(Communicated by Andreas Fischer)

(Communicated by Xinwei Liu)

     The partial covering $0$-$1$ integer program with penalties (PCIPP) is a variant of the covering $0$-$1$ integer program in which at most a fixed number of constraints may be violated, with  violations allowed at the expense of paying corresponding penalties. In this paper,  we propose an approximation  algorithm for  the PCIPP by adopting  methods of  [X. Liu, W. Li and J. Yang, A primal-dual approximation algorithm for the $k$-prize-collecting minimum vertex cover problem with submodular penalties,  Front. Comput. Sci. {17}(3) (2023):  Paper No. 173404].

(Communicated by Xinwei Liu)

(Communicated by Guihua Lin)

     In a cooperative game under an augmenting system (AGS game), players (including null players) play critical connecting roles within the augmenting system. Conventional solutions such as the B value and M value only estimate a single definite value for each infeasible coalition and assign zero payoff to null players. These two values fail to reflect the intermediary and connecting importance of null players in maintaining coalition connectivity, which is unreasonable. To solve this problem, we first propose the $\alpha$-Egalitarian discounted Shapley value (i.e., the $EDS^{\alpha,\delta}$ value) for cooperative games. By introducing an E-potential function, we prove the uniqueness of the $EDS^{\alpha,\delta}$ value and show that it generalises several well-known solutions: the ED value, the $\delta$-discounted Shapley value, the $\alpha$-Egalitarian Shapley value, and the Shapley value. An AGS game can be regarded as an incomplete cooperative game because the values of infeasible coalitions are unknown. Then, we define the lower and upper bounds of coalition values to transform an AGS game into associated cooperative games, namely a lower game, an upper game and a complete extension game associated with the AGS game. On this basis, we extend the \(EDS^{\alpha,\delta}\) value to AGS games and propose the interval \(EDS^{\alpha,\delta}\) value. We further provide an axiomatic characterisation for the interval \(EDS^{\alpha,\delta}\) value. The proposed interval \(EDS^{\alpha,\delta}\) value contains the B value and the M value. Nevertheless, the interval \(EDS^{\alpha,\delta}\) value has inherent limitations: it restricts the feasible range of interval coalition values and cannot fully cover all possible marginal contribution scenarios. To address this drawback, we finally generalise the interval \(EDS^{\alpha,\delta}\) value and propose the Moore-\(EDS^{\alpha,\delta}\) value, which admits a wider interval range. We prove that the interval \(EDS^{\alpha,\delta}\) value is included in the Moore-\(EDS^{\alpha,\delta}\) value.  This study develops an approach to transform AGS games into associated cooperative games with interval coalition values, and proposes interval values to reasonably allocate payoffs to players with connecting roles in augmenting systems.

(Communicated by Shiqian Ma)

     This paper studies contextual chance-constrained programming under multimodal and context-dependent uncertainty. We jointly model uncertain parameters and observable side information using a Gaussian mixture distribution (GMD), yielding a conditional GMD given the observed context. For affine-in-uncertainty constraints, we formulate an exact Value-at-Risk (VaR) reformulation and a convex Conditional Value-at-Risk (CVaR) approximation under the conditional GMD, and adapt an auxiliary-probability-variable reformulation to the contextual setting. Analytical gradients support gradient-based methods for the VaR and CVaR formulations, while an adapted spatial branch-and-bound framework inherits the global convergence guarantee under the stated assumptions. Numerical experiments on a linear benchmark compare solution quality, computational efficiency, and numerical stability. In the empirical setting considered, a stock portfolio study provides evidence that contextual macroeconomic information can improve out-of-sample performance and downside risk control relative to Gaussian and unconditional GMD benchmarks.

(Communicated by Jie Sun)

     In this paper, we study a location-routing problem that incorporates simultaneous pickup and delivery demands with fuzzy time windows. A multi-objective mathematical model is formulated to minimize the total logistics cost while maximizing customer satisfaction. To solve the proposed model, a discrete fireworks algorithm (DFWA) is developed. The DFWA redefines the explosion amplitude and mutation strategies to enhance information exchange among fireworks individuals, thereby alleviating the limitations of the traditional fireworks algorithm (FWA) in multi-objective optimization, such as premature convergence to local optima and limited solution quality. Comprehensive comparative analyses demonstrate that the proposed DFWA can efficiently explore the solution space and achieve competitive optimization performance. Furthermore, the rationality and effectiveness of the mathematical model are validated through a case study.

(Communicated by Fanwen Meng)

    The split feasibility problem (SFP) arises in many applications such as signal processing and image reconstruction. A widely used approach for solving the SFP is the $CQ$ algorithm:

$ x^{k+1} = P_C\big(x^k + \gamma A^T(P_Q - I)Ax^k\big), \ k = 0,1,\dots $

where $C\subset\mathbb{R}^n$ and $Q\subset\mathbb{R}^m$ are nonempty closed convex sets, $A\in\mathbb{R}^{m\times n}$, $\gamma\in \left(0,\frac{2}{L}\right)$, and $L$ represents the maximum eigenvalue of $A^T A$. However, this update requires computing exact projections with high accuracy, and carrying out such exact computations can be very expensive. 

(Communicated by Shiqian Ma)

     The multiple-input multiple-output (MIMO) detection problem is a key challenge in modern digital communications. It can be formulated as a complex quadratic optimization problem that includes unit-modulus constraints and discrete phase-angle constraints. Various approximation algorithms based on semidefinite relaxations have been proposed in the literature to tackle this hard problem. However, in scenarios with low signal-to-noise ratios, some existing semidefinite relaxations often fail to provide tight bounds. In this paper, we first review some existing semidefinite relaxations and discuss their relationship, and then introduce a new semidefinite relaxation that aims to further reduce the relaxation gap. The proposed semidefinite relaxation is tightened by incorporating some valid cuts, named as extended node-edge cuts. Our numerical results demonstrate that this newly proposed semidefinite relaxation is significantly tighter than some previous ones, while preserves relative low computational complexity if a two-stage method is applied to solve the relaxation, highlighting the effectiveness of the new valid inequalities in reducing the relaxation gap.

(Communicated by Chen Ling)

      In this paper, we determine the upper bounds for the product of multiple eigenvector components, as well as the upper bounds of the second and the third largest eigenvector components. Further, we establish Brauer-type upper bounds for signless Laplacian $Z$-spectral radius of uniform hypergraphs, which theoretically improve existing results. Numerical examples are provided to demonstrate the effectiveness of our conclusions.

(Communicated by Chen Ling)

     In this paper, the $n$-quaternion matrix equation $AXB=C$ is studied based on semi-tensor product of matrices. First, we present the left real element representation and the right real element representation of quaternion. Then combining these representations with the semi-tensor product of matrices to convert the $n$-quaternion matrix equation to the real linear system, we obtain the general solution of the $n$-quaternion matrix equation $AXB=C$. We further derive the $\eta$-anti-Hermitian solution of the $n$-quaternion matrix equation $AXB=C$ with $\mathcal{GH}$-representation for special matrices. Finally, we demonstrate the effectiveness of this method through numerical experiments and the proposed method is applied to color image encryption and decryption.

(Communicated by Jein-Shan Chen)

     This work develops a feasible primal-dual interior-point method for semidefinite optimization based on the parameterized kernel function

$$\psi_p(t) = \frac{t^2 - 1}{2} - p \log(t) + (1 - p) \left( e^{1/t - 1} - 1 \right), \qquad 0 \leq p \leq 1, \quad t > 0. \tag{1}$$

The construction combines logarithmic and exponential barrier terms and is used to define the search direction and proximity measure within a Nesterov--Todd scaled primal-dual framework. The manuscript establishes the kernel properties needed for the convergence and complexity analysis, including strict convexity-related inequalities and the estimates used to control the decrease of the proximity function during inner iterations.

     For the large-update method, the analysis reported in the manuscript gives the worst-case iteration bound

$$\mathcal{O}\left( \sqrt{n} (\log n)^2 \log \frac{\operatorname{tr}(X^0 S^0)}{\varepsilon} \right),$$

while the small-update method has the bound

$$\mathcal{O}\left( \sqrt{n} \log \frac{\operatorname{tr}(X^0 S^0)}{\varepsilon} \right).$$

The paper argues that these bounds match or improve the relevant established kernel-function bounds, with the small-update result retaining the best-known order cited in the manuscript. The parameter $p$ also provides a practical tuning mechanism without changing the stated worst-case complexity order.

