(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.$                                                      (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 

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

$$
\begin{aligned}
\min \quad & f(x), \\
\text{s.t.} \quad & g(x) \leq 0,
\end{aligned}
\tag{1}
$$

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

$$
\begin{aligned}
\min \quad & f(x), \\
\text{s.t.} \quad & g_{\max}(x) \leq 0
\end{aligned}
\tag{2}
$$

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:

            - $\tau(z, t) = 0$ for $z \leq -\beta v(t)$ with $\beta > 1$,
            - $\tau(z, t) = 1$ for $z \geq -v(t)$,

            - $g_{m+1} = -v(t)$.

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

$$
\tau(z, t) =
\begin{cases} 
0, & \text{for } z \leq -\beta v(t) \text{ with } \beta > 1, \\
1, & \text{for } z \geq -v(t), \\
\bar{\tau}(z, t), & \text{otherwise},
\end{cases}
\tag{3}
$$

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

$$
\bar{\tau}(-\beta v(t), t) = 0, \quad \frac{\partial \bar{\tau}(-\beta v(t), t)}{\partial z} = 0, \quad \frac{\partial^2 \bar{\tau}(-\beta v(t), t)}{\partial z^2} = 0, \quad \frac{\partial^3 \bar{\tau}(-\beta v(t), t)}{\partial z^3} = 0,
$$

$$
\bar{\tau}(-v(t), t) = 1, \quad \frac{\partial \bar{\tau}(-v(t), t)}{\partial z} = 0, \quad \frac{\partial^2 \bar{\tau}(-v(t), t)}{\partial z^2} = 0, \quad \frac{\partial^3 \bar{\tau}(-v(t), t)}{\partial z^3} = 0.
$$

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

$$
\bar{\tau}(z, t) = \frac{-20(z+v(t))^7}{(\beta v(t)-v(t))^7} + \frac{-70(z+v(t))^6}{(\beta v(t)-v(t))^6} + \frac{-84(z+v(t))^5}{(\beta v(t)-v(t))^5} + \frac{-35(z+v(t))^4}{(\beta v(t)-v(t))^4} + 1. \tag{4}
$$

    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

$$
\hat{g}_{(\theta, \alpha_1, \alpha_2, \beta)}(x, t) = g_{i_1}(x) + \sum_{l=1}^{k-1} c_l \left( l \tau(g_{i_{l+1}}(x), t) g_{i_{l+1}}(x) - \sum_{j=1}^{l} \tau(g_{i_j}(x), t) g_{i_j}(x) + \tau(g_{i_{l+1}}(x), t) \theta t \right)^3, \quad \text{for } (g_i(x))_{i \in I_v(x, t)} \in \Delta_{i_1 \dots i_k}(\theta t). \tag{5}
$$

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 $\Delta_{i_1 \dots i_k} (\theta t).$

$$
\begin{cases} 
g_{i_l}(x) - g_{i_{l+1}}(x) \geq 0, \text{ when } 1 \leq l < k, \\ 
(k-1)g_{i_k}(x) - \sum_{j=1}^{k-1} g_{i_j}(x) + \theta t \geq 0, \\ 
k g_{i_l}(x) - \sum_{j=1}^{k} g_{i_j}(x) + \theta t \leq 0, \text{ when } k+1 \leq l \leq \tilde{k}.
\end{cases}
$$

    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.

---

Algorithm 1   Stochastic Variance Reduced Proximal Difference-of-Convex Algorithm (SVRPDCA)

---

Input:  Let $S$ be an arbitrary positive integer,  initial point $x_0\in \mathbb{R}^n$, set $q = b = [m^{\frac 1 2}]$, $N = Sq$, $\alpha=\frac{1}{2L}.$

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 $\{1,\cdots, N-1\}$.

---

    We also establish that the proposed method attains an $\epsilon$-equilibrium point with a gradient complexity bounded by $O(\sqrt{m}\epsilon^{-2} + m)$ under conditions, where $m$ represents the number of data samples. Numerical experiments validate the efficiency and practical relevance of the proposed algorithm.

(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.

