This paper delves into second-order optimality conditions for a strict minimal solution of a set optimization problem in terms of a kind of generalized second-order radial derivatives. Firstly, we propose a notion of strict minimal solution related to set criterion and discuss some of its properties. Secondly, we introduce a new class of generalized second-order derivatives for set-valued maps by utilizing generalized radial sets, which are defined by Minkowski difference. Then, we discuss the relationships between the introduced derivatives and some other existing derivatives, and establish several of their properties. Moreover, based on the introduced derivatives, the optimality conditions and a scalarization theorem for strict minimal solution are established. Finally, we give an application to a robust multi-objective programming. Several concrete examples are given to exemplify the main results.

The paper explores the existence of subdifferentials for set-valued maps involving Minkowski difference, a concept introduced by Karaman (Optimization 24 (3) (2020) 709–722), and elaborates on it through specific examples and theoretical analysis. Firstly, we introduce several lemmas as a foundation for the subsequent proofs. Then, we present a detailed proof of the existence of subdifferentials, employing a novel proof method that does not rely on the assumptions of cone-convexity, upper boundedness, and upper semicontinuity. Finally, we provide illustrative examples to demonstrate that the existence of subdifferentials established in this paper generalizes and extends the conclusions in previous literature.

This paper is devoted to the investigation of the proper strict efficient solutions to a set optimization problem with a partial set order relation. Firstly, the notion of proper strict efficient solution defined by the Minkowski difference is introduced, and it is worth mentioning that the introduced strict efficiency is different from those in the existing literature. Secondly, a class of generalized contingent derivatives for set-valued maps is proposed, which are characterized in terms of a set criterion. Finally, the necessary and sufficient optimality conditions and a scalarization theorem for proper strict efficiency are established. Some concrete examples are given to illustrate the obtained results.

This paper focuses on second-order optimality conditions for a weak minimal solution of a set optimization problem by using a new kind of generalized second-order radial derivatives. Firstly, we introduce a new notion of generalized second-order outer radial derivative for set-valued maps by utilizing Minkowski difference, discuss its relationships to some existed derivatives, and obtain some of its properties, such as sum and chain rules. Then, based on the introduced derivative, we establish the optimality conditions for a weak minimal solution. Finally, we apply the results to the problems of uncertain multi-objective programming and shortest path. Some of our results improve and imply the corresponding ones in the recent literature.

The aim of this paper is to study the semicontinuity of the sets of approximate solutions for parametric set optimization problems (PSOPs). We use the generalized Hiriart-Urruty oriented distance function to define a metric in the Hausdorff sense, which allows us to examine the continuity of parametric scalarization functions (PSFs). Furthermore, we explore the relationship between the solution sets for the PSOPs and the parametric equilibrium problems (PEPs). We demonstrate that the weak l-minimal approximate solution to the PSOPs is equivalent to the approximate solution of the PEPs. Finally, the semicontinuity of the solution mappings of the PSOPs is obtained by the scalarization methods.

In this note, we show that there are some gaps in Proposition 3.1 and Theorem 4.2 of Han and Yu [Optimization 74(1), 219-238(2023)]. We present the refined versions of [Han WY, Yu GL. Directional derivatives in set optimization with the set order defined by M :: minkowski difference. Optimization. 2023;74:219-238, Proposition 3.1 and Theorem 4.2]. Some examples are provided to illustrate the main results.

The twin extreme learning machine (TELM), based on the hinge-loss function, demonstrates significant potential for pattern classification tasks. However, the hinge-loss function, which minimizes the shortest distance between sets, results in classifiers that are sensitive to noise and unstable in the presence of overfitting. To enhance TELM's performance, a novel learning framework, termed adaptive fractional loss TELM (AFTELM), is proposed. This framework incorporates an adaptive fractional loss (AF-loss) function, offering improved robustness to noise compared to TELM with hinge loss. A theoretical analysis is provided to examine the noise insensitivity of AFTELM. The concave-convex procedure (CCCP) is employed for efficient optimization. Extensive experiments on benchmark datasets validate the superior performance of AFTELM, demonstrating its robustness to noise and enhanced classification ability.

Based on subdifferentials and conjugate functions, we obtain conjugate duality theorems of set optimization problem under set-order relations. This paper has two main purposes. One is to put forward a new notion of subdifferential of set-valued maps based on m-order relations, and establish some properties of the subdifferential, such as convexity, closedness and homogeneity. The other is to propose new conjugate and biconjugate maps of set-valued maps via the Minkowski difference, obtain some relations among the maps and the subdifferential, and establish optimality conditions, weak and strong conjugate duality theorems of set-order solutions to set optimization problem. Finally, we apply the main results of the paper to uncertain optimization problems. communicated by Tuyen Van Nguyen.

In this paper, a novel learning model, namely the Lp-norm sparse Blinex Twin Extreme Learning Machine (PBLTELM), is proposed, which is efficient and accurate for relatively large-scale data classification. The proposed framework incorporates three key innovations: the robust Blinex loss function is integrated to enhance generalization, Lp-norm (0 < p < 1) sparsity constraints are adopted to approximate L0-norm solutions while ensuring computational tractability, and a dual-layer optimization strategy that combines iterative weight updates with the adaptive moment estimation (Adam) algorithm is developed to address the resulting non-convex and non-smooth problem. Theoretical analysis is conducted to verify convergence to local stationary points, which ensures both computational efficiency and model accuracy and properties particularly critical for relatively large-scale applications. Comprehensive empirical evaluations are performed across diverse benchmarks, including a two-dimensional artificial dataset, the CMU facial expression dataset, 12 UCI datasets, and 7 relatively large-scale libsvm datasets, to assess the performance of PBLTELM. The results indicate statistically significant improvements in classification accuracy and computational speed compared with state-of-the-art methods, confirming that PBLTELM serves as a scalable and competitive solution for relatively large-scale classification tasks.

A novel e-zone-insensitive two-parameter pinball loss function tailored for large-scale binary classification is in troduced in this study. By integrating this loss with a capped L2,p-metric, a robust sparse classification framework termed CL2,p-TPSP-TSVM is proposed, which is designed to jointly optimize computational efficiency and outlier robustness. Support vector cardinality of the model is dynamically regulated via parametric adaptation of S, s, and e, which allows for scalable processing of high-dimensional data. To suppress outlier interference, a mecha nism for minimizing intra-class distance dispersion under the capped L2p-norm is incorporated into the model. To address the inherent non-convexity and non-smoothness of the optimization problem, a convergent iterative algo rithm is devised, with the property of monotonic descent guaranteed. Each iteration is decomposed into sequential convex subproblems with closed-form solutions, which ensures computational tractability. Empirical evaluations conducted on 10 large-scale benchmark datasets show statistically significant improvements in classification accuracy and computational efficiency, while the model retains robustness in comparison with state-of-the-art methods. This framework provides support for the advancement of scalable, high-performance machine learning in noisy, high-dimensional regimes.

Given a vector x and a closed convex cone C in an n-dimensional inner product space. If x is not in the dual cone of C, then the maximal angle between x and C is greater than pi/2 . In this case, a formula regarding the maximal angle between x and C is given in terms of the metric projection of -x on C. Critical angles between two convex cones that are greater than or equal to pi/2 are shown to be Nash angles by using this formula. Furthermore, some properties of critical pairs of the cone that is the sum of the n x n positive semidefinite cone and the cone of all n x n symmetric nonnegative matrices are presented. Since the n x n copositive cone is the same as the sum of the n x n positive semidefinite cone and the cone of all n x n symmetric nonnegative matrices for n <= 4, a detailed discussion on how to obtain the angular spectrum of the copositive cone of order 3 is given using the results proved in this paper.

Aiming at the problem that the traditional Twin Support Vector Machine (TSVM) is sensitive to noise and outliers, this paper proposes a twin support vector machine model based on generalized adaptive Huber loss function (GAHTSVM). Via dynamically adjusting the robust parameters s , the model can effectively suppress the adverse effects of noisy data points on the decision function, and improve the sparsity of the model by introducing insensitive region design. For non-convex optimization problems, concave-convex process (CCCP) is adopted to avoid quadratic programming problems and significantly improve the efficiency of the algorithm. Experiments show that GAHTSVM performs well in the scenarios where Gaussian noise is added to real-world datasets and outliers are introduced to artificial datasets: it is significantly better than other algorithms in low noise environment; In the high noise environment, it still maintains a leading position, and most datasets are ranked in the top; It performs well on two artificial datasets and is significantly superior to other algorithms.In conclusion, the model effectively overcomes the noise and outlier interference by dynamically adjusting the outlier penalty, and provides an efficient solution to the pattern recognition problem in high noise environment.

This paper presents an enhanced inertial Tseng’s extragradient method designed to address variational inequality problems involving pseudomonotone operators, along with fixed point problems governed by quasi-nonexpansive operators in real Hilbert spaces. Provided that the parameters satisfy appropriate conditions, the proposed method is shown to converge strongly. Finally, we provide computational results and illustrate their utility through optimal control applications. These aim to show the efficacy and superiority of the proposed algorithm compared with some existing algorithms.