     The numerical section compares the proposed kernel with several established kernels under a common primal-dual implementation. On the two diagonal benchmark structures, tuning $p$ improves over the purely logarithmic choice $p=1$, with the best grid value depending on the problem. In the higher-dimensional diagonal tests, the proposed kernel attains the lowest or tied-lowest total iteration count for every tested size. At $(m,n)=(250,500)$, the manuscript reports 271 total iterations for Problem 1 and 273 for Problem 2 for the proposed kernel, while the simple-power comparator reaches 702 and 685 iterations, respectively. The inner-iteration comparisons show the same ordering.

      The dense feasible tests show a similar pattern. At $n=30$, the proposed kernel requires 231 total iterations, compared with 238 for the classical logarithmic and Bai exponential kernels and 295 for the simple-power kernel. The manuscript therefore concludes that the proposed parameterized kernel combines competitive theoretical complexity with a consistent reduction in iteration counts across the reported test problems.

(Communicated by Xinwei Liu)

     Based on the equivalence between convex-constrained nonlinear monotone equations and split feasibility problems, this paper proposes a modified Hager-Zhang method with approximate projection. The proposed method avoids computing the largest eigenvalue of the matrix $A^\top A$ and does not require multiple expensive projection operations. Moreover, the proposed search direction is independent of any line-search and satisfies both the sufficient descent condition and the trust-region property. Under relatively mild assumptions, the global convergence of the proposed method is established. Numerical experiments on benchmark split feasibility problems demonstrate the competitiveness of the proposed method, and further experiments in signal and image recovery illustrate its potential practical applications.

(Communicated by Guihua Lin)

     With the continuous development of intelligent manufacturing and sustainable production, green manufacturing has become an important direction for modern industrial systems. Flexible job shop scheduling problems (FJSPs) provide an effective mathematical framework for describing complex manufacturing processes with flexible machine selection and diverse operation sequences. In recent years, considerable efforts have been devoted to improving production efficiency and reducing resource consumption in FJSPs. However, most existing studies mainly focus on traditional objectives, such as makespan and energy consumption, while other environmental factors, especially noise emission, have received relatively limited attention. In practical manufacturing environments, excessive industrial noise may negatively affect workers' health, production safety, and the overall working environment. Therefore, integrating noise emission into scheduling optimization is of great significance for achieving sustainable manufacturing.

      This study investigates a multi-objective green flexible job shop scheduling problem considering three conflicting objectives simultaneously, including makespan, energy consumption, and noise emission. Compared with traditional single-objective or bi-objective scheduling problems, the considered problem is more challenging because of the complex interactions among different objectives. For example, reducing completion time usually requires increasing machine utilization, which may result in higher energy consumption and noise levels. Therefore, obtaining a set of high-quality Pareto optimal solutions that balance production efficiency and environmental performance remains a challenging task.

      A comprehensive mathematical optimization model is established for the considered problem. The model integrates several important manufacturing characteristics, including operation precedence constraints, flexible machine assignment, processing time calculation, energy consumption estimation, and noise emission evaluation. Three objectives are simultaneously optimized. The first objective minimizes the maximum completion time of all jobs to improve production efficiency. The second objective minimizes total energy consumption during the manufacturing process to reduce resource usage. The third objective minimizes noise emission to improve environmental sustainability. The formulated model aims to provide decision-makers with a set of alternative scheduling solutions under different preferences among production efficiency and environmental objectives.

      To efficiently solve the formulated multi-objective optimization problem, an Improved Non-dominated Sorting Genetic Algorithm II (INSGA-II) is developed. The proposed algorithm is based on the classical NSGA-II framework and incorporates several problem-oriented improvement strategies. First, a two-layer encoding mechanism is designed to represent machine assignment decisions and operation sequencing decisions separately. This encoding method improves the flexibility of solution representation and ensures that different scheduling components can be effectively optimized during the evolutionary process.

      Second, a hybrid population initialization strategy is proposed. Unlike random initialization methods that may generate low-quality individuals, the proposed strategy introduces problem-specific information to generate better initial solutions while preserving population diversity. A high-quality initial population can provide better search guidance and improve the convergence speed of the evolutionary process.

      Third, improved genetic operators are designed to enhance the exploration and exploitation abilities of the algorithm. The improved crossover and mutation mechanisms increase the possibility of generating promising offspring and help avoid premature convergence during the evolutionary process. Furthermore, a multi-objective local search mechanism is incorporated to further optimize promising individuals. This mechanism improves the refinement ability of the algorithm and contributes to obtaining a better-distributed Pareto solution set.

      Extensive computational experiments are conducted to verify the effectiveness of the proposed INSGA-II. Several representative multi-objective evolutionary algorithms, including NSGA-II, MOPSO, MOEA/D, and IMOEA/D-HS, are selected as benchmark algorithms. All algorithms are evaluated under identical experimental conditions to ensure fairness. Multiple performance indicators are employed to analyze optimization performance from different perspectives, including convergence capability, diversity preservation, and solution stability.

      In addition to standard scheduling instances, large-scale experiments are performed to further examine the scalability and robustness of the proposed algorithm in more complex manufacturing scenarios. Moreover, ablation studies are conducted to investigate the contribution of different improvement strategies. By comparing different algorithm configurations, the effectiveness of the proposed initialization strategy, evolutionary operators, and local search mechanism can be quantitatively analyzed.

      The experimental results demonstrate that the proposed INSGA-II achieves competitive optimization performance compared with benchmark algorithms. The obtained Pareto solutions show better convergence characteristics and improved distribution diversity, enabling more balanced trade-offs among makespan, energy consumption, and noise emission. The statistical analysis results further verify the stability and effectiveness of the proposed method. These findings indicate that the proposed algorithm can effectively address the challenges of multi-objective optimization in green flexible job shop scheduling.

      In conclusion, this study develops an effective optimization framework for green flexible job shop scheduling considering noise emission. By integrating problem-specific encoding, initialization, evolutionary operators, and local search strategies into the NSGA-II framework, the proposed INSGA-II provides an efficient approach for solving complex scheduling problems involving multiple conflicting objectives. The proposed method contributes to sustainable manufacturing decision-making by considering both production performance and environmental impacts. Future research will focus on extending the proposed framework to more complex industrial environments, including dynamic scheduling, uncertain manufacturing conditions, and additional practical constraints.

(Communicated by Chen Ling)

     An $r$-pattern $P$ is  an ordered pair $P=([l],E)$, where $l$ is a positive integer and $E$ is a set of $r$-multisets with elements from $[l]$. An $r$-graph $H$ is said to be  $P$-colorable if there is a homomorphism  $\phi$: $V(H)\rightarrow [l]$ such that  $\{\phi(v_{1}),\ldots,\phi(v_{r})\}\in E$ for every edge $\{v_{1},\ldots,v_{r}\}\in E(H)$. Let $\mathrm{Col}(P)$ denote the family of all $P$-colorable $r$-graphs. This paper focuses on spectral extremal  problems  for $\alpha$-spectral radius of hypergraphs via  analytic techniques.

     For an $r$-uniform hypergraph $H$ and a family of $r$-graphs $\mathcal{F}$, $H$ is $\mathcal{F}$-free if no member $F \in \mathcal{F}$ is a subgraph of $H$. Let $e(H)$ denote the number of edges (called size) of $H$. The \textit{Turán number} $ex(n, \mathcal{F})$ is the maximum size of an $\mathcal{F}$-free $r$-graph on $n$ vertices. A property of $r$-graphs is a family of $r$-graphs closed under isomorphisms. A property is called ${hereditary}$ if it is closed under taking induced subgraphs. For a property $\mathcal{P}$, denoted by $\mathcal{P}_n$ the collection of $r$-graphs in $\mathcal{P}$ of order $n$. It is known that the limit

$\hat{\pi}(\mathcal{P}) := \lim_{n \to \infty} \frac{ex(\mathcal{P}, n)}{\binom{n}{r}}$

     For an $r$-graph $G$ on $n$ vertices and $\alpha\geq1$, the \emph{$\alpha$-spectral radius} of $G$ is defined as

$$\lambda^{(\alpha)}(G) := \max_{\|\mathbf{x}\|_\alpha = 1} r! \sum_{\{i_1, \dots, i_r\} \in E(G)} x_{i_1} \cdots x_{i_r}, \tag{1}$$

where $\mathbf{x}=(x_{1},\ldots,x_{n})\in \mathbb{R}^{n}$  and $\|\mathbf{x}\|_{\alpha}:=(|x_{1}|^{\alpha}+\cdots +|x_{n}|^{\alpha})^{1/\alpha}$.  

