arXiv subjects

Di Hou

Publications and source records attributed to Di Hou.

8 recordsLinked to original sources

arXiv2026

Sparsity-Cone SDP Relaxations and Applications to Variable Fixing for Sparse Quadratic Programs

Quadratic programs (QPs) with sparsity constraint are generally NP-hard, and their efficient global solution depends crucially on tractable tight convex relaxations. In this paper, we propose a sparsity-cone semidefinite programming (SC-SDP) relaxation for sparse (indefinite) QPs. Unlike standard SDP liftings, such as the SDP--RLT relaxation, which involve a $(2n+1)$-dimensional semidefinite matrix, the proposed SC-SDP formulation uses only a $(n+1)$-dimensional matrix together with a single sparsity-cone constraint $\mathcal{K}$ to handle the relaxation of the $\ell_0$-norm constraint. We prove that SC-SDP is equivalent in strength to the SDP--RLT relaxation. We further study the sparsity cone $\mathcal{K}$, deriving structural characterizations and showing that projection onto $\mathcal{K}$ can be computed efficiently via a one-dimensional subproblem. Building on the dual of SC-SDP, we derive explicit presolving mechanisms, including a dual-fixing rule for individual variables, a screening-cut rule for excluding larger support patterns, and a dual-refinement step for improving presolving certificates. To solve the resulting relaxation SC-SDP efficiently, we develop a two-phase Riemannian-based augmented Lagrangian method and exploits the structured projection subproblems. Numerical experiments on several classes of sparse QPs show that SC-SDP preserves the bound quality of SDP--RLT while offering substantial computational advantages and practically effective presolving capabilities.

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arXiv2026

On the efficient computation of proximal operators of affine-constrained nonconvex functions

Proximal operators with affine constraints arise in numerous models in nonconvex projection, composite optimization, and structured regularization. However, their efficient computation remains challenging due to the simultaneous presence of affine constraints and nonsmooth, possibly nonconvex objectives. In this work, we develop a unified dual-representability framework for analyzing and computing affine-constrained proximal mappings. Specifically, we introduce a multiplier inclusion formulation that connects the primal affine-constrained proximal problem to an unconstrained convex dual problem. Based on this formulation, we prove that, whenever the associated dual inclusion problem admits a solution, strong duality holds. For convex functions and a broad class of prox-regular nonconvex functions, we establish that dual representability holds under a simple subdifferential sum rule, and further develop a hierarchy of verifiable regularity conditions that guarantee this sum rule. In addition, we analyze the smoothness and strong convexity properties of the dual objective, providing a rigorous foundation that guarantees fast local convergence rates for efficient first- and second-order methods. Numerical experiments demonstrate that the proposed dual reformulation enables the reliable computation of globally optimal solutions for a range of large-scale nonconvex proximal and projection problems using existing convex optimization solvers.

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arXiv2025

A Low-rank Augmented Lagrangian Method for Polyhedral-SDP and Moment-SOS Relaxations of Polynomial Optimization

Polynomial optimization problems (POPs) can be reformulated as geometric convex conic programs, as shown by Kim, Kojima, and Toh (SIOPT 30:1251-1273, 2020), though such formulations remain NP-hard. In this work, we prove that several well-known relaxations can be unified under a common polyhedral-SDP framework, which arises by approximating the intractable cone by tractable intersections of polyhedral cones with the positive semidefinite matrix cone. Although effective in providing tight lower bounds, these relaxations become computationally expensive as the number of variables and constraints grows at the rate of $\Omega(n^{2\tau})$ with the relaxation order $\tau$. To address this challenge, we propose RiNNAL-POP, a low-rank augmented Lagrangian method (ALM) tailored to solve large-scale polyhedral-SDP relaxations of POPs. To efficiently handle the $\Omega(n^{2\tau})$ nonnegativity and consistency constraints, we design a tailored projection scheme whose computational cost scales linearly with the number of variables. In addition, we identify a hidden facial structure in the polyhedral-SDP relaxation, which enables us to eliminate a large number of linear constraints by restricting the matrix variable to affine subspaces corresponding to exposed faces of the semidefinite cone. The latter enables us to efficiently solve the factorized ALM subproblems over the affine subspaces. At each ALM iteration, we additionally carry out a single projected gradient step with respect to the original matrix variable to automatically adjust the rank and escape from spurious local minima when necessary. We also extend our RiNNAL-POP algorithmic framework to solve moment-SOS relaxations of POPs. Extensive numerical experiments on various benchmark problems demonstrate the robustness and efficiency of RiNNAL-POP in solving large-scale polyhedral-SDP relaxations.

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arXiv2025

RiNNAL+: a Riemannian ALM Solver for SDP-RLT Relaxations of Mixed-Binary Quadratic Programs

Doubly nonnegative (DNN) relaxation usually provides a tight lower bound for a mixed-binary quadratic program (MBQP). However, solving DNN problems is challenging because: (1) the problem size is $\Omega((n+l)^2)$ for an MBQP with $n$ variables and $l$ inequality constraints, and (2) the rank of optimal solutions cannot be estimated a priori due to the absence of theoretical bounds. In this work, we propose RiNNAL+, a Riemannian augmented Lagrangian method (ALM) for solving DNN problems. We prove that the DNN relaxation of an MBQP, with matrix dimension $(n+l+1)$, is equivalent to the SDP-RLT relaxation (based on the reformulation-linearization technique) with a smaller matrix dimension $(n+1)$. In addition, we develop a hybrid method that alternates between two phases to solve the ALM subproblems. In phase one, we apply low-rank matrix factorization and random perturbation to transform the feasible region into a lower-dimensional manifold so that we can use the Riemannian gradient descent method. In phase two, we apply a single projected gradient step to update the rank of the underlying variable and escape from spurious local minima arising in the first phase if necessary. To reduce the computation cost of the projected gradient step, we develop pre-processing and warm-start techniques for acceleration. Unlike traditional rank-adaptive methods that require extensive parameter tuning, our hybrid method requires minimal tuning. Extensive experiments confirm the efficiency and robustness of RiNNAL+ in solving various classes of large-scale DNN problems.

