Mathematical Programming最新文献

Abstract

We present an auction algorithm using multiplicative instead of constant weight updates to compute a ((1-varepsilon )) -approximate maximum weight matching (MWM) in a bipartite graph with n vertices and m edges in time (O(mvarepsilon ^{-1})) , beating the running time of the fastest known approximation algorithm of Duan and Pettie [JACM ’14] that runs in (O(mvarepsilon ^{-1}log varepsilon ^{-1})) . Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a ((1-varepsilon )) -approximate maximum weight matching under (1) one-sided vertex deletions (with incident edges) and (2) one-sided vertex insertions (with incident edges sorted by weight) to the other side. The total time time used is (O(mvarepsilon ^{-1})) , where m is the sum of the number of initially existing and inserted edges.

Abstract

We show how to round any half-integral solution to the subtour-elimination relaxation for the TSP, while losing a less-than (-)  1.5 factor. Such a rounding algorithm was recently given by Karlin, Klein, and Oveis Gharan based on sampling from max-entropy distributions. We build on an approach of Haddadan and Newman to show how sampling from the matroid intersection polytope, combined with a novel use of max-entropy sampling, can give better guarantees.

We develop new tools to study landscapes in nonconvex optimization. Given one optimization problem, we pair it with another by smoothly parametrizing the domain. This is either for practical purposes (e.g., to use smooth optimization algorithms with good guarantees) or for theoretical purposes (e.g., to reveal that the landscape satisfies a strict saddle property). In both cases, the central question is: how do the landscapes of the two problems relate? More precisely: how do desirable points such as local minima and critical points in one problem relate to those in the other problem? A key finding in this paper is that these relations are often determined by the parametrization itself, and are almost entirely independent of the cost function. Accordingly, we introduce a general framework to study parametrizations by their effect on landscapes. The framework enables us to obtain new guarantees for an array of problems, some of which were previously treated on a case-by-case basis in the literature. Applications include: optimizing low-rank matrices and tensors through factorizations; solving semidefinite programs via the Burer–Monteiro approach; training neural networks by optimizing their weights and biases; and quotienting out symmetries.

A key problem in mathematical imaging, signal processing and computational statistics is the minimization of non-convex objective functions that may be non-differentiable at the relative boundary of the feasible set. This paper proposes a new family of first- and second-order interior-point methods for non-convex optimization problems with linear and conic constraints, combining logarithmically homogeneous barriers with quadratic and cubic regularization respectively. Our approach is based on a potential-reduction mechanism and, under the Lipschitz continuity of the corresponding derivative with respect to the local barrier-induced norm, attains a suitably defined class of approximate first- or second-order KKT points with worst-case iteration complexity (O(varepsilon ^{-2})) (first-order) and (O(varepsilon ^{-3/2})) (second-order), respectively. Based on these findings, we develop new path-following schemes attaining the same complexity, modulo adjusting constants. These complexity bounds are known to be optimal in the unconstrained case, and our work shows that they are upper bounds in the case with complicated constraints as well. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained non-convex optimization problems.

We present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider (i) classes of optimization problems of finite-sum form with (possibly strongly) convex and possibly smooth functional components, (ii) first-order methods that can be written as a linear system on state-space form in feedback interconnection with the subdifferentials of the functional components of the objective function, and (iii) quadratic Lyapunov inequalities that can be used to draw convergence conclusions. We present a necessary and sufficient condition for the existence of a quadratic Lyapunov inequality within a predefined class of Lyapunov inequalities, which amounts to solving a small-sized semidefinite program. We showcase our methodology on several first-order methods that fit the framework. Most notably, our methodology allows us to significantly extend the region of parameter choices that allow for duality gap convergence in the Chambolle–Pock method when the linear operator is the identity mapping.