     If $\mathbf{x}\in \mathbb{R}^{n}$ is a vector such that $\|\mathbf{x}\|_{\alpha}=1$ and the equality (1) holds, then $\mathbf{x}$ is called an \emph{eigenvector} of $G$ corresponding to $\lambda^{(\alpha)}(G)$. 

     Let $\mathbb{PE}_{\alpha}(G)$ denote the set of all nonnegative eigenvectors for $\lambda^{(\alpha)}(G)$. Define

$$x_{\text{min}}^{(\alpha)}(G):=\inf_{\mathbf{x} \in \mathbb{PE}_{\alpha}(G)}\mathbf{x}_{\text{min}},$$

where $\mathbf{x}_{\text{min}}$ represents the minimum component of the vector $\mathbf{x}$. We simply write $x_{\text{min}}(G)$ instead of $x_{\text{min}}^{(\alpha)}(G)$   when no confusion occurs.

     Let $\alpha\geq 1$. If $\mathcal{P}$ is a hereditary property of $r$-graphs, let

$\lambda^{(\alpha)}(\mathcal{P},n):=\max_{G\in \mathcal{P}_{n}}\lambda^{(\alpha)}(G)$

and we show that the limit

$$\lambda^{(\alpha)}(\mathcal{P}):=\lim\limits_{n\to \infty}\lambda^{(\alpha)}(\mathcal{P},n)n^{r/\alpha-r}$$

exists. 

     Let $\mathcal{P}$ be a hereditary  property of $r$-graphs with $\hat{\pi}(\mathcal{P})>0$. Suppose that $G_{n}\in \mathcal{P}_{n}$ is an $r$-graph satisfying $\lambda^{(\alpha)}(G_{n})=\lambda^{(\alpha)}(\mathcal{P},n)$ for $\alpha>1$. We first prove that there exist infinitely many $n$ such that

$$(x_{\text{min}}(G_n))^{\alpha}\geq\frac{1}{n}\Big(1-\frac{\alpha}{(\alpha-1)r\log n}\Big).$$

    Then we can show that the hypergraph attaining the maximum $\alpha$-spectral radius in $\mathrm{Col}(P)$ is asymptotically regular: 

  Let $0<\varepsilon<1$, $0\leq\varepsilon'<\varepsilon \hat{\pi}(\mathcal{P})/(r-1)$, and $\mathbf{x}\in \mathbb{R}^{n}$ be a nonnegative eigenvector  of $G_{n}$. If $n$ is sufficiently large and

$(\mathbf{x}_{\text{min}})^{\alpha}\geq \frac{1}{n}(1-\varepsilon'),$

then

    Recall that $\mathrm{Col}(P)$  denotes  the family of all $P$-colorable $r$-graphs.

    Let  $P=([l],E)$ be an  $r$-pattern, and  $G_{n}\in \mathrm{Col}(P)_{n}$ satisfy $\lambda^{(\alpha)}(G_{n})=\lambda^{(\alpha)}(\mathrm{Col}(P),n)$. If $\alpha>1$, then there exist $n_{1}\in \mathbb{N}$ and a constant $M>0$  such that for any $n\geq n_{1}$, the following hold:

• $(1)$ $\lambda^{(\alpha)}(G_{n+1}) \ge  \big(1 + r \big(1 - \frac{1}{\alpha} - \frac{l}{\alpha(n - l r + l)}\big) ( x_{\text{min}}(G_n) )^{\alpha} \big)\lambda^{(\alpha)}(G_n).$
• $(2)$  $(x_{\text{min}}(G_n))^{\alpha} \ge \frac{1}{n}\big( 1 - \frac{\hat{\pi}(\mathrm{Col}(P)) M}{r n} \big).$
• $(3)$ $\delta(G_n) \ge \hat{\pi}(\mathrm{Col}(P)) \left(1 - \frac{M}{n}\right) \binom{n}{r-1}.$

    Consequently, we show that for any family $\mathcal{F}$ of $r$-graphs that is degree-stable with respect to $\mathrm{Col}(P)$, spectral Tur\'an-type  problems can be  reduced to spectral extremal problems within $\mathrm{Col}(P)$:

     Let $\alpha>1$ and $r\geq 2$. Let $P$ be an $r$-pattern, and let $\mathcal{F}$ be  a family of $r$-graphs that is degree-stable with respect to  $\mathrm{Col}(P)$. Then for all sufficiently large $n$ and any $n$-vertex $\mathcal{F}$-free $r$-graph $G$, we have \[\lambda^{(\alpha)}(G)\leq\lambda^{(\alpha)}(\mathrm{Col}(P),n),\] where equality holds only if $G$ is $P$-colorable.

    As an application, we determine the maximum $\alpha$-spectral radius ($\alpha\geq1$) among all $n$-vertex $F^{(r)}$-free $r$-graphs, where $F^{(r)}$ is the $r$-expansion of the color-critical graph $F$. This provides a powerful reduction tool for handling  spectral Tur\'{a}n-type problems for hypergraphs. Finally, leveraging the spectral method, we derive a corresponding edge Tur\'an extremal result. More precisely, we show that if $\mathcal{F}$ is degree-stable with respect to $\mathrm{Col}(P)$, then every $\mathcal{F}$-free edge extremal hypergraph must be a $P$-colorable hypergraph.

(Communicated by Liqun Qi)

    Given a set of algebraic numbers $\alpha_1,\ldots,\alpha_n$, any rational expression $\gamma =P(\alpha_1,\ldots,\alpha_n)/ \mathbb{Q}(\alpha_1,\ldots,\alpha_n)$ with rational coefficients is also an algebraic number. Determining the minimal polynomial of such an expression is a fundamental problem in computational algebra, with applications in algebraic number theory, Galois theory, and symbolic computation. Classical methods based on resultants or field extensions typically produce an annihilating polynomial that may contain extraneous factors, requiring further factorization to recover the minimal polynomial. This paper presents an alternative approach that directly yields the minimal polynomial using elimination theory and Gröbner bases.

    The key idea is to encode $\gamma$ into a polynomial ideal. Let $f_i(x_i)\in\mathbb{Q}[x_i]$ be the minimal polynomial of $\alpha_i$ for $i=1,\ldots,n$, and assume that $f_{k+1}$ is irreducible over $\mathbb{Q}(\alpha_1,\ldots,\alpha_k)$ for $k=1,\ldots,n-1$. Introduce a new variable $z$ and form the ideal 

$I = \langle f_1(x_1), \dots, f_n(x_n), \, Q(x_1, \dots, x_n) z - P(x_1, \dots, x_n) \rangle$

in the ring $\mathbb{Q}[x_1,\ldots,x_n,z]$. It is shown that $I$ is a prime ideal and that every polynomial $h(z)\in\mathbb{Q}[z]$ vanishing on $\gamma$ belongs to $I$. Consequently, the elimination ideal $I\cap\mathbb{Q}[z]$ is a nonzero principal ideal, and its monic generator is exactly the minimal polynomial of $\gamma$.

    Building on this characterization, an algorithm is presented to compute the minimal polynomial of an arbitrary $\gamma$. It begins by computing the minimal polynomial $f_i$ of each $\alpha_i$ and verifying the irreducibility condition, before constructing the polynomial ideal $I$. Then a Gröbner basis of $I$ is calculated with respect to a lexicographic order eliminating $x_1,\ldots,x_n$, and the univariate polynomial $m(z)$ is extracted. If any $f_{k+1}$ is reducible, $m(z)$ is factored to isolate the true minimal polynomial. The effectiveness of this technique is illustrated by four nontrivial examples, which demonstrate how the Gröbner‑based elimination directly yields the minimal polynomial for expressions involving several nested radicals and roots of unity.

    Overall, this paper provides a rigorous and systematic algorithm for computing minimal polynomials of arbitrary rational expressions in algebraic numbers, filling a gap in the literature where such explicit algebraic‑geometric methods were not fully documented.

(Communicated by Xinmin Yang)

    The sum of affine ratios problem (SOARP), as a special type of nonlinear programming problem, has been widely studied. In the paper, the following sum of affine ratios problem is studied:

(SOARP):$\begin{cases} \min \sum_{i=1}^p \frac{g_i^T z + g_{0i}}{e_i^T z + e_{0i}} \\ \text{s.t. } z \in \mathcal{Z} = \{ z \in \mathbb{R}^n | Az - b \leq 0, z \geq 0 \}, \end{cases}$

where $p$ is a natural number satisfying $p>1$; $A\in \mathbb{R}^{m\times n}$, $b\in \mathbb{R}^{m}$; for any $i=1,2,\cdots,p$, $g_i, e_i\in \mathbb{R}^{n}$, $g_{0i}, e_{0i}\in \mathbb{R}$; $\mathcal{Z}$ is a nonempty and bounded set; for any $z\in \mathcal{Z}$, $g_i^{\top}z+g_{0i}$ and $e_i^{\top}z+e_{0i}\ (i=1,\cdots,p)$ are affine functions, and $e_i^{\top}z+e_{0i}\neq 0$, $\forall z\in \mathcal{Z}$. Let $E=\{1,\cdots,p\}$, since $e_i^{\top}z+e_{0i}\neq 0$ for each $i\in E$, it shows that the function $e_i^{\top}z+e_{0i}$ satisfies either $e_i^{\top}z+e_{0i}>0$ or $e_i^{\top}z+e_{0i}<0$. Let $E^+=\{i|e_i^{\top}z+e_{0i}>0,\ \forall z\in \mathcal{Z},\ i=1,2,\cdots,p\}$ and $E^-=\{i|e_i^{\top}z+e_{0i}<0,\ \forall z\in \mathcal{Z},\ i=1,2,\cdots,p\}$, we have $E=E^-\bigcup E^+$. In the paper, we can assume that $e_i^{\top}z+e_{0i}> 0$ for any $z\in \mathcal{Z}$ in the paper, that is, $E^-=\emptyset$ and $E=E^+$.

    In order to solve problem (SOARP), we present a rectangular branch-deletion-bound algorithm (RBDBA) based on a standard splitting operation and a region deletion rule.  First of all, the Charnes-Cooper transformation is used to obtain the corresponding maximum and minimum values for each fractional function $\frac{g_i^{\top}z+g_{0i}}{e_i^{\top}z+e_{0i}}\ (i\in E)$. That is, $\underline{h}_{i}^{0}=\min_{z\in \mathcal{Z}}\frac{g_i^{\top}z+g_{0i}}{e_i^{\top}z+e_{0i}}$ as the minimum value and $\overline{h}_{i}^{0}=\max_{z\in \mathcal{Z}}\frac{g_i^{\top}z+g_{0i}}{e_i^{\top}z+e_{0i}}$ as maximum value for each fractional function $h_i=\frac{g_i^{\top}z+g_{0i}}{e_i^{\top}z+e_{0i}}$ over the $z\in \mathcal{Z}$. Thus, an initial rectangle $\mathcal{H}^0=\big[\underline{h}^{0},\overline{h}^{0}\big]\triangleq\big\{h\in \mathbb{R}^{p}\mid \underline{h}_{i}^{0}\leqslant h_i\leqslant \overline{h}_{i}^{0},\ i\in E\big\}$ is obtained. Meanwhile, problem (EP$(\mathcal{H}^0)$) can be recast as follows:

(EP$(\mathcal{H}^0)$):$\begin{cases} \min \sum_{i=1}^p h_i \\ \text{s.t.} \quad h_i = \frac{g_i^T z + g_{0i}}{e_i^T z + e_{0i}}, \quad i \in E, \\ \quad z \in \mathcal{Z}, \quad h \in \mathcal{H}^0. \end{cases}$

(LRP$(\mathcal{H}^0)$): $\min \sum_{i=1}^p h_i $

s.t. $h_i \leq \frac{(g_i^\top z + g_{0i}) - (e_i^\top z + e_{0i})\underline{h}_i + l_i \underline{h}_i}{l_i}, \quad i \in E,$

$h_i \leq \frac{(g_i^\top z + g_{0i}) - (e_i^\top z + e_{0i}) \bar{h}_i + u_i \bar{h}_i}{u_i}, \quad i \in E,$

$h_i \geq \frac{(g_i^\top z + g_{0i}) - (e_i^\top z + e_{0i})\underline{h}_i + u_i \underline{h}_i}{u_i}, \quad i \in E,$

$ h_i \geq \frac{(g_i^\top z + g_{0i}) - (e_i^\top z + e_{0i}) \bar{h}_i + l_i \bar{h}_i}{l_i}, \quad i \in E,$

$z \in \mathcal{Z}, \quad h \in \mathcal{H}.$

By calculating the approximate error between problems (LRP) and (EP), that is, it can be guaranteed that problem (LRP) gives a trustworthy lower bound for the optimal value of problem (EP). 

    Next, based on the standard splitting operation and the simple region deletion rule, the algorithm (RBDBA) is designed in the paper. 

    Then, theoretical analysis of the algorithm (RBDBA) is provided in terms of convergence and complexity. From the point of convergence, for the given termination error $\epsilon > 0$, the rectangle $\mathcal{H}^k$ is generated by Algorithm (RBDBA) at the $k$-th iteration, let $z^k$ is an optimum solution to problem (SOARP) with $\mathcal{H}^k=\prod_{i=1}^{p}\mathcal{H}^k_{i}$. Meanwhile, the current upper bound $UB_{k}$ and lower bound $LB(\mathcal{H}^k)$ are obtained over $\mathcal{H}^k$, which satisfies

$\big|UB_{k}-LB(\mathcal{H}^k)\big|\leqslant 4p\max_{i\in E}\Big\{\frac{u_{i}}{l_{i}}\times\big|\overline{h}^k_{i}- \underline{h}^k_{i}\big|\Big\}.$

From the point of convergence, given the termination error $\epsilon>0$, the maximum number of iterations of Algorithm (RBDBA) is

$\left\lceil \log_2 \frac{(4p\zeta)^p \prod_{i=1}^p \left| \bar{h}_i^0 - \underline{h}_i^0 \right|}{\epsilon^p} \right\rceil$

in the worst case.

    Finally, to demonstrate the computational performance of Algorithm (RBDBA), numerical experiments are conducted for some problems in the Appendix, by comparing Algorithm (RBDBA) with other existing algorithms, the advantages of Algorithm (RBDBA) are obvious. Moreover, our algorithm is also useful for the world example, and our algorithm also has great application prospects in practical applications.

(Communicated by Chen Ling)

    Community detection is fundamental to understanding the structure of networks. Most existing methods rely on network topology or low-rank representations, but they are designed for clean multi-layer networks without noise. In this paper, we investigate the problem of community detection in noisy multi-layer networks. We propose an optimization model for community detection based on a unified matrix and a low-rank transition probability tensor (LTPT). To solve this model, we design an efficient ADMM-based algorithm and derive analytical solutions for all subproblems. Additionally, we analyze the computational complexity of the algorithm and prove its convergence. We conducted numerical experiments on both synthetic and real-world networks. The results demonstrate the effectiveness of the proposed algorithm and its advantages over existing methods in the literature.

(Communicated by Xinmin Yang)

    In recent decades, issues related to directions have  emerged in mathematical programming and decision-making. In the last few years,  for a constrained set-valued optimization problem,  directional Pareto minimality, directional strict efficiency and directional weak minimality with respect to one set or two sets have been introduced and discussed.  To  eliminate ineffective solutions in directional set-valued optimization problems, directional Benson proper minimality with respect to two directional sets in set-valued optimization problems are investigated. 

