Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.27727

Forest Characterization of When SDP-RLT Strengthens RLT for Bilinear Terms

Hayato Watanabe·Takami Sato·Kazuhide Nakata·Ken Kobayashi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study two relaxations for bilinear terms in box-constrained variables: the reformulation-linearization technique (RLT) relaxation and the SDP-RLT relaxation, which combines RLT with semidefinite programming (SDP). We represent the pattern of bilinear terms by an undirected graph, with one vertex for each variable and one edge for each bilinear term. A point feasible for the RLT relaxation fixes only part of the lifted matrix of the SDP-RLT relaxation, leaving the remaining entries to be chosen subject to the diagonal RLT inequalities, and the two relaxations differ exactly when these entries cannot always be chosen so that the matrix becomes positive semidefinite. Formulating this as a positive semidefinite matrix completion problem, we show that the feasible regions of the two relaxations coincide if and only if the graph is a forest. We also measure how large the difference can be when the graph contains a complete bipartite subgraph: a bilinear objective given by a Hadamard matrix yields optimal values whose ratio grows with the size of that subgraph.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hayato Watanabe, Takami Sato, Kazuhide Nakata, Ken Kobayashi. 2026-10-20. Forest Characterization of When SDP-RLT Strengthens RLT for Bilinear Terms. https://arxiv.org/abs/2610.27727

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Accelerated Primal-Dual Proximal Gradient Splitting Methods for Convex-Concave Saddle-Point Problems

In this paper, based a novel primal-dual dynamical model with adaptive scaling parameters and Bregman divergences, we propose new accelerated primal-dual proximal gradient splitting methods for solving bilinear saddle-point problems with optimal nonergodic convergence rates. For the first, using the spectral analysis, we show that a naive extension of acceleration to a quadratic game is unstable. Motivated by this, we present an accelerated primal-dual gradient flow which combines acceleration with careful velocity correction. To work with non-Euclidean distances, we also equip our continuous model with general Bregman divergences and prove the exponential decay of a Lyapunov function. Then, new primal-dual splitting methods are developed based on proper semi-implicit Euler schemes of the continuous model, and the theoretical convergence rates are nonergodic and optimal with respect to the matrix norms, Lipschitz constants and convexity parameters. Moreover, we introduce efficient restart variants to further improve the proposed methods and provide some numerical results to validate the practical performance.

math.OC↗
arXiv2026

An Inexact Variable Metric Proximal Gradient-subgradient Algorithm for a Class of Fractional Optimization Problems

In this paper, we study a class of fractional optimization problems, in which the numerator of the objective is the sum of a convex function and a differentiable function with a Lipschitz continuous gradient, while the denominator is a nonsmooth convex function. This model captures ratio-type formulations arising in sparse learning and related applications. To address this class of problems, we propose an inexact variable metric proximal gradient-subgradient algorithm (iVPGSA), which, to the best of our knowledge, is the first inexact proximal algorithm specifically designed for such type of fractional problems. By incorporating a variable metric proximal term and allowing for approximate subproblem solutions under a flexible error criterion, the proposed algorithm is highly adaptable to a broader range of problems while achieving favorable computational efficiency. Under suitable assumptions, we establish that any accumulation point of the generated sequence is a critical point of the target problem. Moreover, we develop a Kurdyka-Łojasiewicz (KL)-based analysis framework, relying only on the classical KL property and its associated exponent, to prove the global convergence of the entire sequence and characterize its convergence rate, \textit{without} requiring a strict sufficient descent property. Our results clarify how the classical KL exponent and inexactness jointly influence the convergence rate. Finally, numerical experiments on the $\ell_1/\ell_2$ Lasso problem and the constrained $\ell_1/\ell_2$ sparse optimization problem demonstrate the computational advantages of the iVPGSA over existing representative algorithms.

math.OC↗
arXiv2026

Stackelberg stopping games

We study a Stackelberg variant of the classical discrete-time Dynkin game, in which Player 1 (the leader) commits to a stopping strategy first and Player 2 (the follower) responds optimally. This leader-follower structure induces an optimal control problem for the leader and gives rise to intrinsic time-inconsistency. We first clarify notions of precommitment and equilibrium strategies in the Stackelberg setting, and contrast them with the Nash equilibrium in the standard Dynkin game using a finite-horizon example. We then consider an infinite-horizon framework with a time-homogeneous Markov process on a general Polish state space. We characterize the leader's value function under randomized precommitment strategies and show that randomized exact equilibrium strategies may fail to exist via a counterexample. Motivated by this nonexistence phenomenon, we introduce an entropy-regularized Stackelberg stopping game. The regularization induces a continuous response rule and yields the existence of randomized regular equilibria. We further show that these regular equilibria induce epsilon-equilibria for the original Stackelberg stopping game when the regularization parameter is sufficiently small. In the finite-state setting, we also establish a limiting result as the regularization parameter converges to zero.

math.OC↗