(Communicated by Chen Ling)

In this paper, we consider a class of tensor optimization problem

$\frac{1}{2} \| \hat{\mathcal{X}} \times_{1} A_{1} + \hat{\mathcal{X}} \times_{2} A_{2} + \cdots + \hat{\mathcal{X}} \times_{N} A_{N} - \mathcal{B} \|^{2}$

$=\frac{1}{2} \min_{\mathcal{X} \in \mathbb{R}_{+}^{J_{1} \times J_{2} \times \cdots \times J_{N}}}\| \mathcal{X} \times_{1} A_{1} + \mathcal{X} \times_{2} A_{2} + \cdots + \mathcal{X} \times_{N} A_{N}- \mathcal{B} \|^{2},$

where the tensor $\mathcal{B} \in  \mathbb{R}^{J_{1} \times J_{2} \times \cdots \times J_{N}}$, and the coefficient matrices  $A_{n} \in \mathbb{R}^{J_{n} \times J_{n}},  n = 1, 2, \cdots , N$. In order to find a nonnegative tensor $\hat{\mathcal{X}} \in  \mathbb{R}_{+}^{J_{1} \times J_{2} \times \cdots \times J_{N}}$, we design a nonmonotonic descent stepsize and construct a new gradient projection algorithm to solve this problem by making use of the BB stepsize. The convergence analysis of the new algorithm is also given. Numerical experiments are performed to illustrate the feasibility and effectiveness of the new algorithm, including tests on synthetic data, solutions to microscopic heat transport problems, and image restoration tasks. Comparisons with some previous methods are also given.

(Communicated by Jie Sun)

With the increasing complexity of the problems in practical fields, traditional numerical algorithms are not very ideal for solving non-smooth non-Lipschitz optimization problems. We focus on a broader class of non-smooth non-Lipschitz optimization problems, in which the objective function is the sum of a non-smooth non-convex function and a non-Lipschitz function. Based on the smoothing approximation technique and projection neural network, we propose a smooth inertial projection neural network (SIPNN) by introducing inertial terms into the neural network. Under certain conditions we show that the solution of SIPNN is convergent to the stationary point of non-Lipschitz optimization problems. Compared with other methods employing differential inclusions for solving non-smooth problems, the smoothing approximation technique can effectively overcome the difficulty of choosing subdifferentials, and the introduction of inertial terms can control variables which allows us to explore globally the optimal solution of the original optimization problems. The simulation results are presented to illustrate the performance and effectiveness of the proposed method.

(Communicated by Robert Csetnek )

      In this paper, a new approach to solving the considered optimization problem is proposed for both differential inclusions (DFIs) and inequality constraints. The problem is reduced to a problem with one DFI defined by the intersection of two set-valued mappings. The optimality conditions for such a problem are expressed as the sum of two locally adjoint mappings (LAMs). In turn, this also requires calculating the argmaximum sets and subdifferentials of Hamiltonian functions. Then it is natural that LAM can be computed for a set-valued mapping generated by a system of inequality constraints. Finally, we consider some applications in a linear optimal control problem with polyhedral constraints and a constraint given by a convex cone.

(Communicated by Jie Sun)

      This study develops a data-driven distributionally robust two-stage stochastic programming model with second-order stochastic dominance (SSD) constraints. Uncertainty is modeled within a 1-Wasserstein ball centered at an empirical distribution. Two cases are analyzed based on how uncertainty affects the second stage. In the first case, where uncertainty impacts only the objective

function, the problem can be reformulated as a convex program and further simplified into a tractable conic program. In the second case, where uncertainty influences constraints, the problem is generally NP-hard, as feasibility verification educes to a norm maximization problem over a polytope. However, under specific conditions, it remains tractable. Numerical experiments on a supply chain problem compare the proposed method with Sample Average Approximation (SAA). Results show that for large sample sizes, the proposed approach exhibits greater robustness and stability against sample fluctuations.

(Communicated by Chen Ling)

    This paper studies the real multivariate eigenvalue problem for nonsymmetric tensor tuple. We formulate the tensor real multivariate eigenvalue problem equivalently as polynomial optimization. We generate a random objective function that achieves different values at different multivariate values. Each of them can be computed by Lasserre's hierarchy of semidefinite relaxations if there are finitely many ones. Under suitable conditions, we prove that the semidefinite relaxation has finite convergence for nonsymmetric tensor tuples. Numerical experiments are presented to show the efficiency of the proposed method.

(Communicated by Jein-Shan Chen)

    This paper investigates the minimal planning horizon required to determine optimal production schedules in dynamic lot-sizing models under batch changeover constraints. The classical dynamic lot-sizing problem assumes the ability to make setup decisions each period independently. However, in many industrial environments such as semiconductor manufacturing, pharmaceutical production, or automotive component assembly, batch-based changeovers are required. This introduces nontrivial temporal coupling between decision periods, rendering classical planning horizon results inapplicable.