    Let $X$ and $Y$ be Banach spaces, $F: X\rightrightarrows Y$ be a proper set-valued mapping,  $\Omega\subseteq X$ be a nonempty closed subset, $C\subseteq Y$ be a closed convex pointed cone, $L\subseteq S_{X}$ and $M\subseteq S_{Y}$, where $S_{X}$ and $ S_{Y}$ are the unit spheres  of $X$ and $Y$, respectively. Let $\bar{x}\in \Omega$ and $\bar{y}\in F(\bar{x})$.  Consider the following constrained optimization problem:

$$C-\min F(x), \text{ subject to } x \in \Omega. \tag{1}$$

    Let $D_{D}^{L, M}F(\bar{x}, \bar{y})$  denote  the directional Dini lower derivative of $F$ at $ ( \bar{x}, \bar{y})$ with respect to  $L$ and $M$, and let $T_{B}^{L}(\Omega, \bar{x})$ denote the Bouligand tangent cone to $\Omega$ at $\bar{x}$ with respect to $L$.  Let $ C^{*\#}$ be the quasi-interior of the dual cone of $C$, and  let $ M^{*\#}$ be the quasi-interior of the dual cone of $M$. 

    The main contributions of this work are summarized as follows:

    (i) The concepts of  directional Benson  proper minimum point for problem (1) and directional Benson openness  with respect to $L$ and $M$ for $F$ are introduced.  The incompatibility between them  is established when $\Omega=X$.  

    (ii) On primal spaces,    optimality conditions  for the  directional Benson proper minimality of  problem (1) are established.

   Theorem A. If $\left( \bar{x}, \bar{y} \right )$ is a local directional Benson proper minimum point of  problem (1)  with respect to $L$ and $M$, then

$\mathrm{cl}\left[D_{D}^{L, -M}F(\bar{x}, \bar{y} )\left(T_{B}^{L}(\Omega, \bar{x}) \right)+C \right ]\cap  (-C )\subseteq\{0\}.$

    Conversely, let  $\tilde{F}(\cdot)=F(\cdot)+C$. Suppose that $F$ is a $C$-convex set-valued mapping, $\Omega$ is a closed convex set, $F(\cdot)\cap(\bar{y}-\mathrm{cone}M)$ is lower semicontinuous on $\Omega\cap (\bar{x}+\mathrm{cone}L)$,  and 

$\mathrm{cl}\left[D_{D}^{L, -M}\tilde{F}(\bar{x}, \bar{y} )\left(T_{B}^{L}(\Omega, \bar{x}) \right)+C \right ]\cap  (-C )\subseteq\{0\}.$

Then $( \bar{x}, \bar{y})$ is a  directional Benson proper minimum point of problem (1)  with respect to $L$ and $M$. 

     (iii)  Scalar characterizations for the  directional Benson  proper minimality of  problem (1) are established by using    $ C^{*\#}$ and $ M^{*\#}$.

   Theorem B   Suppose that(a) the norm topology  gives $Y$ as the topological dual space of  $Y^{*}$ and $C$ has a bounded base, or (b) $C$ is locally compact.  Let $F-\bar{y}$ be nearly $C$-subconvexlike on $\Omega$ at $(\bar{x}, 0)$ with respect to $L$ and $M$, and $(\bar{x}, \bar{y})$ be a    directional Benson proper minimum point of  problem (1)  with respect to $L$ and $M$.    Then there exists $ c^{*}\in C^{*\#} $ such that 

$$\langle c^{*}, y \rangle\geq  \langle c^{*}, \bar{y} \rangle, \ \ \ \ \forall ~y\in F (\Omega\cap(\bar{x}+\mathrm{cone}L) )\cap ( \bar{y} -\mathrm{cone}M). \tag{2}$$

Suppose further that $F(\Omega\cap(\bar{x}+\mathrm{cone}L))\cap(\bar{y}-\mathrm{cone}M)\neq \{\bar{y}\}$.  Then  $ c^{*}\in C^{*\#}\setminus M^{*\#} $.

    Conversely,  suppose that there exists $ c^{*}\in C^{*\#}\cup  M^{*\#} $ such that (2) holds. Then  $(\bar{x}, \bar{y})$ is  a    directional Benson proper minimum point of  problem (1)  with respect to $L$ and $M$.

     (iv) Directional Benson proper minimal elements of sets have  geometric significance.  Optimality conditions for them are established by $T_{B}^{L}(\Omega, \bar{x})$ and $C^{*\#}$.  

(Communicated by Xinwei Liu)

    This paper addresses multiobjective optimization problems constrained by the intersection of affine constraints and a closed convex set. The main challenge in such problems lies in computational expensive of projection onto the full feasible region. To integrate the advantages of projection based methods and the augmented Lagrangian method, we propose an augmented Lagrangian projected gradient algorithm. This method combines multiobjective steepest descent directions with penalty and multiplier updates to handle the affine constraints, while projections are performed only onto the closed convex set. This design avoids direct projection onto the complex constrain set. By considering the first-order approximation of the objective function, feasibility improvement, and multiplier stabilization, we introduce a Lyapunov-type potential function, which decreases along the iterates, yielding global convergence to Pareto stationary points. Moreover, we establish a non-asymptotic complexity bound, showing that the number of iterations required to reach a prescribed accuracy grows at most quadratically with respect to the inverse of the tolerance. Numerical experiments on various test problems demonstrate the efficiency of the proposed method, particularly in high-dimensional settings, compared with the standard mutliobjective augmented Lagrangian method and the mutliobjective projected gradient algorithm.

(Communicated by Huifu Xu)

    This paper addresses the nonlinear programming problem of the following form

where $x\in \mathbb R^{n}$, $f:\mathbb R^{n}\rightarrow \mathbb R$ and $g:\mathbb R^{n}\rightarrow \mathbb R^{m}$. Here,$m$ is large, while $n$ is medium-sized. This program has numerous applications, such as the discretized semi-infinite programming problem, the finite minimax problem involving the maximum of a large number of complex function, and the discretized semi-infinite minimax problem.

    The nonlinear programming problem (1) is equivalent to

where $g_{\max}(x) = \max_{1 \leq i \leq m} \{ g_i(x) \}$ represents a single, but nonsmooth, constraint.

    Next, we present the flattened spline smoothing function. 

    Let $\upsilon(t)=\alpha_{1}t+\alpha_{2}$, $\alpha_{1}>0$, $\alpha_{2}>0$, $0<t\leq1$ are some adjusting parameters. Let $\tau(z,t):\mathbb R\times[0,1]\rightarrow[0,1]$ be  a bivariate function of class $C^{2,1}$ that satisfies the following conditions:

            - $g_{m+1} = -v(t)$.

   According to its definition, $\tau(z,t)$ can be expressed as follows:

where, for any $t\in[0,1]$, the function $\bar{\tau}(z,t)$ satisfies the following conditions:

There exist numerous functions for $\bar{\tau}(z,t)$,  including polynomial functions and spline functions. Due to their straightforward mathematical formulations and numerous advantageous properties, throughout this paper, $\bar{\tau}(z,t)$ is chosen as the following polynomial function

    Let $I_{\upsilon}(x,t)=\{i|g_{i}(x)>-\beta\upsilon(t)\}$, and let $\tilde{k}$ denote the cardinality of the set $I_{\upsilon}(x,t)$. The flattened spline smoothing function can be written as

where $c_1 = 1/(6t^2), \quad c_l / c_{l+1} = (l+2)/l, \quad 1 \leq l \leq \tilde{k}.$ The region defined by the following inequalities is the cell $\Delta_{i_1 \dots i_k} (\theta t).$

    In this study, we propose a novel homotopy method that employs the flattened spline smoothing function $\hat g_{(\theta,\alpha_{1},\alpha_{2},\beta)}(x,t)$ to uniformly approximate $g_{\max}(x)$ for solving the nonlinear programming problem (1). In contrast to the spline smoothing homotopy method, the proposed approach not only uses a spline smoothing function that uniformly approximates the constrained maximum function with fewer constrained functions, but also reduces the dimension of the homotopy map. Additionally, the flattened spline smoothing function offers significant advantages: it focuses on constraints whose function values are close to zero or are even constant, requiring the evaluation of gradients and Hessians for only a few constraints, or sometimes just one. As a result, the proposed method significantly reduces the computational burden associated with calculating the gradients and Hessians of the flattened spline smoothing function. The global convergence of a path determined by the flattened spline smoothing homotopy equation is proven under conditions similar to those of other homotopy methods. Numerical examples demonstrate that the proposed method is more effective compared to some existing methods.

(Communicated by K.L. Teo)

   In this paper, we investigate a class of nonlinear time-delay optimal control problems subject to general equality and inequality constraints. Such problems are particularly challenging due to the nonlocal dependence of delayed states and controls, which complicates both numerical discretization and gradient evaluation. The multiple shooting (MS) method is well known for its robustness and flexibility in handling complex constraints; however, its direct application to time-delay systems is hindered by the need to accurately reconstruct historical states across shooting intervals, often leading to reduced accuracy or numerical instability. To address these difficulties, we propose a hybrid computational framework that integrates the multiple shooting method with a collocation technique. The control is parameterized over each shooting interval, while delayed states and controls are approximated through a collocation scheme, resulting in a finite-dimensional nonlinear optimization problem. Unlike approaches based on finite-difference approximations, we derive an explicit gradient formula for the discretized problem and establish its validity through a rigorous variational analysis. Three numerical examples are presented to demonstrate the effectiveness of the proposed approach. The results show that the proposed method consistently achieves improved objective values compared with existing techniques, while maintaining a reasonable computational cost.

(Communicated by Xinmin Yang)

    This paper is devoted to the investigation of optimality conditions and saddle-point theorems for an Uncertain Semi-infinite Multiobjective Interval-valued Optimization Problem, denoted by (USMIOP). The model simultaneously involves interval-valued objectives, infinite number of constraints, and uncertain parameters, which widely exists in engineering design, decision analysis and robust control.

    Firstly, the concepts of robust $\leq_{UC}$-efficient solution and robust weak $<_{UC}$-efficient solution for problem (USMIOP) are proposed based on the UC-order relation, which can overcome the limitation that the traditional LU-order cannot compare fully overlapping intervals. It is proved that the set of robust $\leq_{UC}$-efficient solutions is a subset of the set of robust $\leq_{LU}$-efficient solutions, and the similar inclusion relation also holds for weak efficient solutions.

    Secondly, under the Abadie constraint qualification, a necessary optimality condition for robust weak $<_{UC}$-efficient solution is established by using the Clarke subdifferential. Different from the existing literature, the obtained result does not require the convexity assumption of the lower and upper bound functions of interval-valued objectives, and is applicable to non-differentiable and uncertain semi-infinite cases.

     Thirdly, a class of generalized interval-valued invex functions, namely $(p,r)-(l^{U},l^{C})-(\eta,\theta)$-invex functions, is introduced by combining the UC-order and $(p,r)-\rho-(\eta,\theta)$-invexity. This definition extends the existing generalized invexity for interval-valued functions. Under the above invexity assumption, a sufficient optimality condition for the robust weak $<_{UC}$-efficient solution is derived.

    Finally, an interval-valued Lagrange function associated with problem (USMIOP) is constructed, and a corresponding saddle-point definition is given under the UC-order. Then, two saddle-point theorems are deduced, which establish the equivalence between the robust weak $<_{UC}$-efficient solution and the saddle-point of the Lagrange function. The obtained results improve and extend the corresponding ones in the previous literature.

    Several examples are provided to illustrate the obtained optimality conditions and saddle-point theorems. The research of this paper enriches the optimality theory of non-smooth interval-valued optimization, and provides a theoretical support for the study of uncertain semi-infinite multiobjective optimization problems with interval-valued objectives.

(Communicated by Xinmin Yang)

    This paper develops optimality conditions for nonsmooth multiobjective optimization problems involving sparsity constraints through error bound and scalarization technique. 

    First, the original  multiobjective optimization problem is reformulated into a scalarized form using the Pascoletti–Serafini scalarization approach. 

    Next, by exploiting an error bound of the sparsity set together with the local Lipschitz continuity of the objective functions, a penalty-based reformulation is introduced, yielding an equivalent unconstrained optimization model. We establish that the local minimizers of this unconstrained model correspond precisely to the locally weakly efficient solutions of the original constrained problem. Furthermore, necessary optimality conditions are derived by using limiting subdifferentials, and sufficient conditions are provided under suitable convexity assumptions. 

    In contrast to existing studies on single-objective sparse optimization, this work extends the theoretical framework to the nonsmooth multiobjective setting. Based on the theoretical results, a numerical algorithm is designed to compute sparse weakly efficient solutions for the multiobjective problem. Numerical experiments are conducted to illustrate the effectiveness and practical applicability of the proposed approach.

(Communicated by Lingchen Kong)

    We propose a neural network model for a special case of the extended vertical linear complementarity problem (EVLCP) where three linear functions are involved per row. The EVLCP seeks $x\in\mathbb{R}^n$ such that

$\min\{M_1x+q_1,\;M_2x+q_2,\;M_3x+q_3\}=0,$

with $M_i\in\mathbb{R}^{n\times n}, q_i\in\mathbb{R}^n$. This problem appears in game theory, market equilibrium, and contact mechanics. Real-time solution is often required but conventional algorithms face speed limitations. To overcome this, we reformulate the EVLCP using a novel nonsmooth function based on the Fischer–Burmeister (FB) function. Define $\phi(a, b) = a + b - \sqrt{a^2 + b^2}$ and extend it to three variables by

$\phi(a, b, c) = \phi(a, b) + c - \sqrt{\phi(a, b)^2 + c^2}, \quad (a, b, c) \in \mathbb{R}^3.$

It holds that  $\phi(a, b) =0$ if and only if $a \geq 0, \, b \geq 0, \, c \geq 0, \, abc = 0.$ Consequently, the EVLCP is equivalent to the system $\Phi(x)=0$ where $\Phi$ is an $n$-dimensional vector whose $j$-th component is $\phi(M_{1j}^\top x + q_{1j}, M_{2j}^\top x + q_{2j}, M_{3j}^\top x + q_{3j}).$ Letting $f(x) = \frac{1}{2} \|\Phi(x)\|^2$, the EVLCP reduces to the unconstrained minimization $\min_{x \in \mathbb{R}^n} f(x).$ Although $\Phi$ is nonsmooth, $f$ is continuously differentiable with gradient $(\nabla f(x)=V^\top\Phi(x))$ for any $(V\in\partial\Phi(x))$, the Clarke subdifferential. Moreover, any $(V\in\partial\Phi(x))$ can be expressed as $(V=D_aM_1+D_bM_2+D_cM_3)$ with diagonal matrices $(D_a,D_b,D_c)$ whose entries come from the subdifferential of $(\phi)$ row-wise.

    We propose the steepest-descent neural network

$\frac{dx(t)}{dt} = -\tau \nabla f(x(t)), \quad x(0) = x_0, \quad \tau > 0.$

The equilibrium points of this dynamical system coincide with EVLCP solutions under the row $W$property (all row representative matrices have determinants of the same sign). Using $f$ as a Lyapunov function, we prove that every isolated equilibrium is asymptotically stable. If, in addition, the matrix set $\{ M_1, M_2, M_3 \}$ possesses the row $W$ property, the unique equilibrium is exponentially stable: there exist constants $\omega < 0, \kappa > 0, \delta > 0$ such that whenever $\|x(t_0) - x^*\| <\delta$,

$\|x(t) - x^*\| \leq \kappa e^{\omega t} \|x(t_0) - x^*\|, \quad t \geq t_0.$

The $R_0$ property $(\min\{M_1x, M_2x, M_3x\} = 0 \Rightarrow x = 0)$ guarantees that all trajectories are bounded and converge to an equilibrium point, thus ensuring global convergence.

    Numerical experiments validate the theoretical findings. First, we consider a generalized bimatrix game from the literature, which reduces to an EVLCP of dimension four. The neural network converges to the Nash equilibrium with high precision; larger scaling factors accelerate convergence while preserving stability. In comparison, a smoothing neural network from a previous study stagnates at a moderate accuracy regardless of parameter tuning, and the smoothing Newton method requires an extremely small smoothing parameter to achieve comparable precision. Next, a two-dimensional EVLCP is solved with various scaling factors; the trajectory converges to the origin rapidly with errors below machine precision. The method also handles a market equilibrium problem for two complementary products, yielding equilibrium prices that satisfy all complementarity conditions and converge monotonically. Finally, we test a scalable $n$-dimensional EVLCP constructed from tridiagonal matrices with dimensions ranging up to several hundreds. The neural network achieves consistently high accuracy (errors on the order of $10^{-8}$ to $10^{-12}$ while computation time scales roughly quadratically with the problem size. All experiments confirm that larger scaling factors accelerate convergence without sacrificing stability, in accordance with the exponential stability guarantee.

