This paper unified Wolfe and Mond–Weir-type higher-order symmetric dual multiobjective programs over arbitrary cones. The usual duality theorems are proved for unified dual programs under the assumptions of higher-order [Formula: see text]-[Formula: see text]-convexity. Several known results are obtained as special cases. An example is given to show how higher-order Wolfe and Mond–Weir-type symmetric duals can be extracted from our model. An example is presented to show the existence of higher-order [Formula: see text]-[Formula: see text]-convex functions.

In this paper, we first establish some new properties of a kind of nonlinear scalarization function. Second, based on set approaches, we introduce a class of new definitions of robustness for an uncertain multi-objective optimization problem with a variable domination structure. Finally, by applying the scalarization method, we obtain the necessary and sufficient optimality conditions of the robustness for the uncertain multi-objective optimization problem with a variable domination structure in a more general setting. Some of the obtained results extend and imply the corresponding ones in recent literature.

In this paper, we consider the optimization problems with 0/1-loss and sparsity constraints (0/1-LSCO) that involve two blocks of variables. First, we define a [Formula: see text]-stationary point of 0/1-LSCO, according to which we analyze the first-order necessary and sufficient optimality conditions. Based on these results, we then develop a gradient descent Newton pursuit algorithm (GDNP), and analyze its global and locally quadratic convergence under standard assumptions. Finally, numerical experiments on 1-bit compressed sensing demonstrate its superior performance in terms of a high degree of accuracy.

In this paper, we establish several proper separation theorems for an element and a convex set and for two convex sets in terms of their quasi-relative interiors. Then, we prove that the separation theorem given by [Cammaroto, F and B Di Bella (2007). On a separation theorem involving the quasi-relative. Proceedings of the Edinburgh Mathematical Society, 50(3), 605–610] in Theorem 2.5, is in fact a proper separation theorem for two convex sets in which the classical interior is replaced by the quasi-relative interior. Besides, we extend some known results in the literature, such as [Adán, M and V Novo (2004). Proper efficiency in vector optimization on real linear spaces. Journal of Optimization Theory and Applications, 121, 515–540] in Theorem 2.1 and [Edwards, R (1965). Functional Analysis: Theory and Applications. New York: Reinhart and Winston] in Corollary 2.2.2, through the quasi-relative interior and the quasi-interior, respectively. As an application, we provide Karush–Kuhn–Tucker multipliers for quasi-relative solutions of vector optimization problems. Several examples are given to illustrate the obtained results.

With the implementation of new environmental policies such as “carbon peak” and “carbon neutrality”, reducing carbon emissions through the development of clean technology in the automobile industry has become a key priority. However, the high cost of researching and developing green technology has led to high vehicle prices, which poses a major barrier to expanding the market share of such vehicles. The decision of whether to invest in research and development (R&D) has become a challenging one for automobile manufacturers. In this paper, we propose a game theory analysis scheme to study the R&D investment decisions of two original equipment manufacturers (OEMs) — an electric vehicle manufacturer (EM) and a fuel vehicle manufacturer (FM) — who, respectively, produce electric vehicles (EVs) and fuel vehicles (FVs). Since the manufacturers exhibit bounded rationality and their R&D investment decision-making involves a long-term, continuously learning and adjusting process, we model this dynamic R&D investment decision-making process as an evolutionary game to study manufacturers’ stable evolutionary behaviors in optimal R&D investment strategies. Different from previous literatures, where the prices for vehicles with high or low R&D investment were predetermined, we optimize the price of each vehicle, market shares, and optimal utilities of OEMs using a two-stage Stackelberg game for each investment strategy profile. Additionally, we use the Personal Carbon Trading (PCT) mechanism to help reduce carbon emissions. The main contribution of this paper is exploring the conditions for the evolutionary stable strategies (ESSs) of the evolutionary game based on the optimal utilities of the OEMs under different strategy profiles. The impact of preference parameters and green R&D coefficients on the OEMs’ decisions, as well as consumers’ purchase choices are also discussed. Finally, numerical simulations using real-world data are conducted to verify the theoretical results on ESSs.

This paper investigates the three-stage assembly flow shop scheduling problem, provided that there is a fixed maintenance period (MP) imposed on one of the machines in the first stage, the objective is to minimize the makespan. The starting time and duration of MP are known in advance, during MP no job can be processed on the corresponding machine. Only the non-resumable scenario is considered, i.e., if a job fails to finish before MP, it must restart from the beginning after the machine becomes available again, rather than continuing its processing. The problem generalizes the three-stage assembly flow shop scheduling problem without MPs and is therefore strongly NP-hard. To the best of our knowledge, the problem has not been explored so far. We propose three approximation algorithms for the problem.

In this paper, we develop some three term nonlinear conjugate gradient methods based on the Hestenes–Stiefel (HS), the Polak–Ribière–Polyak (PRP) and the Liu–Storey (LS) methods. The proposed algorithms always generate sufficient descent directions which satisfy [Formula: see text]. When the Wolfe or the Armijo line search is used, we establish the global convergence of the proposed methods in a concise way. Moreover, the linear convergence rate of the methods is discussed as well. The extensive numerical results show the efficiency of the proposed methods.

It is difficult to balance detection speed and detection accuracy in current methods for sampling on feature maps. We combine the parameterless interpolation algorithm used in the traditional method with the algorithm in deep learning to improve shortcomings of the traditional interpolation algorithm by introducing learnable parameters into the algorithm. The new algorithm guarantees the features of the original image by calculating the correlation between feature map channels and pixels to ensure the correct restoration of the image features during up-sampling. In this paper, we propose a new up-sampling operation model — Reparametric Blend Up-sampling (RBU), which uses the method of rearranging pixels to obtain a feature map that matches the up-sampling size, uses blending to compute inter-pixel and inter-channel correlations and back-propagates the model to learn the appropriate parameters to ensure feature extraction. In the field of object detection, this model replaces the original up-sampling operation and improves the mAP mean value by 1.1% and the detection of small target objects by 3.1% in the VOC dataset with the richest application scenarios.

In this paper, we consider how to maximize the liquid welfare in a combinatorial auction where bidders have monotone valuations and are budget constrained. We study the setting that budgets are public information and present a universally truthful, budget feasible and computationally-efficient randomized [Formula: see text]-approximate mechanism, where [Formula: see text] is the number of items and [Formula: see text] is the number of bidders, respectively.

For a positive real number [Formula: see text], the [Formula: see text]-Sombor index of a graph [Formula: see text], introduced by Réti et al, is defined as [Formula: see text] where [Formula: see text] denotes the degree of the vertex [Formula: see text] in [Formula: see text]. By the definition, [Formula: see text] is exactly the Sombor index of [Formula: see text], while [Formula: see text] is the first Zagreb index of [Formula: see text]. In this paper, for [Formula: see text] we present the extremal values of the [Formula: see text]-Sombor index of trees with some given parameters, such as matching number, the number of pendant vertices, diameter. This generalizes the relevant results on Sombor index due to Chen, Li and Wang ((2002). Extremal values on the Sombor index of trees, MATCH Communications in Mathematical and in Computer Chemistry, 87, 23–49). Handling [Formula: see text] appears to be different for [Formula: see text] in contrast to the case when [Formula: see text]. To demonstrate this, we also characterize the extremal trees with respect to the [Formula: see text] with matching number, the number of pendant vertices and diameter. In addition, three relevant conjectures are proposed.