Mathematical Programming最新文献

Optimization problems involving minimization of a rank-one convex function over constraints modeling restrictions on the support of the decision variables emerge in various machine learning applications. These problems are often modeled with indicator variables for identifying the support of the continuous variables. In this paper we investigate compact extended formulations for such problems through perspective reformulation techniques. In contrast to the majority of previous work that relies on support function arguments and disjunctive programming techniques to provide convex hull results, we propose a constructive approach that exploits a hidden conic structure induced by perspective functions. To this end, we first establish a convex hull result for a general conic mixed-binary set in which each conic constraint involves a linear function of independent continuous variables and a set of binary variables. We then demonstrate that extended representations of sets associated with epigraphs of rank-one convex functions over constraints modeling indicator relations naturally admit such a conic representation. This enables us to systematically give perspective formulations for the convex hull descriptions of these sets with nonlinear separable or non-separable objective functions, sign constraints on continuous variables, and combinatorial constraints on indicator variables. We illustrate the efficacy of our results on sparse nonnegative logistic regression problems.

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to a row-equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse row-equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the (ell _1)-norm of the Graver basis is bounded by a function of the maximum (ell _1)-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix row-equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such a row-equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the (ell _1)-norm of the Graver basis of the constraint matrix, when parameterized by the (ell _1)-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix.

This paper considers the basic problem of scheduling jobs online with preemption to maximize the number of jobs completed by their deadline on m identical machines. The main result is an O(1) competitive deterministic algorithm for any number of machines (m >1).

Abstract

This note provides a counterexample to a theorem announced in the last part of the paper (Vicente and Custódio Math Program 133:299–325, 2012). The counterexample involves an objective function  (f: mathbb {R}rightarrow mathbb {R}) which satisfies all the assumptions required by the theorem but contradicts some of its conclusions. A corollary of this theorem is also affected by this counterexample. The main flaw revealed by the counterexample is the possibility that a directional direct search method (dDSM) generates a sequence of trial points  ((x_k)_{k in mathbb {N}}) converging to a point  (x_*) where f is discontinuous, lower semicontinuous and whose objective function value  (f(x_*)) is strictly less than  (lim _{krightarrow infty } f(x_k)) . Moreover the dDSM generates trial points in only one of the continuity sets of f near  (x_*) . This note also investigates the proof of the theorem to highlight the inexact statements in the original paper. Finally this work introduces a modification of the dDSM that allows, in usual cases, to recover the properties broken by the counterexample.

We consider the oracle complexity of computing an approximate stationary point of a Lipschitz function. When the function is smooth, it is well known that the simple deterministic gradient method has finite dimension-free oracle complexity. However, when the function can be nonsmooth, it is only recently that a randomized algorithm with finite dimension-free oracle complexity has been developed. In this paper, we show that no deterministic algorithm can do the same. Moreover, even without the dimension-free requirement, we show that any finite-time deterministic method cannot be general zero-respecting. In particular, this implies that a natural derandomization of the aforementioned randomized algorithm cannot have finite-time complexity. Our results reveal a fundamental hurdle in modern large-scale nonconvex nonsmooth optimization.

Composite minimization is a powerful framework in large-scale convex optimization, based on decoupling of the objective function into terms with structurally different properties and allowing for more flexible algorithmic design. We introduce a new algorithmic framework for complementary composite minimization, where the objective function decouples into a (weakly) smooth and a uniformly convex term. This particular form of decoupling is pervasive in statistics and machine learning, due to its link to regularization. The main contributions of our work are summarized as follows. First, we introduce the problem of complementary composite minimization in general normed spaces; second, we provide a unified accelerated algorithmic framework to address broad classes of complementary composite minimization problems; and third, we prove that the algorithms resulting from our framework are near-optimal in most of the standard optimization settings. Additionally, we show that our algorithmic framework can be used to address the problem of making the gradients small in general normed spaces. As a concrete example, we obtain a nearly-optimal method for the standard (ell _1) setup (small gradients in the (ell _infty ) norm), essentially matching the bound of Nesterov (Optima Math Optim Soc Newsl 88:10–11, 2012) that was previously known only for the Euclidean setup. Finally, we show that our composite methods are broadly applicable to a number of regression and other classes of optimization problems, where regularization plays a key role. Our methods lead to complexity bounds that are either new or match the best existing ones.

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.