    In summary, this work introduces a nonsmooth neural network for solving EVLCP with three functions per row. The approach eliminates smoothing parameters, provides rigorous stability guarantees (asymptotic and exponential), and demonstrates superior numerical efficiency and precision. Future extensions include the general case $m\ge3$ and nonlinear EVLCP.

(Communicated by Andreas Fischer)

    In this paper, we consider a class of nonsmooth, nonconvex and non-Lipschitz optimization problem  which arises in many important applications such as signal processing, variable selection and image restoration. To solve this problem, we propose to apply linearized alternating direction method of multipliers(LADMM) to a more general form of problem, its objective function consists of three blocks with linear constraint. When the augmented lagrangian function satisfies the Kurdyka–\L ojasiewicz property and some parameter constraints, as well as some matrix conditions, we prove that the sequence generated by LADMM converges to the critical point of problem. To conclude, we firstly derive the stopping criteria of the algorithm, then we validate the convergence and efficiency of the algorithm with numerical experiments.

(Communicated by Lingchen Kong)

    This paper considers a bilevel program, which has many applications in practice. To develop effective numerical algorithms, it is generally necessary to transform the bilevel program into a single-level optimization problem. The most popular approach is to replace the lower-level program by its KKT conditions and then the bilevel program can be reformulated as a mathematical program with equilibrium constraints (MPEC for short). However, since the MPEC does not satisfy the Mangasarian-Fromovitz constraint qualification at any feasible point, the well-developed nonlinear programming theory cannot be applied to MPECs directly. In this paper, we apply the Wolfe duality to show that, under very mild conditions, the bilevel program is equivalent to a new single-level reformulation (WDP for short) in the globally and locally optimal sense. We give an example to show that, unlike the MPEC reformulation, WDP may satisfy the Mangasarian-Fromovitz constraint qualification at its feasible points. We give some properties of the WDP reformulation and the relations between the WDP and MPEC reformulations. We further propose a relaxation method for solving WDP and investigate its limiting behavior. Comprehensive numerical experiments indicate that, although solving WDP directly does not perform very well in our tests, the relaxation method based on the WDP reformulation is quite efficient.

(Communicated by Zheng-Hai Huang)

    The tensor split feasibility problem (TSFP), as a high-order extension of the classical split feasibility problem (SFP), has important applications in areas such as information retrieval. Its objective is to find a vector $x \in \mathbb{R}^n$ such that

$x\in C~~\text{and}~~\mathcal A x^{m-1}\in Q,$

where $C \subseteq \mathbb{R}^n$ and $Q \subseteq \mathbb{R}^p$ are nonempty closed convex sets, and $\mathcal{A}$ is a real tensor of order $m$ with dimensions $p \times n \times \cdots \times n$. Existing numerical methods, including the projection algorithm and the Levenberg-Marquardt (LM) algorithm, suffer from several limitations: their performance often depends on the initial point lying within a traditional convergence region, and the inner--outer iterative structure may lead to increased computational cost.

    To enhance the efficiency of solving TSFP, we reformulate the problem as the merit function

$p(x)=\tfrac12\|P_C(x)-x\|^2+\tfrac12\|P_Q(\mathcal{A}x^{m-1})-\mathcal{A}x^{m-1}\|^2,$

whose gradient is

$g(x)=(I-P_C)x+J(x)^\top (I-P_Q)\mathcal{A}x^{m-1},\qquad$

$J(x)=(m-1)\mathcal{A}x^{m-2}.$

This leads to the fixed-point formulation

$x=P_\Omega(x-\alpha g(x)),$

where $\Omega$ is a closed convex set containing the TSFP solution set. Based on this fixed-point formulation, we introduce a novel second order dynamical system based on the gradient projection method, which yields the following continuous-time algorithm

$$\begin{cases}\ddot{x}(t)+\gamma(t)\dot{x}(t)+\lambda(t)\bigl(x(t)-P_\Omega(x(t)-\alpha g(x(t)))\bigr)=0,\\[2mm]x(0)=u_0,\quad \dot{x}(0)=v_0,\end{cases}$$

where $\alpha>0$, $\gamma(t)$ and $\lambda(t)$ are Lebesgue measurable functions.

    In the theoretical analysis, we show that the merit function $p(x)$, its gradient $g(x)$, the tensor mappings $\mathcal{A}x^{m}$, $\mathcal{A}x^{m-1}$, and the Jacobian $J(x)$ are Lipschitz continuous on bounded closed sets, thereby ensuring the well-posedness of the dynamical system. It is further proved that the proposed algorithm has a unique solution under mild conditions. Furthermore, the corresponding trajectory of the algorithm always converges to the unique solution.

    Compared with up-to-date methods, numerical experiments are given to verify the effectiveness and competitiveness of the proposed algorithm. The experimental results show that the proposed algorithm consistently outperforms the projection algorithm and the LM algorithm in terms of computational efficiency.

(Communicated by Xinmin Yang)

This note is to analyze the linear convergence rate of the algorithm proposed in [Li, Q., Zheng, B.,  Zheng, Y.T.: An efficient nonmonotone adaptive cubic regularization method with line search for unconstrained optimization problem. Appl. Math. Lett.  98, 74-80 (2019)] under the assumption that the objective function is strongly convex.

(Communicated by Jie Sun)

    In addressing the challenges of nonconvex, nonsmooth, and nonseparable optimization, this paper introduces a novel symmetric inertial Bregman Alternating Direction Method of Multipliers (SIBADMM). This algorithm extends the symmetric ADMM by incorporating an inertial mechanism and Bregman divergence within the x-subproblem, aiming to accelerate convergence. The asymptotic convergence is established under the assumption of boundedness in the sequence generated by the algorithm, utilizing the Kurdyka-Łojasiewicz property. The practical utility of SIBADMM is demonstrated through its application to various linear regression problems with different regularization terms, thereby showcasing its effectiveness.

(Communicated by Jie Sun)

    Consider the nonconvex and nonsmooth optimization problem

                   $\min\limits_{x\in \mathbb{R}^n }\ \psi(x)= \frac{1}{m} \sum \limits_{i=1}^m f_i(x)+r_1(x)-r_2(x),$                                                  (1)

where for each $ i \in \{1, 2, \ldots, m\} $, the function $f_i: \mathbb{R}^n \to (-\infty, \infty] $ is smooth. The functions $r_1: \mathbb{R}^n \to (-\infty, \infty] $ and $ r_2: \mathbb{R}^n \to (-\infty, \infty] $ are nonsmooth convex functions. In this paper, we propose a stochastic variance-reduced proximal difference-of-convex algorithm (SVRPDCA) for $(1)$, i.e.

for  $k=1:N-1$  do

S1  if $\mod(k,q)==0$, calculate the full gradient $v_k = \nabla f(x_k),$

S2  else

S3  Randomly select a subset $\mathcal{M}_k\subseteq \{1,2,\ldots,n\}$ such that $|\mathcal{M}_k|=b$, and compute

$v_{k}=\frac{1}{b} \sum_{i \in \mathcal{M}_{k}} \nabla f_{i}(x_{k})-\nabla f_{i}(x_{k-1})+v_{k-1},$

S4  end if

S5  Compute $\xi_{k} \in \partial r_{2}(x_{k})$ and update $x_{k+1}$  by

$x_{k+1}=\arg\min _{x \in \mathbb{R}^{n}}\left\langle x-x_{k}, v_{k}-\xi_{k}\right\rangle+r_{1}(x)+\frac{1}{2 \alpha}\|x-x_{k}\|^{2}.$                             (2)

end for

Output: Return $x_{R}$, where $R$ is chosen uniformly at random from $\{1,\cdots, N-1\}$.

(Communicated by Akiko Yoshise)

    In this paper, we present a predictor-corrector interior-point algorithm (PC IPA) with new search directions to solve $P_*(\kappa)-$weighted horizontal linear complementary problem (WHLCP). $P_*(\kappa)-$WHLCP includes monotone WLCP, $P_*(\kappa)-$horizontal linear complementary problem (HLCP), $P_*(\kappa)-$linear complementary problem (LCP), monotone LCP and convex quadratic programming as special cases, and could model a wide range of equilibrium problems in scientific engineering and economic management. The main idea of our PC IPA is transforming the centering equations of the central path by the algebraic equivalent transformation (AET) technique based on a power function with an arbitrary positive integer $q$. Upon analyzing the effect of different $q$ on the transformed system, we select a power function $\varphi(t) = t^{\frac{5}{2}}$ in order to get the search direction. The feasibility and global convergence of the proposed algorithm are verified under appropriate conditions. Additionally, the iteration bound of our algorithm is comparable to the best-known bounds for such available IPAs. The efficacy of the proposed algorithm is demonstrated through the presentation of numerical results.