    We study a generalized dynamic lot-sizing model where products are subject to joint setup constraints over contiguous planning periods---so-called \emph{batch changeovers}. Our main contribution is the derivation of sufficient conditions under which finite minimal planning horizons exist, allowing for optimal production decisions to be made without requiring infinite forward-looking forecasts.

    Let $T$ denote the length of the planning horizon, and suppose demands $\{d_t\}_{t=1}^T$ are deterministic. Each product incurs a per-unit production cost $c_t$, a holding cost $h_t$, and a fixed batch setup cost $S$. The novelty in our model lies in the constraint that setup decisions apply to batches of periods, i.e., production is constrained to occur only if a batch-wide setup is activated.

    We show that under reasonable assumptions (bounded holding and production costs, stable demand), the optimal policy structure retains certain properties akin to the classical Wagner-Whitin model, such as the existence of \emph{non-speculative production}. However, the batch setup constraint introduces a new layer of complexity: the optimal solution may depend on multi-period patterns that require careful treatment to determine horizon sufficiency.

    To formalize this, we derive a minimal planning horizon bound $T^*$, such that for any $T \geq T^*$, the truncated problem yields the same initial-period decision as the infinite-horizon problem. This is achieved by constructing a cost-to-go function and showing it satisfies monotonicity and submodularity properties under batch constraints.

    Numerical experiments on benchmark instances demonstrate that the proposed minimal planning horizon bound is tight in practice and significantly reduces computational burden without compromising optimality. Moreover, we propose a dynamic programming algorithm tailored to exploit the batch structure, improving runtime efficiency by an order of magnitude compared to standard MILP solvers.

    Our results contribute to the theoretical understanding of planning horizons in complex lot-sizing settings and provide practical tools for industries facing setup-intensive operations with limited forecast visibility.

(Communicated by Jie Sun)

      This paper presents a novel multi-objective location-routing problem with time windows for the effective management of multimodal transportation networks at two levels: the upper level addresses multimodal transportation issues, whereas the lower one focuses on the location-routing aspects. In our model, two objectives are included referring to maximizing customer satisfaction and minimizing total costs, aggregating transportation, transit, and location costs. To solve the problem efficiently, we introduce a new approach based on a squirrel search algorithm using three crossover and four mutation operators designed to handle Pareto optimality in multi-objective optimization. Besides, we present a greedy clustering algorithm combined with Metropolis criterion to generate high-quality initial solutions and accelerate convergence. We evaluate the performance of proposed approach using various scale instances in multimodal transportation networks. Our experimental results suggest the efficiency and scalability of our approach.

(Communicated by Guanglu Zhou)

     In this paper, we focus on the mixed-constrained semi-infinite programming and present an adaptive augmented Lagrangian filter method. Continuous infinite inequality constraints are smoothed and transformed into equivalent finite constraints in integral form, which are then combined with the objective function via adaptive parameters. Compared with the existing methods, the new approach is a penalty-function-free method that employs a filter instead of forcing sequences to make the optimality measure approach to zero. Our algorithm also includes a feasibility recovery phase to quickly detect infeasible problems. The global convergence is proved under some suitable conditions. Numerical results demonstrate that the proposed method is effective.

(Communicated by Jie Sun)

       In this paper, we consider the 3-block linearly constrained difference-of-convex (DC) optimization problems:

$\min_{x,y,z}\ \{ \ f(x) + g (y) + h(z) \mid A x +By+Cz =b \ \},$                                                       (1)

