Table of Content

30 June 2026, Volume 14 Issue 2

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Linear Convergence of ISTA and FISTA

Bo-Wen Li, Bin Shi, Ya-Xiang Yuan

2026, 14(2):  345-363.  doi:10.1007/s40305-024-00561-0

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In this paper, we revisit the iterative shrinkage-thresholding algorithm (ISTA) for solving linear inverse problems with sparse representation, commonly found in signal and image processing. We conduct numerical experiments and observe that the convergence behavior of ISTA tends to be linear instead of logarithmic, resulting in a flat approximation on the logarithmic-scale ordinate. Through meticulous observations, we find that assuming the smooth part to be strongly convex, rather than just convex, is more reasonable for the least-square model, even when dealing with ill-conditioned image matrices. We also improve the pivotal inequality for composite optimization by considering the smooth part to be strongly convex instead of general convex, which was first introduced in Li et al. (Proximal Subgradient Norm Minimization of ISTA and FISTA, arXiv:2211.01610, 2022). By utilizing this pivotal inequality, we extend linear convergence to composite optimization, both in terms of the objective value and the square of the proximal subgradient norm. To support our findings, we use a simple ill-conditioned matrix that allows for easy computation of the singular values, instead of the original blur matrix. The new numerical experiment demonstrates that the fast iterative shrinkage-thresholding algorithm for strongly convex functions has a faster linear convergence rate compared to ISTA. Furthermore, based on the tighter pivotal inequality, we also generalize the faster linear convergence rate to composite optimization, considering both the objective value and the square of the proximal subgradient norm. We achieve this by leveraging a well-constructed Lyapunov function with slight modifications and the phase-space representation based on the high-resolution ordinary differential equation framework from the implicit-velocity scheme.

On the Analysis of Model-Free Methods for the Linear Quadratic Regulator

Ze-Yu Jin, Johann Michael Schmitt, Zai-Wen Wen

2026, 14(2):  364-396.  doi:10.1007/s40305-024-00546-z

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Many reinforcement learning methods achieve great success in practice but lack theoretical foundation. In this paper, we study the convergence analysis of the model-free methods for the Linear Quadratic Regulator by treating the underlying system as a black box. The global linear convergence properties and sample complexities are established for several popular algorithms such as the temporal differences (TD)-learning method, the policy gradient algorithm, and the actor-critic (AC) algorithm. Our analysis shows that the actor-critic algorithm can reduce the sample complexity compared with the policy gradient algorithm. Although our analysis is still preliminary, it still explains the benefit of AC algorithm in a certain sense.

Approximation Algorithms for k-Submodular Maximization Subject to a Knapsack Constraint

Hao Xiao, Qian Liu, Yang Zhou, Min Li

2026, 14(2):  397-412.  doi:10.1007/s40305-024-00539-y

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In this paper, we study the problem of maximizing $k$-submodular functions subject to a knapsack constraint. For monotone objective functions, we present a $\frac{1}{2}\left(1-\mathrm{e}^{-2}\right) \approx$ 0.432 greedy approximation algorithm, improving the previous best-known ratio $\frac{1}{2}(1- \left.\mathrm{e}^{-1}\right) \approx 0.316$. We also consider the non-monotone knapsack problem and provide two algorithms. The first is a greedy-type combinatorial algorithm with approximation ratio $\frac{1}{3}\left(1-\mathrm{e}^{-3}\right) \approx 0.317$, while the second is a multilinear-extension-based algorithm with approximation ratio $\frac{1}{3}-\varepsilon$, where $\varepsilon>0$.

Forecasting VaR and Returns Distribution Using the Real-Time GARCH Models with Standardized Two-Sided Lindley Distribution

Zhi-Min Wu, Guang-Hui Cai

2026, 14(2):  413-451.  doi:10.1007/s40305-024-00564-x

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The Real-time GARCH models which see the conditional volatility of financial returns as a mixture of past and current information have been confirmed as more effective models on modeling and forecasting financial volatility and risks, thus attracting abundant attention. However, the existing Real-time GARCH models only assume innovations follow the standard Normal distribution and fail to take into account the asymmetry of its distribution. In this paper, we consider the standardized two-sided Lindley (STSL) distribution as the distribution of innovations and then propose the Real-time GARCH-STSL models to estimate and forecast conditional density and VaR of financial returns. Besides, some important properties of the new models are discussed including the conditional density function and VaR of returns, the conditions of weak stationarity, and the maximum likelihood estimation method. Furthermore, Monte Carlo simulations are also conducted to observe the asymptotic performance of these proposed models. Finally, empirical results show that the new models are superior to competitors considering the Normal distribution for innovations in terms of the in-sample empirical fitting, out-of-sample multi-step-ahead returns density and extreme VaR predictions.

