Mathematical Programming最新文献

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.

Abstract

The Frank–Wolfe (FW) method, which implements efficient linear oracles that minimize linear approximations of the objective function over a fixed compact convex set, has recently received much attention in the optimization and machine learning literature. In this paper, we propose a new FW-type method for minimizing a smooth function over a compact set defined as the level set of a single difference-of-convex function, based on new generalized linear-optimization oracles (LO). We show that these LOs can be computed efficiently with closed-form solutions in some important optimization models that arise in compressed sensing and machine learning. In addition, under a mild strict feasibility condition, we establish the subsequential convergence of our nonconvex FW-type method. Since the feasible region of our generalized LO typically changes from iteration to iteration, our convergence analysis is completely different from those existing works in the literature on FW-type methods that deal with fixed feasible regions among subproblems. Finally, motivated by the away steps for accelerating FW-type methods for convex problems, we further design an away-step oracle to supplement our nonconvex FW-type method, and establish subsequential convergence of this variant. Numerical results on the matrix completion problem with standard datasets are presented to demonstrate the efficiency of the proposed FW-type method and its away-step variant.

We study piecewise affine policies for multi-stage adjustable robust optimization (ARO) problems with non-negative right-hand side uncertainty. First, we construct new dominating uncertainty sets and show how a multi-stage ARO problem can be solved efficiently with a linear program when uncertainty is replaced by these new sets. We then demonstrate how solutions for this alternative problem can be transformed into solutions for the original problem. By carefully choosing the dominating sets, we prove strong approximation bounds for our policies and extend many previously best-known bounds for the two-staged problem variant to its multi-stage counterpart. Moreover, the new bounds are—to the best of our knowledge—the first bounds shown for the general multi-stage ARO problem considered. We extensively compare our policies to other policies from the literature and prove relative performance guarantees. In two numerical experiments, we identify beneficial and disadvantageous properties for different policies and present effective adjustments to tackle the most critical disadvantages of our policies. Overall, the experiments show that our piecewise affine policies can be computed by orders of magnitude faster than affine policies, while often yielding comparable or even better results.

In generalized malleable scheduling, jobs can be allocated and processed simultaneously on multiple machines so as to reduce the overall makespan of the schedule. The required processing time for each job is determined by the joint processing speed of the allocated machines. We study the case that processing speeds are job-dependent (M ^{natural })-concave functions and provide a constant-factor approximation for this setting, significantly expanding the realm of functions for which such an approximation is possible. Further, we explore the connection between malleable scheduling and the problem of fairly allocating items to a set of agents with distinct utility functions, devising a black-box reduction that allows to obtain resource-augmented approximation algorithms for the latter.

Abstract

We investigate the concept of adjustability—the difference in objective values between two types of dynamic robust optimization formulations: one where (static) decisions are made before uncertainty realization, and one where uncertainty is resolved before (adjustable) decisions. This difference reflects the value of information and decision timing in optimization under uncertainty, and is related to several other concepts such as the optimality of decision rules in robust optimization. We develop a theoretical framework to quantify adjustability based on the input data of a robust optimization problem with a linear objective, linear constraints, and fixed recourse. We make very few additional assumptions. In particular, we do not assume constraint-wise separability or parameter nonnegativity that are commonly imposed in the literature for the study of adjustability. This allows us to study important but previously under-investigated problems, such as formulations with equality constraints and problems with both upper and lower bound constraints. Based on the discovery of an interesting connection between the reformulations of the static and fully adjustable problems, our analysis gives a necessary and sufficient condition—in the form of a theorem-of-the-alternatives—for adjustability to be zero when the uncertainty set is polyhedral. Based on this sharp characterization, we provide two efficient mixed-integer optimization formulations to verify zero adjustability. Then, we develop a constructive approach to quantify adjustability when the uncertainty set is general, which results in an efficient and tight poly-time algorithm to bound adjustability. We demonstrate the efficiency and tightness via both theoretical and numerical analyses.