The progressive decoupling algorithm solves problems of optimization by iteratively suppressing complications coming from linear linkage relations. Here its formulation is significantly extended and a sharper theory of convergence is provided. Minimization steps no longer have to be executed exactly, and proximal terms in broadened form, potentially for numerical preconditioning, can vary from one iteration to the next. Linear convergence, for which no results were previously available, is tied precisely to a condition of metric subregularity at the solution, with its rate given by a formula based on the modulus of that property. It is established that, for problems with piecewise linear-quadratic objectives, linear convergence is guaranteed, and that for convex general objectives the lack of it is, in a perturbational sense, rare. The methodology is shown to be extendable also to problem decomposition with nonconvex objectives, but with shortcomings so far in practical implementability.
Abstract This paper deals with the following question: Suppose that there exist an integer or a non-negative integer solution $$\varvec{x}$$ x to a system $$\varvec{A}\varvec{x}= \varvec{b}$$ A x = b , where the number of non-zero components of $$\varvec{x}$$ x is n . The target is, for a given natural number $$k < n$$ k < n , to approximate $$\varvec{b}$$ b with $$\varvec{A}\varvec{y}$$ A y where $$\varvec{y}$$ y is an integer or non-negative integer solution with at most k non-zero components. We establish upper bounds for this question in general. In specific cases, these bounds are tight. If we view the approximation quality as a function of the parameter k , then the paper explains why the quality of the approximation increases exponentially as k goes to n . This paper is a complete version of an extended abstract that appeared at the 26th International Conference on Integer Programming and Combinatorial Optimization (IPCO) [28].
In this paper, we propose a quasi-Newton method for solving smooth and monotone nonlinear equations, including unconstrained minimization and minimax optimization as special cases. For the strongly monotone setting, we establish two global convergence bounds: (i) a linear convergence rate that matches the rate of the celebrated extragradient method, and (ii) an explicit global superlinear convergence rate that provably surpasses the linear convergence rate after at most 𝒪(d) iterations, where d is the problem’s dimension. In addition, for the case where the operator is only monotone, we prove a global convergence rate of 𝒪(min{1/k,√(d)/k^1.25}) in terms of the duality gap. This matches the rate of the extragradient method when k = 𝒪(d^2) and is faster when k = (d^2) . These results are the first global convergence results to demonstrate a provable advantage of a quasi-Newton method over the extragradient method, without querying the Jacobian of the operator. Unlike classical quasi-Newton methods, we achieve this by using the hybrid proximal extragradient framework and a novel online learning approach for updating the Jacobian approximation matrices. Specifically, guided by the convergence analysis, we formulate the Jacobian approximation update as an online convex optimization problem over non-symmetric matrices, relating the regret of the online problem to the convergence rate of our method. To facilitate efficient implementation, we further develop a tailored online learning algorithm based on an approximate separation oracle, which preserves structures such as symmetry and sparsity in the Jacobian matrices.
Inspired by Shapiro et al. [74], we consider a stochastic optimal control (SOC) and Markov decision process (MDP) under simultaneous epistemic and aleatoric uncertainties using Bayesian composite risk (BCR) measures. The proposed BCR-SOC/MDP model evaluates the risk of stagewise cost via a two-layer framework: the inner risk measure tackles aleatoric uncertainty conditional on a latent environment parameter, while the outer risk measure deals with the epistemic uncertainty of the inner risk under the Bayesian posterior. The resulting time-varying risk evaluation induced by Bayesian updating enables an information-adaptive risk-sensitive decision framework. Unlike [74], our policies are allowed to depend explicitly on the posterior belief, reflecting that accumulated information about epistemic uncertainty can influence the assessment of future aleatoric uncertainty and, consequently, the decision maker’s actions [79]. The new modeling paradigm subsumes several classical SOC/MDP formulations, including risk-averse and distributionally robust SOC/MDPs as well as partially observed and Bayes-adaptive MDPs, and generates so-called preference robust SOC/MDP models. Moreover, we derive conditions under which the BCR-SOC/MDP model is well-defined, show that finite-horizon BCR-SOC/MDP models can be solved via dynamic programming, and extend the analysis to the infinite-horizon case. Under standard conditions, we establish asymptotic convergence of the optimal values and optimal policies as data accumulate, and provide quantitative error bounds for several representative classes of risk measures. To enhance computational tractability, we develop a hyper-parameter discretization approach for the posterior belief space. Finally, we carry out numerical tests on a spread betting problem and an inventory control problem, demonstrating the effectiveness of the proposed model and numerical schemes.
The single-source unsplittable flow (SSUF) problem asks to send flow from a common source to terminals with unrelated demands, each terminal being served through a single path. The classical SSUF objective is to minimize the violation of some given arc capacities. A seminal result of Dinitz, Garg, and Goemans showed that, whenever a fractional flow exists respecting the capacities, then there is an unsplittable one violating the capacities by at most the maximum demand. Goemans conjectured a natural cost version of the same result, where the unsplittable flow is required to be no more expensive than the fractional one. Intriguingly, there are arguably no non-trivial graph classes for which it is known to hold. We show that a slight weakening of it holds for planar graphs, by exploiting a connection to a highly structured discrepancy problem. Moreover, our techniques extend to simultaneous upper and lower bounds on the flow values. This affirmatively answers a conjecture of Morell and Skutella for planar SSUF. Finally, we show that our approach can be extended to general (non-planar) graphs with a capacity violation that depends on the genus.