Time-Consistent Strategies for DC Pension Plans with the Return of Premiums Clauses in Stochastic Environments

Xiao-Jia Li, Hao Chang, Xing-Jiang Chen

2026, 14(2):  452-488.  doi:10.1007/s40305-024-00554-z

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This paper studies a defined contribution (DC) pension plan with stochastic interest rate and stochastic volatility in a mean-variance framework. In the DC pension plans, a part of premiums are often returned to the members who died during the accumulation phase in order to protect the rights of the plan members, while the survival members can share the difference between the accumulated wealth and the returned premiums. From the survival members’ point of view, they hope to maximize the expectation of terminal wealth and to minimize the volatility of terminal wealth. In this paper, we formulate this problem as a continuous-time mean-variance model in a game theoretical framework. By using the extended stochastic optimal control theory, we obtain the time-consistent equilibrium strategy and the efficient frontier in explicit form. In addition, the characteristics of the demand for stocks and bonds are discussed and some special cases are also derived in detail. Our theoretical results display that the characteristic and structure of the time-consistent equilibrium strategy and the efficient frontier are considerably different from those obtained with constant interest rate and constant volatility. Finally, our results are illustrated by a numerical simulation and some economic implications are revealed.

A Best Possible Online Algorithm for Single-Machine Scheduling with Non-delayed Processing Constraint and Bounded Delivery Times

Yan-Jun Qu, Ji Tian, Ru-Yan Fu, Kai-Jie Ge

2026, 14(2):  489-501.  doi:10.1007/s40305-024-00541-4

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We investigate the problem of the online scheduling with non-delayed processing constraint and bounded delivery times on a single machine where jobs arrive over time. The non-delayed processing (NDP) constraint means that the available jobs cannot be delayed for processing when a machine is idle. Each job's information, such as processing time and delivery time, becomes known at its release time. Once the processing of a job is completed on the machine, we deliver it to the destination by a vehicle. The objective is to minimize the time by which all jobs have been delivered. In this paper, we assume that all jobs have bounded delivery times, i.e., $\beta q_j \leqslant p_j$ for each job $J_j$, where $p_j$ and $q_j$ denote the processing time and delivery time of $J_j$, respectively, and $\beta$ is a given non-negative real number. We present a best possible online algorithm with a competitive ratio of $1+\frac{1}{\beta+2}$.

An Alternating Proximal Gradient Algorithm for Nonsmooth Nonconvex-Linear Minimax Problems with Coupled Linear Constraints

Hui-Ling Zhang, Zi Xu

2026, 14(2):  502-520.  doi:10.1007/s40305-024-00550-3

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In this paper, we propose an alternating proximal gradient algorithm for solving nonsmooth nonconvex-linear minimax problems with coupled linear constraints, which have attracted wide attention in machine learning, signal processing and many other fields in recent years. The iteration complexity of the proposed algorithm is proved to be $\mathcal{O}\left(\varepsilon^{-3}\right)$ to reach an $\varepsilon$-stationary point. To our knowledge, this is the first algorithm with iteration complexity guarantee for solving nonsmooth nonconvex-linear minimax problems with coupled linear constraints.

On the O(1/K²) Ergodic Convergence of ADMM with Dual Step Size from 0 to 2

Tao Zhang

2026, 14(2):  521-532.  doi:10.1007/s40305-024-00568-7

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We initially establish the O(1/K²) (K represents the number of iterations) ergodic convergence rate of the alternating direction method of multipliers (ADMM) with dual step size from 0 to 2 and dynamically updating the penalty parameter. The convergence rate is derived under the assumption that the two objective functions involved are linear and strongly convex, respectively. In contrast, there is no convergence rate analysis for ADMM with dual step size ranging from 0 to 2 and dynamically updating the penalty parameter. Furthermore, we analyze the convergence to the solution.

On Error Bounds for the Extended Vertical Linear Complementarity Problem

Shi-Liang Wu, He-Hui Wang

2026, 14(2):  533-546.  doi:10.1007/s40305-024-00573-w

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In this paper, by using a general equivalent form of the minimum function, some new error bounds of the extended vertical linear complementarity problem under appropriate conditions are obtained, which cover some existing results. Not only that, these new error bounds skillfully avoid the inconvenience caused by the row rearrangement technique for error bounds.

The Unbounded Parallel-Batching Bicriteria Scheduling with Two-Component Jobs

Cheng He, Jing Wu, Hao Lin, Yuan Zhang, Yan Zhao

2026, 14(2):  547-564.  doi:10.1007/s40305-024-00545-0

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This paper studies a bicriteria scheduling problem on a parallel-batching machine to minimizemaximum cost andmakespan simultaneously. Each job has two components: standard component and specific component. Standard components are processed in batches. Specific components are processed individually. The processing order of two components of a job has no constraint. A job is completed only when its two components are completed. For the simultaneous optimization scheduling problem, we design an O(n⁴)-time algorithm.

Expected Residual Minimization Method for a Class of Stochastic Tensor Variational Inequalities

Jian-Xun Liu, Zhao-Feng Lan, Sheng-Jie Li

2026, 14(2):  565-590.  doi:10.1007/s40305-024-00569-6

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This paper considers the expected residual minimization (ERM) formulation for a class of stochastic tensor variational inequalities (STVI) where the involved set contains 0. Initially, we derive some theoretical results regarding the H-eigenvalues of tensors and formulate a class of stochastic multi-person nonoperative games as an STVI. Subsequently, we transform the STVI into an ERM problem by using the regularized gap function and explore the properties of the object function. Furthermore, we use the quasi-Monte Carlo method to address the ERM problem and conduct convergence analysis. Ultimately, we conduct numerical experiments to validate our theoretical findings.

An Approximation Theorem and Generic Uniqueness of Weakly Pareto-Nash Equilibrium for Multiobjective Population Games

Hua-Xin Chen, Wen-Sheng Jia

2026, 14(2):  591-602.  doi:10.1007/s40305-024-00565-w

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We mainly study an approximation theorem and generic uniqueness of weakly ParetoNash equilibrium (PNE) for multiobjective population games. First, we define the concept of approximate weakly PNE of multiobjective population games. Then, an approximation theorem is proved under very mild conditions. Furthermore, we prove the generic uniqueness of weakly PNE for multiobjective population games by establishing a function space. Finally, we obtain the generic uniqueness of weakly PNE for multiobjective population games in the case of function perturbation. The above results are new and improve the related results of the recent literature.

Does Symbiosis Matter in Marine Innovation Ecosystems? An Examination Using Lotka-Volterra Model

Qian Pan, Pei-De Liu, Bao-Ying Zhu

2026, 14(2):  603-631.  doi:10.1007/s40305-025-00637-5

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The evolutionary mode of innovation agents within the marine innovation ecosystem (MIE) holds significant implications for the formulation of marine innovation policies and the facilitation of high-quality development within the marine economy. The aim of this paper is to investigate the symbiosis modes of innovation agents that can drive the innovative energy of MIE. First, considering the marine innovation enterprise population (EP) and marine innovation academic research institution population (AP) in China as innovation agents, we propose the concept of evolution of two agents in MIE based on symbiosis theory and construct a two-agent evolutionary model based on the Lotka-Volterra model. Then, we explore the symbiosis mode of the two agents in the MIE using the innovation size data of EP and AP in the MIE from 1997 to 2022. Furthermore, considering the incompleteness of the current data, the influence mechanism of EP and AP and the evolution path in different environments are explored by computer simulation technology. The findings indicate that mutualistic symbiosis among innovation agents is more likely to drive the MIE, as opposed to independence among these agents. Finally, some policy implications are suggested on how to keep the mutualistic symbiosis mode in the process of symbiosis dy

An Alternating Gradient Projection Algorithm with Momentum for Nonconvex-Concave Minimax Problems

Jue-You Li, Tao Xie

2026, 14(2):  632-652.  doi:10.1007/s40305-024-00540-5

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The growing interest in addressing minimax optimization problem has been fueled by recent applications in machine learning. Although extensively studied in the convex-concave regime, where a global solution can be efficiently computed, this paper delves into the minimax problem within the nonconvex-concave setup. We propose an alternating gradient projection algorithm with momentum (M-AGP), belonging to single-loop algorithms that not only are easier to implement but also require only the computation of gradient projection updates. We demonstrate that the proposed algorithm identifies an ε-stationary point of the nonconvex-strongly concave minimax problem in O(ε^-2) iterations, representing the best-known rate in the literature. Finally, we utilize two test problems, namely robust nonlinear regression and an image classification problem, to showcase the efficacy of the proposed algorithm.

Single-Machine Online Scheduling with Non-delayed Processing Constraint and Deterioration Effect in the Steel Rolling Process

Lan-Meng Meng, Ran Ma, Yu-Zhong Zhang

2026, 14(2):  653-670.  doi:10.1007/s40305-024-00552-1

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This paper focuses on the online production scheduling of the steel rolling processes in a single-machine environment with objective to minimize the maximum delivery completion time of all of the jobs, subject to the job deterioration effect and nondelayed processing constraint. The deterioration is reflected in the processing time of the job. Specifically, the job's processing time is a linear function of its start time and can be denoted as $p_j=a_j(A+B t)$, where $A>0, B>0$ and $a_j>0$ represents the job's processing deterioration rate. For this problem, we firstly show that the competitive ratio of any deterministic online algorithm is not less than $1+B a_{\text {max }}$. Then, we design an online algorithm called Modified-Largest Delivery Time (M-LDT) and show the algorithm M-LDT is $(1+\alpha)\left(1+B a_{\max }\right)$-competitive, where $\alpha$ is the positive root of $\alpha^2-\alpha-1=0$. Finally, we use the combination of graphs and tables to give the simulation results of multiple instances, and then verify the correctness and effectiveness of our proposed online algorithm.