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arXiv2025

A low-rank augmented Lagrangian method for doubly nonnegative relaxations of mixed-binary quadratic programs

Doubly nonnegative (DNN) programming problems are known to be challenging to solve because of their huge number of $\Omega(n^2)$ constraints and $\Omega(n^2)$ variables. In this work, we introduce RNNAL, a method for solving DNN relaxations of large-scale mixed-binary quadratic programs by leveraging their solutions' possible low-rank property. RNNAL is a globally convergent Riemannian augmented Lagrangian method (ALM) that penalizes the nonnegativity and complementarity constraints while preserving all other constraints as an algebraic variety. After applying the low-rank decomposition to the ALM subproblem, its feasible region becomes an algebraic variety with favorable geometric properties. Our low-rank decomposition model is different from the standard Burer-Monteiro (BM) decomposition model in that we make the key improvement to equivalently reformulate most of the quadratic constraints after the BM decomposition into fewer and more manageable affine constraints. This modification is also important in helping us to alleviate the violation of Slater's condition for the primal DNN problem. Moreover, we make the crucial step to show that the metric projection onto the algebraic variety, although non-convex, can be transformed into a solvable convex optimization problem under certain regularity conditions, which can be ensured by a constraint-relaxation strategy. RNNAL is able to handle general semidefinite programming (SDP) with additional polyhedral cone constraints, thus serving as a prototype algorithm for solving general DNN problems. Numerous numerical experiments are conducted to validate the efficiency of the proposed RNNAL method.

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arXiv2023

A Sparse Smoothing Newton Method for Solving Discrete Optimal Transport Problems

The discrete optimal transport (OT) problem, which offers an effective computational tool for comparing two discrete probability distributions, has recently attracted much attention and played essential roles in many modern applications. This paper proposes to solve the discrete OT problem by applying a squared smoothing Newton method via the Huber smoothing function for solving the corresponding KKT system directly. The proposed algorithm admits appealing convergence properties and is able to take advantage of the solution sparsity to greatly reduce computational costs. Moreover, the algorithm can be extended to solve problems with similar structures including the Wasserstein barycenter (WB) problem with fixed supports. To verify the practical performance of the proposed method, we conduct extensive numerical experiments to solve a large set of discrete OT and WB benchmark problems. Our numerical results show that the proposed method is efficient compared to state-of-the-art linear programming (LP) solvers. Moreover, the proposed method consumes less memory than existing LP solvers, which demonstrates the potential usage of our algorithm for solving large-scale OT and WB problems.

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arXiv2015

CHAOS: Accurate and Realtime Detection of Aging-Oriented Failure Using Entropy

Even well-designed software systems suffer from chronic performance degradation, also named "software aging", due to internal (e.g. software bugs) and external (e.g. resource exhaustion) impairments. These chronic problems often fly under the radar of software monitoring systems before causing severe impacts (e.g. system failure). Therefore it's a challenging issue how to timely detect these problems to prevent system crash. Although a large quantity of approaches have been proposed to solve this issue, the accuracy and effectiveness of these approaches are still far from satisfactory due to the insufficiency of aging indicators adopted by them. In this paper, we present a novel entropy-based aging indicator, Multidimensional Multi-scale Entropy (MMSE). MMSE employs the complexity embedded in runtime performance metrics to indicate software aging and leverages multi-scale and multi-dimension integration to tolerate system fluctuations. Via theoretical proof and experimental evaluation, we demonstrate that MMSE satisfies Stability, Monotonicity and Integration which we conjecture that an ideal aging indicator should have. Based upon MMSE, we develop three failure detection approaches encapsulated in a proof-of-concept named CHAOS. The experimental evaluations in a Video on Demand (VoD) system and in a real-world production system, AntVision, show that CHAOS can detect the failure-prone state in an extraordinarily high accuracy and a near 0 Ahead-Time-To-Failure (ATTF). Compared to previous approaches, CHAOS improves the detection accuracy by about 5 times and reduces the ATTF even by 3 orders of magnitude. In addition, CHAOS is light-weight enough to satisfy the realtime requirement.

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arXiv2014

Bio-inspired Mechanism and Model Exploration of Software Aging

Software systems situated in network environment may experience performance degradation, availability decrease and even crash during long time running, which is called software aging. This phenomenon has been studied for more than 15 years, but most of the literatures studied software as a black box, none of them uncovered the fundamental and widely accepted mechanism of software aging as far as we know. Through analyzing the characteristics between biological aging and software aging, we find some interesting common points and bridge the gap between these two seemingly unrelated phenomena. The free radical aging theory in biological studies is also applicative to explore the mechanism and model of software aging. This paper finds an equivalent concept named `software free radical' in software aging to free radical in biological aging. In our study, the accumulation of `software free radical' is a root cause of software aging. Using the free radical modeling methodology in biological aging, we give a model for describing the kinetic of software aging based on feedback loops. Although this paper doesn't give enough theoretical proof of the modeling method, the practical results show that the feedback loop model can describe the kinetic of software aging precisely. To further validate the aging mechanism, we propose several software rejuvenation strategies focusing on cleaning the `software free radical'. The results show that software aging can be mitigated effectively by strengthening negative feedback loop or weakening positive feedback loop. This paper is the first try to answer the question `How software ages' through interdisciplinary studies. Leveraging the conclusions in this paper, people can design better software systems or keep their systems at a high performance level during long time running.

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