Mathematical Programming最新文献

A rational number is dyadic if it has a finite binary representation $p/2^k$ , where p is an integer and k is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is dyadic if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case.

Trust-region methods (TR) can converge quadratically to minima where the Hessian is positive definite. However, if the minima are not isolated, then the Hessian there cannot be positive definite. The weaker Polyak-Łojasiewicz (PŁ) condition is compatible with non-isolated minima, and it is enough for many algorithms to preserve good local behavior. Yet, TR with an exact subproblem solver lacks even basic features such as a capture theorem under PŁ. In practice, a popular inexact subproblem solver is the truncated conjugate gradient method (tCG). Empirically, TR-tCG exhibits superlinear convergence under PŁ. We confirm this theoretically. The main mathematical obstacle is that, under PŁ, at points arbitrarily close to minima, the Hessian has vanishingly small, possibly negative eigenvalues. Thus, tCG is applied to ill-conditioned, indefinite systems. Yet, the core theory underlying tCG is that of CG, which assumes a positive definite operator. Accordingly, we develop new tools to analyze the dynamics of CG in the presence of small eigenvalues of any sign, for the regime of interest to TR-tCG.

There has been significant work recently on integer programs (IPs) $min\{c^{\top }x:Ax\le b,x\in Z^n\}$ with a constraint marix A with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant $\Delta \in Z_{>0}$ , $\Delta$ -modular IPs are efficiently solvable, which are IPs where the constraint matrix $A\in Z^{m\times n}$ has full column rank and all $n\times n$ minors of A are within $\{-\Delta ,\cdots ,\Delta \}$ . Previous progress on this question, in particular for $\Delta =2$ , relies on algorithms that solve an important special case, namely strictly $\Delta$ -modular IPs, which further restrict the $n\times n$ minors of A to be within $\{-\Delta ,0,\Delta \}$ . Even for $\Delta =2$ , such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly $\Delta$ -modular IPs. Prior advances were restricted to prime $\Delta$ , which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly $\Delta$ -modular IPs in strongly polynomial time if $\Delta \le 4$ .

The Matroid Secretary Conjecture is a notorious open problem in online optimization. It claims the existence of an O(1)-competitive algorithm for the Matroid Secretary Problem (MSP). Here, the elements of a weighted matroid appear one-by-one, revealing their weight at appearance, and the task is to select elements online with the goal to get an independent set of largest possible weight. O(1)-competitive MSP algorithms have so far only been obtained for restricted matroid classes and for MSP variations, including Random-Assignment MSP (RA-MSP), where an adversary fixes a number of weights equal to the ground set size of the matroid, which then get assigned randomly to the elements of the ground set. Unfortunately, these approaches heavily rely on knowing the full matroid upfront. This is an arguably undesirable requirement, and there are good reasons to believe that an approach towards resolving the MSP Conjecture should not rely on it. Thus, both Soto (SIAM Journal on Computing 42(1): 178-211, 2013.) and Oveis Gharan and Vondrák (Algorithmica 67(4): 472-497, 2013.) raised as an open question whether RA-MSP admits an O(1)-competitive algorithm even without knowing the matroid upfront. In this work, we answer this question affirmatively. Our result makes RA-MSP the first well-known MSP variant with an O(1)-competitive algorithm that does not need to know the underlying matroid upfront and without any restriction on the underlying matroid. Our approach is based on first approximately learning the rank-density curve of the matroid, which we then exploit algorithmically.

Explorable heap selection is the problem of selecting the nth smallest value in a binary heap. The key values can only be accessed by traversing through the underlying infinite binary tree, and the complexity of the algorithm is measured by the total distance traveled in the tree (each edge has unit cost). This problem was originally proposed as a model to study search strategies for the branch-and-bound algorithm with storage restrictions by Karp, Saks and Widgerson (FOCS '86), who gave deterministic and randomized $n·exp\left(O\left(\sqrt{logn}\right)\right)$ time algorithms using $O(log{\left(n\right)}^{2.5})$ and $O\left(\sqrt{logn}\right)$ space respectively. We present a new randomized algorithm with running time $O(nlog{\left(n\right)}^3)$ against an oblivious adversary using $O(logn)$ space, substantially improving the previous best randomized running time at the expense of slightly increased space usage. We also show an $\Omega (log(n)n/log(log\left(n\right)\left)\right)$ lower bound for any algorithm that solves the problem in the same amount of space, indicating that our algorithm is nearly optimal.

While there is strong evidence that maternal smallpox infection can cause foetal loss, it is not clear whether smallpox infections were a demographically important cause of stillbirths historically. In this paper, we use parish-level data from the Swedish Tabellverket data set for 1780-1839 to test the effect of smallpox on stillbirths quantitatively, analysing periods before and after the introduction of vaccination in 1802. We find that smallpox infection was not a major cause of stillbirths before 1820, because most women contracted smallpox as children and were therefore not susceptible during pregnancy. We do find a small, statistically significant effect of smallpox on stillbirths from 1820 to 1839, when waning immunity from vaccination put a greater share of pregnant women at risk of contracting smallpox. However, the reduced prevalence of smallpox in this period limited its impact on stillbirths. Thus, smallpox was not an important driver of historical stillbirth trends.

Optimization algorithms can see their local convergence rates deteriorate when the Hessian at the optimum is singular. These singularities are inescapable when the optima are non-isolated. Yet, under the right circumstances, several algorithms preserve their favorable rates even when optima form a continuum (e.g., due to over-parameterization). This has been explained under various structural assumptions, including the Polyak–Łojasiewicz condition, Quadratic Growth and the Error Bound. We show that, for cost functions which are twice continuously differentiable ((textrm{C}^2)), those three (local) properties are equivalent. Moreover, we show they are equivalent to the Morse–Bott property, that is, local minima form differentiable submanifolds, and the Hessian of the cost function is positive definite along its normal directions. We leverage this insight to improve local convergence guarantees for safe-guarded Newton-type methods under any (hence all) of the above assumptions. First, for adaptive cubic regularization, we secure quadratic convergence even with inexact subproblem solvers. Second, for trust-region methods, we argue capture can fail with an exact subproblem solver, then proceed to show linear convergence with an inexact one (Cauchy steps).

If a sparse semidefinite program (SDP), specified over (ntimes n) matrices and subject to m linear constraints, has an aggregate sparsity graph G with small treewidth, then chordal conversion will sometimes allow an interior-point method to solve the SDP in just (O(m+n)) time per-iteration, which is a significant speedup over the (varOmega (n^{3})) time per-iteration for a direct application of the interior-point method. Unfortunately, the speedup is not guaranteed by an O(1) treewidth in G that is independent of m and n, as a diagonal SDP would have treewidth zero but can still necessitate up to (varOmega (n^{3})) time per-iteration. Instead, we construct an extended aggregate sparsity graph (overline{G}supseteq G) by forcing each constraint matrix (A_{i}) to be its own clique in G. We prove that a small treewidth in (overline{G}) does indeed guarantee that chordal conversion will solve the SDP in (O(m+n)) time per-iteration, to (epsilon )-accuracy in at most (O(sqrt{m+n}log (1/epsilon ))) iterations. This sufficient condition covers many successful applications of chordal conversion, including the MAX-k-CUT relaxation, the Lovász theta problem, sensor network localization, polynomial optimization, and the AC optimal power flow relaxation, thus allowing theory to match practical experience.

Robust combinatorial optimization with budgeted uncertainty is one of the most popular approaches for integrating uncertainty into optimization problems. The existence of a compact reformulation for (mixed-integer) linear programs and positive complexity results give the impression that these problems are relatively easy to solve. However, the practical performance of the reformulation is quite poor when solving robust integer problems, in particular due to its weak linear relaxation. To overcome this issue, we propose procedures to derive new classes of valid inequalities for robust combinatorial optimization problems. For this, we recycle valid inequalities of the underlying deterministic problem such that the additional variables from the robust formulation are incorporated. The valid inequalities to be recycled may either be readily available model constraints or actual cutting planes, where we can benefit from decades of research on valid inequalities for classical optimization problems. We first demonstrate the strength of the inequalities theoretically, by proving that recycling yields a facet-defining inequality in many cases, even if the original valid inequality was not facet-defining. Afterwards, we show in an extensive computational study that using recycled inequalities can lead to a significant improvement of the computation time when solving robust optimization problems.

We consider a class of stochastic smooth convex optimization problems under rather general assumptions on the noise in the stochastic gradient observation. As opposed to the classical problem setting in which the variance of noise is assumed to be uniformly bounded, herein we assume that the variance of stochastic gradients is related to the “sub-optimality” of the approximate solutions delivered by the algorithm. Such problems naturally arise in a variety of applications, in particular, in the well-known generalized linear regression problem in statistics. However, to the best of our knowledge, none of the existing stochastic approximation algorithms for solving this class of problems attain optimality in terms of the dependence on accuracy, problem parameters, and mini-batch size. We discuss two non-Euclidean accelerated stochastic approximation routines—stochastic accelerated gradient descent (SAGD) and stochastic gradient extrapolation (SGE)—which carry a particular duality relationship. We show that both SAGD and SGE, under appropriate conditions, achieve the optimal convergence rate, attaining the optimal iteration and sample complexities simultaneously. However, corresponding assumptions for the SGE algorithm are more general; they allow, for instance, for efficient application of the SGE to statistical estimation problems under heavy tail noises and discontinuous score functions. We also discuss the application of the SGE to problems satisfying quadratic growth conditions, and show how it can be used to recover sparse solutions. Finally, we report on some simulation experiments to illustrate numerical performance of our proposed algorithms in high-dimensional settings.