An Improved Space Semi-Streaming Algorithm for Submodular Maximization under b-Matching Constraint

Shu-Yu Bao, Qing-Qin Nong, Su-Ning Gong

2026, 14(2):  671-686.  doi:10.1007/s40305-024-00543-2

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Let $G=(V, E)$ be a multi-graph without self-loops where each vertex $v \in V$ has a capacity $b_v \in \mathbb{Z}_{+}$. A $b$-matching is a subset of edges $M \subseteq E$ such that each vertex $v$ is incident with at most $b_v$ edges in $M$ and each edge can be selected at most once. We study monotone submodular maximization subject to $b$-matching constraint (Monotone $\mathrm{MS}_{\mathrm{B}} \mathrm{M}$ ) in the semi-streaming model. There is a trade-off between the space complexity and the approximation ratio of the algorithms of the Monotone $\mathrm{MS}_{\mathrm{B}} \mathrm{M}$. In a recent breakthrough, Levin and Wajc give a 5.828 approximate algorithm for this problem and it needs to store $O\left(\sum_{v \in V} b_v \log n\right)$ edges in memory. In this paper, we present an improved space algorithm that contains $O\left(\sum_{v \in V} \max \left\{b_v \cdot \log \left(\frac{b_v}{\varepsilon}\right), b_v^{\frac{3}{2}} \cdot \log \left(\frac{1}{\varepsilon}\right)\right\}\right)$ edges in memory by introducing queues with size limits for the vertices and its approximation ratio is 6.2.

Approximation Scheme for the Single-Client Capacitated Facility Location Problem With Operational Cost Budget Constraint

Lu Chen, Zheng Chen, Long Wan, Wen-Chang Luo

2026, 14(2):  687-699.  doi:10.1007/s40305-024-00555-y

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We investigate the single-client capacitated facility location problem with operational cost budget constraint. Specifically speaking, we are given a single client with demand, a set of potential facilities with capacities, and a positive integer specifying the operational cost budget. For each open facility, one should pay its opening cost and operational cost. For each open facility to service the client, one should pay the service cost. The objective is to open enough facilities serving the client’s demand and satisfying the capacity and operational cost budget constraints while minimizing the sum of opening and service costs. In this paper, based on dynamic programming and sparse techniques, we derive a fully polynomial time approximation scheme (FPTAS) with violating the operational cost budget constraint at most an arbitrarily small factor or claim that there is no feasible solution.

On Smoothing l₁ Exact Penalty Function for Nonlinear Constrained Optimization Problems

Yu-Fei Ren, You-Lin Shang

2026, 14(2):  700-718.  doi:10.1007/s40305-024-00566-9

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The penalty function method is a significant method for solving nonlinear constrained optimization problems (COP). In this paper, a new quadratic continuous differentiable smooth penalty function is proposed for the l₁ exact penalty function. The error estimations between the objective function values of the smooth penalty problem, the penalty problem and the original problem are also studied. Furthermore, based on the smoothed penalty function, an algorithm for solving COP is proposed, and the convergence of the algorithm is proved. Finally, several numerical examples are given to illustrate the effectiveness of the proposed algorithm.

Direct Product Multicommodity Max-Concurrent-Flow Min-Sparse-Cut Theorem

Rui Guan, Dong-Yue Liang, Wei Wang, Wei-Hua Yang

2026, 14(2):  719-729.  doi:10.1007/s40305-024-00567-8

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We extend the max-concurrent-flow min-sparse-cut theorem from product multicommodity to direct product multicommodity. We prove that $\Theta(\log k)$ is the tight gap between the max-concurrent-flow and the min-sparse-cut for direct product multicommodity, where $k$ is the number of commodities. Besides, when the network we consider is centralized, we prove that there is a $\frac{1}{\alpha}$-approximation algorithm for the sparsest cut problem, where $\alpha$ is the maximum weight ratio of vertices.

Numerical Computations and Tail Asymptotics for Stationary Indices of a Discrete-time Queue

Hong-Bo Zhang

2026, 14(2):  730-747.  doi:10.1007/s40305-024-00557-w

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This paper considers numerical computations and tail asymptotics for stationary queue length and sojourn time distribution of the Geo/T-IPH/1 queue, where T-IPH denotes the discrete-time phase type distribution defined on a birth and death chain with countably many states. By analysis of probability generating functions, recursive formulas for queue length and sojourn time, and factorial moments of queue length and sojourn time are given. Moreover, a complete characterization of the regions of system parameters for exact tail asymptotic characterizations for such indices is also given, and the results show that there are three types of exact tail asymptotics for the indices in different regions.