To address the current limitation of single value prediction models in ensuring high prediction accuracy, this paper introduces a novel smooth pinball loss twin extreme learning machine quantile regression (STELMQR) model. STELMQR incorporates quantile parameters into two non-parallel hyperplanes, effectively capturing the asymmetric and heterogeneous distribution characteristics of data points across different quantile levels. By utilizing smooth pinball quantile loss, the model minimizes the impact of noise, enhancing its robustness. Furthermore, STELMQR leverage a dual coordinate descent algorithm, which accelerates the model training process and optimizes fewer variables compared to other quantile regression models. To validate the efficacy of the proposed models, numerical experiments were conducted on three types of artificial datasets and nine UCI benchmark datasets. The results demonstrate the validity and superiority of STELMQR in handling complex data distributions and improving prediction accuracy.

In this paper, inertial iterative algorithms based on auxiliary principle are proposed for solving linearly constrained monotone equilibrium problems (LCMEP) via an auxiliary principle, which is to construct an auxiliary equilibrium problem and show that a solution of the auxiliary problem is also a solution to the original problem. The convergence results of the inertial iterative algorithm are established under some mild assumptions. We obtain the worst-case convergence rate O(1/t) of the proposed algorithm in the nonergodic case. Furthermore, we propose an self-adaptive inertial iterative algorithm for solving LCMEP, which can improve the convergence rate and robustness of the non-adaptive inertial iterative algorithm and reduce the uncertainty caused by the selection of fixed inertia parameters. Some customized inertial iterative algorithms are also given by choosing special positive-definite matrix in auxiliary equilibrium problem.

In this paper, we study a class of regularization problems with the smoothly clipped absolute deviation (SCAD) penalty function, which can split into a twice continuously differentiable loss function and the SCAD penalty function. We first establish an expression for the limiting subdifferential of the SCAD penalty function, and further, the graphical derivative of the limiting subdifferential of the SCAD penalty function is derived. This graphical derivative expression is used to establish the necessary and sufficient conditions for strong metric subregularity of limiting subdifferential. Finally, using the second subderivative we provide the necessary and sufficient conditions for the quadratic growth condition of the regularization problems. In particular, we demonstrate the equivalence between the quadratic growth conditions and the strong metric subregularity of the limiting subdifferential at a local minimizer in such regularization problems.

The primary aim of this paper is to explore new ideas regarding second-order subdifferentials for set-valued maps, utilizing the framework of set order relations involving Minkowski difference. We commence by establishing fundamental properties of these subdifferentials, encompassing convexity, closure and the Moreau-Rockafellar theorem. Furthermore, existence theorems of the subdifferentials are derived. In addition, we establish optimality conditions of the m-order robust solutions to uncertain set optimization via the subdifferential. Moreover, we formulate duality theorems between the primal and the Wolfe dual problems. Finally, the paper concludes with an application of our current methodology to the context of two-player zero-sum matrix games.

This paper focuses on optimality conditions for an approximate weak minimal solution of a set optimization problem by using a new kind of Clarke generalized directional derivative. First, by means of a Gerstewitz scalarizing function for a partial set order, we introduce a new notion of generalized directional derivative for set-valued maps, discuss its relationships with an existing derivative, and obtain some of its properties, such as Lipschitz continuity. Then, based on the introduced derivative, we establish the optimality conditions for an approximate weak minimal solution. Some of our results improve the corresponding ones in recent literature.

Given a vector $x$ and a closed convex cone $\mathcal{C}$ in an $n$-dimensional inner product space. If $x$ is not in the dual cone of $\mathcal{C}$, then the maximal angle between $x$ and $\mathcal{C}$ is greater than $\frac{\pi}{2}$. In this case, a formula regarding the maximal angle between $x$ and $\mathcal{C}$ is given in terms of the metric projection of $-x$ on $\mathcal{C}$. Critical angles between two convex cones that are greater than or equal to $\frac{\pi}{2}$ are shown to be Nash angles by using this formula. Furthermore, some properties of critical pairs of the cone that is the sum of the $n\times n$ positive semidefinite cone and the cone of all $n\times n$ symmetric nonnegative matrices are presented. Since the $n\times n$ copositive cone is the same as the sum of the $n\times n$ positive semidefinite cone and the cone of all $n\times n$ symmetric nonnegative matrices for $n\le 4$, a detailed discussion on how to obtain the angular spectrum of the copositive cone of order 3 is given using the results proved in this paper.