(Communicated by Xinwei Liu)

    Due to the significant need for efficiently handling the huge data sets which mostly emerge in the contemporary world models, here we focus on modifying a well-known memoryless continuous optimization algorithm as well as improving a classic data compression model. With these issues at the forefront, firstly we suggest two optimal settings for the parameter of the Dai--Liao conjugate gradient method by approaching the search direction of the method to that of the efficient memoryless BFGS quasi--Newton method in an ellipsoid norm least-squares framework. Then, we deal with modifying the optimization model of the nonnegative matrix factorization problem. More precisely, we add penalty terms to the classic least-squares models of the nonnegative matrix factorization subproblems in the popular alternative solution process, as a plan to control collinearity between the columns/rows of the factorization elements. We put our theoretical assertions to the test on the CUTEr problems as well as several random nonnegative matrix factorization cases, and assess the outputs in various aspects. The outputs generally show the acceptable impact of our modifications.

(Communicated by Xinwei Liu)

    The alternating direction method of multipliers (ADMM) is one of the effective methods for solving two-block separable optimization problems with linear constraints, and has been widely applied in image processing, power systems, sparse learning, and other fields. Its essence is the application of the Douglas-Rachford splitting method to the dual problem. Classical ADMM updates the Lagrange multipliers only once per iteration, while symmetric ADMM achieves dual updates of the Lagrange multipliers in each iteration by introducing an additional multiplier update step, thereby significantly improving the convergence performance and numerical stability of the algorithm. In this paper, we propose a novel symmetric ADMM algorithmic framework for two-block separable nonconvex optimization problems with linear constraints, which introduces two distinct relaxation factors to enhance the flexibility and convergence efficiency of the algorithm. This method has been comprehensively studied for convex problems. However, for nonconvex problems, in the convergence analysis of symmetric alternating direction method of multipliers with two different relaxation factors without introducing Bregman distances or regularization terms, proving the monotonicity of the Lagrangian function remains a challenging problem.

    In terms of theoretical analysis, we establish the convergence theory of the proposed algorithm in this paper. Notably, our convergence proof does not rely on common technical assumptions such as Bregman distances or regularization terms. Specifically, by constructing a novel auxiliary function and under the mild condition that the Kurdyka–Łojasiewicz (KŁ) inequality is satisfied, we prove that the iterative sequence generated by the algorithm converges to a stationary point of the problem.

    The main contributions of this paper can be summarized as follows: First, we design a symmetric ADMM scheme with dual relaxation factors to solve two-block separable nonconvex and nonsmooth optimization problems with linear constraints, and the proposed method allows for a wider range of parameters, which can better adapt to the structural characteristics of different problems through flexible adjustment of relaxation parameters. Second, we establish a concise convergence analysis framework that does not depend on Bregman distances and regularization terms, reducing the complexity of theoretical analysis; moreover, it can degenerate into the classical ADMM. Finally, we validate the practical application effectiveness of the proposed algorithm through numerical experiments, and the experimental results demonstrate that the algorithm outperforms traditional ADMM and its variants in terms of convergence speed and solution accuracy.

(Communicated by Zheng-Hai Huang)

    Tensor low rank approximation is an important tool in tensor data analysis and processing. In the sense of tensor-tensor product (T-product) derived from general invertible transformation, the best low tubal-rank approximation of third order tensors can be obtained through truncated tensor singular value decomposition (T-SVD). In this paper, we first present two deterministic frequent directions type algorithms for near optimal low tubal-rank approximations of third order tensors. Moreover, we propose a randomized frequent directions algorithm for near optimal low tubal-rank approximations of third order tensors. Corresponding relative error bounds for the presented algorithms are derived. The related numerical examples on third order tensors of color image, grayscale video and synthetic data with larger scale illustrate the favorable performance of the presented methods compared to some existing methods.

(Communicated by Guanglu Zhou)

The crucial component of tensor theory is the $H$-tensor. An even order real symmetric nonsingular $H$-tensor is positive definite tensor and it has a significant impact on tensor complementarity theory. In this article, we introduce generalized $S$-Nekrasov tensor, a new subclass of $H$-tensor. We introduce the concept of weak Nekrasov tensor and provide a sufficient condition for a weak Nekrasov tensor to be $H$-tensor. We prove that the solution set of tensor complementarity problem is nonempty and compact for a real generalized $S$-Nekrasov tensor with positive diagonal entries.

(Communicated by Walaa Moursi)

In this paper, a class of three-dimensional subspace conjugate gradient algorithm based on the cubic regularization model is discussed. The regularization parameter in the cubic regularization model are updated by an interpolation function. According to the criterion of approximate model, the algorithm adaptively selects the quadratic approximation or the cubic regularization model to approximate the objective function by adjusting the regularization parameter. The corresponding subspace conjugate gradient algorithm is obtained by minimizing the approximation model of the objective function in the given subspace. The algorithm has global convergence for the general non-convex objective functions, and numerical experiments imply that the algorithm is robust and efficient.

(Communicated by Xinmin Yang)

Lately, a novel swarm intelligence model, namely the consensus-based optimization (CBO) algorithm, was introduced to deal with the global optimization problems. Limited by the conditions of It$\hat{\rm o}$'s formula, the convergence analysis of the previous CBO finite particle system mainly focuses on the problem with smooth objective function. With the help of smoothing method, this paper achieves a breakthrough by proposing an effective CBO algorithm for solving the global solution of a nonconvex, nonsmooth, and possible non-Lipschitz continuous minimization problem with theoretical analysis, which dose not rely on the mean-field limit. We indicate that the proposed algorithm exhibits a global consensus and converges to a common state with any initial data. Then, we give a more detailed error estimation on the objective function values along the state of the proposed algorithm towards the global minimum. Finally, some numerical examples are presented to illustrate the appreciable performance of the proposed method on solving the nonsmooth, nonconvex minimization problems.

(Communicated by Ting Kei Pong)

By utilizing $T$-algebras, we introduce the concepts of $w$-$P$ and $w$-uniqueness properties for nonlinear transformations, and investigate some interconnections among these concepts.

The results above are applied to relaxation transformations and self-adjoint linear transformations. Moreover, we study the finiteness of $w$-solutions for the homogeneous cone complementarity problem.

(Communicated by Fanwen Meng)

Subspace minimization conjugate gradient (SMCG) methods are highly efficient iterative schemes for unconstrained optimization and have garnered increasing attention recently. The inertial step strategy is often utilized to speed up iterative methods. By taking advantage of the information at the recent three iterations further, we exploit a new adaptive inertial step strategy. We design a new update form for regularization parameter, and extend the search direction in the SMCG method based on regularization model to the solution of constrained nonlinear equations by incorporating the resulting adaptive inertial step strategy. Based on the resulting search direction and the projection technique, we present an efficient inertial SMCG method based on regularization model for solving constrained nonlinear monotone equations, where the search direction satisfies the sufficient descent property and the trust region property independent of any line search. The global convergence and improved convergence rate of the proposed method are established under the assumptions without imposing the Lipschitz continuity on the underlying mapping. Numerical experiments demonstrate that the proposed method is very promising.

(Communicated by Chen Ling)

In the fields of computer graphics and robot kinematics, the application of dual quaternions has become a common trend, but in-depth research  on dual quaternion matrix theory remains relatively scarce. This paper aims to propose an efficient strategy for solving the dual quaternion matrix equation $AXB=C$. First, we investigate the  unique theoretical framework of the dual quaternion matrix vector operator and combine it with the complex representation of dual quaternion matrix. This approach provides a powerful tool for deriving equivalent forms of dual quaternion matrix equations, enabling us to obtain the general solution of $AXB=C$. Subsequently, we analyze the structure of the $\eta$-Hermitian matrix and propose the $GH$-representation that significantly simplifies the computation process when solving for the $\eta$-Hermitian solution. Finally, we design corresponding algorithms and validate their effectiveness through numerical experiments.