We study a mixed-integer set (mathcal {S}:={(x,t) in {0,1}^n times mathbb {R}: f(x) ge t}) arising in the submodular maximization problem, where f is a submodular function defined over ({0,1}^n). We use intersection cuts to tighten a polyhedral outer approximation of (mathcal {S}). We construct a continuous extension (bar{textsf{F}}_f) of f, which is convex and defined over the entire space (mathbb {R}^n). We show that the epigraph ({{,textrm{epi},}}(bar{textsf{F}}_f)) of (bar{textsf{F}}_f) is an (mathcal {S})-free set, and characterize maximal (mathcal {S})-free sets containing ({{,textrm{epi},}}(bar{textsf{F}}_f)). We propose a hybrid discrete Newton algorithm to compute an intersection cut efficiently and exactly. Our results are generalized to the hypograph or the superlevel set of a submodular-supermodular function over the Boolean hypercube, which is a model for discrete nonconvexity. A consequence of these results is intersection cuts for Boolean multilinear constraints. We evaluate our techniques on max cut, pseudo Boolean maximization, and Bayesian D-optimal design problems within a MIP solver.

We show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of the lattice isomorphism problem, which is known to be in the complexity class SZK. We achieve this by deploying a result on characteristic vectors by Elkies that gained attention in the context of 4-manifolds and Seiberg-Witten equations, but seems rather unnoticed in the algorithmic lattice community. On the way, we also show that for a given Gram matrix (G in mathbb {Q}^{n times n}), we can efficiently find a rational lattice that is embedded in at most four times the initial dimension n, i.e. a rational matrix (B in mathbb {Q}^{4n times n}) such that (B^intercal B = G).

Consider the closed convex hull K of a monomial curve given parametrically as ((t^{m_1},ldots ,t^{m_n})), with the parameter t varying in an interval I. We show, using constructive arguments, that K admits a lifted semidefinite description by (mathcal {O}(d)) linear matrix inequalities (LMIs), each of size (leftlfloor frac{n}{2} rightrfloor +1), where (d= max {m_1,ldots ,m_n}) is the degree of the curve. On the dual side, we show that if a univariate polynomial p(t) of degree d with at most (2k+1) monomials is non-negative on ({mathbb {R}}_+), then p admits a representation (p = t^0 sigma _0 + cdots + t^{d-k} sigma _{d-k}), where the polynomials (sigma _0,ldots ,sigma _{d-k}) are sums of squares and (deg (sigma _i) le 2k). The latter is a univariate positivstellensatz for sparse polynomials, with non-negativity of p being certified by sos polynomials whose degree only depends on the sparsity of p. Our results fit into the general attempt of formulating polynomial optimization problems as semidefinite problems with LMIs of small size. Such small-size descriptions are much more tractable from a computational viewpoint.

We consider the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm, also known as the first-order Lagrangian method, for constrained optimization problems involving a smooth strongly convex cost and smooth convex constraints. We prove that, for every given compact set of initial conditions, there always exists a sufficiently small stepsize guaranteeing exponential stability of the optimal primal-dual solution of the problem with a domain of attraction including the initialization set. Inspired by the analysis of nonlinear oscillators, the stability proof is based on a non-quadratic Lyapunov function including a nonlinear cross term.

We establish a general condition on the cost function to obtain uniqueness and Monge solutions in the multi-marginal optimal transport problem, under the assumption that a given collection of the marginals are absolutely continuous with respect to local coordinates. When only the first marginal is assumed to be absolutely continuous, our condition is equivalent to the twist on splitting sets condition found in Kim and Pass (SIAM J Math Anal 46:1538–1550, 2014; SIAM J Math Anal 46:1538–1550, 2014). In addition, it is satisfied by the special cost functions in our earlier work (Pass and Vargas-Jiménez in SIAM J Math Anal 53:4386–4400, 2021; Monge solutions and uniqueness in multi-marginal optimal transport via graph theory. arXiv:2104.09488, 2021), when absolute continuity is imposed on certain other collections of marginals. We also present several new examples of cost functions which violate the twist on splitting sets condition but satisfy the new condition introduced here, including a class of examples arising in robust risk management problems; we therefore obtain Monge solution and uniqueness results for these cost functions, under regularity conditions on an appropriate subset of the marginals.