where

$f(x) = f_1(x)-f_2(x)~ ~\text{and}~~  g(y) = g_1(y)-g_2(y),$

$h:\mathbb{R}^{n_3}\rightarrow \mathbb{R}$ be a continuous differentiable convex function with Lipschitz continuous gradient, $A\in \mathbb{R}^{m\times n_1}$, $B\in \mathbb{R}^{m\times n_2}$ and $C\in\mathbb{R}^{m\times n_3}$ are given matrixes, $b\in \mathbb{R}^{m}$ is a vector, with $f_1:\mathbb{R}^{n_1}\rightarrow \mathbb{R}\cup\{+\infty\}$ and $g_1:\mathbb{R}^{n_2}\rightarrow \mathbb{R}\cup\{+\infty\}$  are  proper closed convex functions, $f_2:\mathbb{R}^{n_1}\rightarrow \mathbb{R}\cup\{+\infty\}$ and $g_2:\mathbb{R}^{n_2}\rightarrow \mathbb{R}\cup\{+\infty\}$ are continuous convex functions. We propose a majorized Bregman ADMM to solve the DC problem (1). Compared with the classical Bregman ADMM, the majorized Bregman ADMM only requires solving convex subproblems at each iteration, rather than DC subproblems. We prove that the sequence generated by the proposed method converges to a critical point of the augmented Lagrangian function, under the assumption that the potential function satisfies the Kurdyka-Łojasiewicz (KŁ) property. Preliminary numerical experiments are conducted to support our theoretical analysis.

(Communicated by Lingchen Kong)

      We study a complex Newton-based method with a correction step for homogeneous polynomial optimization on the complex unit sphere. In particular, its constrained stationary point meets the definition for being a US-eigenpair of the corresponding to symmetric complex tensor.

      We analyze the local quadratic convergence rate of the complex Newton-based method with a correction step, provided that a sufficiently close initial point. As two concrete applications, we apply it to two tasks: computing the US-eigenpairs of symmetric complex tensors; and computing the geometric measure of entanglement of quantum multipartite pure states. We provide several numerical examples of our complex Newton-based method with a correction step, and observe that it performs fast and effective on these two tasks.

(Communicated by Fanwen Meng)

    The split feasibility problem (SFP) has many important and wide applications, such as radiotherapy, image reconstruction, and signal processing. Let $C$ and $Q$ be nonempty closed convex sets in $\Re^{n}$ and $\Re^{m}$, respectively, and $A$ an $m \times n$ real matrix, this kind of problem is about finding

$x\in C, ~\mathrm{s.t.}~Ax\in Q,$                                                                             (1)

if such $x$ exists.

    Recent researches on the SFP have been deeply integrated with inertial acceleration method, and some scholars have discovered that these techniques can speed up the process of first-order optimization algorithms. However, the most of them limit the range of inertial factors and cause the sequence $\|x^{k}-z\|(z\in S)$ no longer monotonically non-increasing. The alternating inertial method, expressed as

$w^{k}=\begin{cases}x^{k},&\text{if ~}k\text{~is even},\\ x^{k}+\theta _{k}( x^{k}-x^{k-1}) ,&{\text{if ~}k\text{~is odd}},\end{cases}$                                                      (2)

is a appropriate improvement strategy, which retains the acceleration effect of inertial method while alleviating the degree of oscillation.

    In this paper, we propose three KM-CQ-like alternating inertial algorithms that all expand the range of inertial factors. We use projections onto general closed convex sets in the first version, which exploring the acceleration effects of alternating inertial technique preliminarily. The second version simplifies the computation of procedure by relaxing projections onto half-spaces. The last version further optimizes the previous methods by incorporating double projection technique.

    Our specific contributions in this paper are summarized as follows.

    (i) The proposed three KM-CQ-like algorithms incorporate alternating inertial steps allowing them to improve the convergence of the algorithms without inertial steps, which improve the range of values of inertial factors and simplify the form of limiting condition.

    (ii) The second algorithm also add relaxation effects that allows it to accelerate the convergence of the first algorithm and simplify the calculation of projections. On the other hand, we can see it is faster in Example 4.2.

    (iii) The last algorithm use double projections that are generated by half-spaces in the previous iteration and the current iteration. Similar to the expected conclusions, it converges faster than the second algorithm in Example 4.2.

    (iv) The convergence of the iterative sequences generated by the proposed algorithms is established, with the monotonicity of sequence $\|{x}^{2k}-z\|(z\in S)$, which alleviates the oscillation of the iterative points.

    (v) The performance and advantages of the algorithms proposed in this paper are confirmed by two applications in a simple the SFP and signal processing. On the other hand, the increase of dimension will make the advantages of our algorithms more obvious.

    Under simple parameter settings, these algorithms restore the monotonicity of $\|x^{2k}-z\|$ and achieve the convergence of the algorithm. Experiments demonstrate the feasibility of each algorithm in practical applications and the effectiveness of the acceleration procedure.

