enjoys a global linear convergence rate if and only if the dual objective satisfies the Polyak–Łojasiewicz (PŁ) inequality in the presence of strong convexity. In addition, we give an explicit formula for the linear convergence rate factor. Moreover, we study the R-linear convergence of ADMM under two scenarios.

Abstract

The truncated singular value decomposition (SVD), also known as the best low-rank matrix approximation with minimum error measured by a unitarily invariant norm, has been applied to many domains such as biology, healthcare, among others, where high-dimensional datasets are prevalent. To extract interpretable information from the high-dimensional data, sparse truncated SVD (SSVD) has been used to select a handful of rows and columns of the original matrix along with the best low-rank approximation. Different from the literature on SSVD focusing on the top singular value or compromising the sparsity for the seek of computational efficiency, this paper presents a novel SSVD formulation that can select the best submatrix precisely up to a given size to maximize its truncated Ky Fan norm. The fact that the proposed SSVD problem is NP-hard motivates us to study effective algorithms with provable performance guarantees. To do so, we first reformulate SSVD as a mixed-integer semidefinite program, which can be solved exactly for small- or medium-sized instances within a branch-and-cut algorithm framework with closed-form cuts and is extremely useful for evaluating the solution quality of approximation algorithms. We next develop three selection algorithms based on different selection criteria and two searching algorithms, greedy and local search. We prove the approximation ratios for all the approximation algorithms and show that all the ratios are tight when the number of rows or columns of the selected submatrix is no larger than half of the data matrix, i.e., our derived approximation ratios are unimprovable. Our numerical study demonstrates the high solution quality and computational efficiency of the proposed algorithms. Finally, all our analysis can be extended to row-sparse PCA.

A clutter is a family of sets, called members, such that no member contains another. It is called intersecting if every two members intersect, but not all members have a common element. Dense clutters additionally do not have a fractional packing of value 2. We are looking at certain substructures of clutters, namely minors and restrictions. For a family of clutters we introduce a general sufficient condition such that for every clutter we can decide whether the clutter has a restriction in that set in polynomial time. It is known that the sets of intersecting and dense clutters satisfy this condition. For intersecting clutters we generalize the statement to k-wise intersecting clutters using a much simpler proof. We also give a simplified proof that a dense clutter with no proper dense minor is either a delta or the blocker of an extended odd hole. This simplification reduces the running time of the algorithm for finding a delta or the blocker of an extended odd hole minor from previously ({mathscr {O}}(n^4)) to ({mathscr {O}}(n^3)) filter oracle calls.

The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost (f(cdot )) due to an ordering (sigma ) of the items (say [n]), i.e., (min _{sigma } sum _{iin [n]} f(E_{i,sigma })), where (E_{i,sigma }) is the set of items mapped by (sigma ) to indices [i]. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata et al. (in: APPROX, 2012), using Lovász extension of submodular functions. We show a ((2-frac{1+ell _{f}}{1+|E|}))-approximation for monotone submodular MLOP where (ell _{f}=frac{f(E)}{max _{xin E}f({x})}) satisfies (1 le ell _f le |E|). Our theory provides new approximation bounds for special cases of the problem, in particular a ((2-frac{1+r(E)}{1+|E|}))-approximation for the matroid MLOP, where (f = r) is the rank function of a matroid. We further show that minimum latency vertex cover is (frac{4}{3})-approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest.