Minimization Problem Research Articles

A minimization problem with free boundary for p-Laplacian weakly coupled system

Abstract In this article, we consider a weakly coupled p p -Laplacian system of a Bernoulli-type free boundary problem, through minimization of a corresponding functional. We prove various properties of any local minimizer and the corresponding free boundary.

A Block Lanczos Method for Large-Scale Quadratic Minimization Problems with Orthogonality Constraints

A Block Lanczos Method for Large-Scale Quadratic Minimization Problems with Orthogonality Constraints

How many clues to give? A bilevel formulation for the minimum Sudoku clue problem

It has been shown that any 9 by 9 Sudoku puzzle must contain at least 17 clues to have a unique solution. This paper investigates the more specific question: given a particular completed Sudoku grid, what is the minimum number of clues in any puzzle whose unique solution is the given grid? We call this problem the Minimum Sudoku Clue Problem (MSCP). We formulate MSCP as a binary bilevel linear program, present a class of globally valid inequalities, and provide a computational study on 50 MSCP instances of 9 by 9 Sudoku grids. Using a general bilevel solver, we solve 95% of instances to optimality, and show that the solution process benefits from the addition of a moderate amount of inequalities. Finally, we extend the proposed model to other combinatorial problems in which uniqueness of the solution is of interest.

Generalized minimum 0-extension problem and discrete convexity

AbstractGiven a fixed finite metric space $$(V,\mu )$$ ( V , μ ) , the minimum 0-extension problem, denoted as $$\mathtt{0\hbox {-}Ext}[{\mu }]$$ 0 - Ext [ μ ] , is equivalent to the following optimization problem: minimize function of the form $$\min \nolimits _{x\in V^n} \sum _i f_i(x_i) + \sum _{ij} c_{ij}\hspace{0.5pt}\mu (x_i,x_j)$$ min x ∈ V n ∑ i f i ( x i ) + ∑ ij c ij μ ( x i , x j ) where $$f_i:V\rightarrow \mathbb {R}$$ f i : V → R are functions given by $$f_i(x_i)=\sum _{v\in V} c_{vi}\hspace{0.5pt}\mu (x_i,v)$$ f i ( x i ) = ∑ v ∈ V c vi μ ( x i , v ) and $$c_{ij},c_{vi}$$ c ij , c vi are given nonnegative costs. The computational complexity of $$\mathtt{0\hbox {-}Ext}[{\mu }]$$ 0 - Ext [ μ ] has been recently established by Karzanov and by Hirai: if metric $$\mu $$ μ is orientable modular then $$\mathtt{0\hbox {-}Ext}[{\mu }]$$ 0 - Ext [ μ ] can be solved in polynomial time, otherwise $$\mathtt{0\hbox {-}Ext}[{\mu }]$$ 0 - Ext [ μ ] is NP-hard. To prove the tractability part, Hirai developed a theory of discrete convex functions on orientable modular graphs generalizing several known classes of functions in discrete convex analysis, such as $$L^\natural $$ L ♮ -convex functions. We consider a more general version of the problem in which unary functions $$f_i(x_i)$$ f i ( x i ) can additionally have terms of the form $$c_{uv;i}\hspace{0.5pt}\mu (x_i,\{u,v\})$$ c u v ; i μ ( x i , { u , v } ) for $$\{u,\!\hspace{0.5pt}\hspace{0.5pt}v\}\in F$$ { u , v } ∈ F , where set $$F\subseteq \left( {\begin{array}{c}V\\ 2\end{array}}\right) $$ F ⊆ V 2 is fixed. We extend the complexity classification above by providing an explicit condition on $$(\mu ,F)$$ ( μ , F ) for the problem to be tractable. In order to prove the tractability part, we generalize Hirai’s theory and define a larger class of discrete convex functions. It covers, in particular, another well-known class of functions, namely submodular functions on an integer lattice. Finally, we improve the complexity of Hirai’s algorithm for solving $$\mathtt{0\hbox {-}Ext}[{\mu }]$$ 0 - Ext [ μ ] on orientable modular graphs.

Adaptive finite element methods for scalar double‐well problem

AbstractSome scalar double‐well problems eventually lead to a degenerate convex minimization problem with unique stress. We consider the adaptive conforming and nonconforming finite element methods for the scalar double‐well problem with the reliable a posteriori error analysis. A number of experiments confirm the effective decay rates of the methods.

Supplier Evaluation in Supply Chain Environment Based on Radial Basis Function Neural Network

The comprehensive evaluation and selection of suppliers under the environment of supply chain management has become a key factor affecting the success of supply chain. How to select suppliers and the strategic partnership between suppliers under the environment of supply chain management has become an important challenge. To solve this problem, this paper takes the supplier evaluation and selection of Guangzhou Automobile Toyota Company as the research object, constructs the index system of supplier comprehensive evaluation and selection, uses the RBF neural network algorithm to establish the supplier evaluation and selection model, and makes an experimental study. The results show that radial basis function neural network is a local approximation network, which has a unique and definite solution to the problem, and there is no local minimum problem in BP network. It is a method that enables enterprises and suppliers to have a clear understanding and seek further promotion together. The research provides theoretical data support for enterprise managers to make decisions.

Finding Optimal Solutions with Neighborly Help

AbstractCan we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighbor instances, that is, instances with one local modification? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems, most notably, graph theory’s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted). We focus on two prototypical graph problems, colorability and vertex cover. For example, we show that it is $$\text {NP}$$ NP -hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in $$\text {P}$$ P . We observe that vertex cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level. Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for $$\text {DP}$$ DP (differences of $$\text {NP}$$ NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For vertex cover, we show that recognizing $$\beta $$ β -vertex-critical graphs is complete for $$\Theta _2^\text {p}$$ Θ 2 p (parallel access to $$\text {NP}$$ NP ), obtaining the first completeness result for a criticality problem for this class.

Analysis of the Impact of Racial Discrimination on Real-life Social Interactions

Race has always been one of the core issues focused on throughout the world. With the technological progress and civilization of mankind, people's living standards have improved to a great extent, and civilization has developed to a greater extent, but the problem of racial discrimination in a variety of scenarios has not been eliminated, and in some ways, racial discrimination can have a very profound impact on a country to a company. In some ways, racial discrimination can have a profound effect on a country or a company. In general, in companies where racial discrimination is a minimal problem, there tends to be less conflict and tension among members. Conversely, when prejudice and discrimination are prevalent among individuals within a company, it can lead to the perpetuation of problems such as unfair communication, cooperation and promotion practices. Ultimately, this can have a negative impact on a company's performance and reputation. Therefore, it is crucial to address the issue of racial discrimination. The purpose of this paper is to analyze and mitigate the impact of racial discrimination on our lives and work through collected data and specific case studies.

A nonmonotone accelerated proximal gradient method with variable stepsize strategy for nonsmooth and nonconvex minimization problems

A nonmonotone accelerated proximal gradient method with variable stepsize strategy for nonsmooth and nonconvex minimization problems

Semi-Proximal ADMM for Primal and Dual Robust Low-Rank Matrix Restoration from Corrupted Observations

The matrix nuclear norm minimization problem has been extensively researched in recent years due to its widespread applications in control design, signal and image restoration, machine learning, big data problems, and more. One popular model is nuclear norm minimization with the l2-norm fidelity term, but it is only effective for those problems with Gaussian noise. A nuclear norm minimization problem with the l1-norm fidelity term has been studied in this paper, which can deal with the problems with not only non-Gaussian noise but also Gaussian noise or their mixture. Moreover, it also keeps the efficiency for the noiseless case. Given the nonsmooth proposed model, we transform it into a separated form by introducing an auxiliary variable and solve it by the semi-proximal alternating direction method of multipliers (sPADMM). Furthermore, we first attempt to solve its dual problem by sPADMM. Then, the convergence guarantees for the aforementioned algorithms are given. Finally, some numerical studies are dedicated to show the robustness of the proposed model and the effectiveness of the presented algorithms.

Comparative dynamic optimization study of batch hydrothermal liquefaction of two microalgal strains for economic bio-oil production

This work presents dynamic optimization strategies of batch hydrothermal liquefaction of two microalgal species, Aurantiochytrium sp. KRS101 and Nannochloropsis sp. to optimize the reactor temperature profiles. Three dynamic optimization problems are solved to maximize the endpoint biocrude yield, minimize the final time, and minimize the reactor thermal energy. The biocrude maximization and time minimization problems demonstrated 11% and 6.18% increment in the optimal biocrude yields and reduction of 78.2% and 61.66% in batch times compared to the base cases for the microalgae with higher lipid and protein fractions, respectively. The energy minimization problem revealed a significant reduction in the reactor thermal energies to generate the targeted biocrude yields compared to the biocrude maximization. Therefore, the identified optimal temperature trajectories outperformed the conventional fixed temperature profiles and could improve the overall economics of the batch bio-oil production from the algal-based biorefineries by significantly enhancing the reactor performance.

Energy or Accuracy? Near-Optimal User Selection and Aggregator Placement for Federated Learning in MEC

To unveil the hidden value in the datasets of user equipments (UEs) while preserving user privacy, federated learning (FL) is emerging as a promising technique to train a machine learning model using the datasets of UEs locally without uploading the datasets to a central location. Customers require to train machine learning models based on different datasets of UEs, through issuing FL requests that are implemented by FL services in a mobile edge computing (MEC) network. A key challenge of enabling FL in MEC networks is how to minimize the energy consumption of implementing FL requests while guaranteeing the accuracy of machine learning models, given that the availabilities of UEs usually are uncertain. In this paper, we investigate the problem of energy minimization for FL in an MEC network with uncertain availabilities of UEs. We first consider the energy minimization problem for a single FL request in an MEC network. We then propose a novel optimization framework for the problem with a single FL request, which consists of (1) an online learning algorithm with a bounded regret for the UE selection, by considering various contexts (side information) that influence energy consumption; and (2) an approximation algorithm with an approximation ratio for the aggregator placement for a single FL request. We thirdly deal with the problem with multiple FL requests, for which we devise an online learning algorithm with a bounded regret. We finally evaluate the performance of the proposed algorithms by extensive experiments. Experimental results show that the proposed algorithms outperform their counterparts by reducing at least 13% of the total energy consumption while achieving the same accuracy.

Reformulation for stress topology optimization of continuum structures by floating projection

Stress topology optimization of continuum structures is an important topic for structural design and has been widely investigated. However, stress topology optimization itself is ill-conditioned and the resulting optimized designs highly depend on the used parameters, mesh size, and so on. In this paper, stress minimization and stress-constrained topology optimization problems are reformulated by introducing a global measure of an optimized design, the average solid stress. The floating projection topology optimization method is established with ε-relaxation and the p-norm function. Due to the modified problem formulations, the optimized design becomes insensitive to the selection of p in the p-norm function once p is large enough, such as p=16, to ensure the total elimination of stress concentration. Numerical results show the uniform stress distribution of optimized designs, demonstrating the effectiveness of the proposed method. Some findings on the elimination of stress concentration, mesh-independent optimized topology, equivalent smooth design, and impractical design are presented and discussed.