(Communicated by Kok Lay Teo)

    In this study, we examine the linear quadratic (LQ) optimal control problem for large-scale interconnected systems, with a focus on designing sparse static state-feedback controllers. Traditional LQ control methods often result in dense feedback matrices that require global information sharing among all subsystems. While such dense designs may be optimal in terms of performance, they are often impractical in distributed or resource-constrained environments due to excessive communication and implementation costs. Consequently, this paper seeks to develop a framework that simultaneously promotes sparsity in the controller while maintaining an acceptable level of system performance degradation.

    To achieve this, we formulate a constrained optimization problem where the primary objective is to minimize the number of nonzero entries in the feedback matrix $K$, expressed via the nonconvex and discontinuous $\ell_0-$norm. This sparsity-promoting goal is constrained by two critical system-level considerations: (i) the cost associated with the closed-loop system's response to disturbances must not exceed a predefined threshold above the optimal LQ cost, and (ii) the stability of the closed-loop system must be preserved via a Lyapunov-type equation. The cost of the closed-loop system is defined as the trace of a controllability Gramian $P$, which satisfies the Lyapunov-type equation involving the feedback matrix $K$. The constraint on system performance introduces a tradeoff parameter $\gamma > 0$, which allows the system designer to control the degree of allowable performance loss relative to the optimal centralized LQ controller. This parameter directly impacts the achievable sparsity of the feedback matrix: larger values of $\gamma$ encourage sparser solutions at the expense of higher communication cost, and vice versa.

    Given the nonconvexity introduced by the $\ell_0-$norm and the complexity of the Lyapunov-based equality constraint, solving this problem directly is computationally challenging. To address this, we adopt an augmented Lagrangian framework that transforms the constrained optimization into an unconstrained minimization problem. The resulting augmented Lagrangian function comprises three components, namely the $\ell_0-$norm of the feedback matrix, an indicator function representing the constraint on the cost function, and a matrices quadratic equation associated with the Lyapunov-tpe equation, involving a matrix of Lagrange multipliers. This formulation separates the objective into two nonsmooth components, corresponding to the sparsity and cost constraints, coupled through a smooth term arising from the system dynamics and Lyapunov relation. To solve the augmented problem, we propose a Proximal Linearized Minimization (PLM) method tailored to the two-block structure of the augmented objective function. Each iteration alternates between updating $K$ and $P$ using a proximal-gradient step designed for nonsmooth and nonconvex functions. The convergence of our method is analyzed using the Kurdyka-Łojasiewicz (KŁ) inequality, a powerful framework for analyzing nonconvex and nonsmooth optimization algorithms. Under mild regularity conditions, we prove that the PLM algorithm converges to a critical point of the augmented objective function. This analysis guarantees stability and consistency of the proposed framework, making it suitable for control system applications where reliable performance is essential.

    We demonstrate the effectiveness of our approach through a series of numerical experiments on interconnected systems. The results show that our method can identify sparse feedback matrices that achieve comparable closed-loop performance to traditional optimal control designs, but with significantly reduced controller complexity and communication requirements.

(Communicated by Jie Sun)

    This paper investigates a class of optimal control problems characterized by piecewise constant time delays, which frequently arise in engineering, biological, and networked systems. These problems pose significant challenges due to the discontinuous nature of the delays and the resulting complexity in computing gradients required for optimization. Within the control parameterization framework, the original infinite-dimensional problem is transformed into a finite-dimensional nonlinear programming problem, enabling the application of gradient-based optimization techniques. However, the traditional variational method for computing gradients of cost and constraint functions becomes increasingly inefficient as the control discretization is refined. Although the co-state method is well known for its computational efficiency in delay-free optimal control problems, its application to systems with piecewise constant time delays has been hindered by the absence of an explicit co-state system. In this work, we rigorously derive the co-state system corresponding to such problems, facilitating efficient and accurate gradient computation. Building on this derivation, we propose a computational framework that significantly accelerates the solution process without compromising accuracy. Numerical experiments demonstrate that the co-state method offers substantial improvements in computational efficiency over the variational approach, particularly in cases involving fine control discretization.

(Communicated by Maria Josefa Canovas)

     We propose a vector linear programming formulation for a non-stationary, finite-horizon Markov decision process with vector-valued rewards. Pareto efficient policies are shown to correspond to efficient solutions of the linear program, and vector linear programming theory allows us to fully characterize deterministic efficient policies. An algorithm for enumerating all efficient deterministic policies is presented then tested numerically in an engineering application.