An improved proportional topology optimization method combining a polarized material interpolation scheme and Heaviside threshold function

An improved proportional topology optimization (IPTO) method is proposed in this work. The main improvement of this method is that the conventional solid isotropic material with penalization (SIMP)-based material interpolation scheme is replaced by a polarized material interpolation scheme, and the Heaviside threshold function is adopted based on the original proportional topology optimization (PTO) method. By using this approach, the minimum compliance problem can be solved without requiring the numerical derivation of the sensitivity function. To verify the feasibility and effectiveness of the proposed method, two-dimensional (2D) and three-dimensional (3D) cantilevers and L-bracket beams are used as examples. The 2D results obtained by the IPTO method are compared with those obtained by the PTO and SIMP methods. Numerical examples demonstrate that IPTO can acquire better objective function values and more ideal topology structures compared to PTO and SIMP. Furthermore, IPTO offers significant advantages over PTO and SIMP in terms of convergence speed and the ability to suppress intermediate density elements. Additionally, this method enables topology optimization design under multiple working conditions. Therefore, it provides an effective approach for structural topology optimization in research and engineering applications. With appropriate adjustment, this method can also be applied to composite material design and heat conduction design.

Minimum Energy Demands of Energy Storages for Fast Frequency Response: Formulation, Solution, and Implementation

Energy storage (ES) is a kind of promising but costly fast-frequency-response (FFR) resource in low-inertia power systems. This paper addresses the minimum demand of a power system for energy capacities of ES to providing sufficient frequency support, including the formulation, the optimal solution, and its practical implementation. First, the minimum energy control problem of ESs for providing FFR with transient frequency constraints is formulated and then converted into an optimal frequency trajectory planning counterpart. This transformation makes the transient frequency constraint tractable and allows to identify the optimal frequency trajectory that minimizes energy demand for ESs, which leads to an analytical expression of the system minimum demand for ESs energy capacities to ensure frequency security. Moreover, by unfolding the optimal frequency trajectory, the optimal control strategy of ESs is derived explicitly for the first time. The theoretical result directs to a practical controller implementation, which is referred to as the decentralized optimal energy frequency control (DOEFC). Finally, simulations on the modified IEEE 39- bus system verify our results and demonstrate the superiority of the DOEFC in reducing the energy capacity demand of ESs, defending successive disturbances, and hedging against parameter errors.

Влияние кучного выщелачивания золотосодержащего сырья на окружающую среду

The issue of rational use of metal-containing raw materials goes hand in hand with the problem of minimization of environmental hazards caused by new technologies, which hampers the development of metal leaching, including gold leaching. The objective of this paper is the increase the mineral reserves of gold-mining enterprises via the application of leaching technology in stacks on the condition of minimization of damage to the natural environment. The results of the study of the impact of heap leaching complex on the surrounding ecosystems are provided based on the example of the stack of the «Kazakhaltyn» mine (the Republic of Kazakhstan), formed by tails of enrichment. The technological scheme of gold leaching and the parameters of its implementation, considering the minimization of consequences potentially hazardous for ecosystems, have been recommended. The calculation of parameters of hypothetically hazardous impact of gold leaching processes is provided for a stack of 6 m height; the time of leaching is 90 days; for four sorption stages, the coefficient of through-extraction of gold is 0.65. The specifics of the version of technology implying the use of recycled water supply to prevent contact with the environment are described. The processes of leaching affecting the environment and the preventive measures have been systematized. It has been demonstrated that emergencies are local, whereas sodium cyanide under the impact of atmospheric air and solar energy reacts with the generation of volatile short-lived hydrogen cyanide. It has been proved that the process of demetallization of mining and processing tails in the stack is a measure of environmental protection; it is not more hazardous than the traditional technology of handling processing tails. Such demetallization process is worth spreading.

Learning-Based Cloud Server Configuration for Energy Minimization Under Reliability Constraint

Cloud computing has attracted wide attention from both academia and industry, since it can provide flexible and on-demand hardware and software resources as services. Energy consumption of cloud servers is the main concern of cloud service providers since reducing energy consumption can bring them a lower operation cost (and hence a higher profit) and alleviate carbon footprints to the environment. Typically, the common power management techniques for enhancing energy efficiency would make cloud servers more vulnerable to soft errors and hence adversely impact the quality of services. Thus, reliability cannot be ignored in the design of methodologies for improving the energy efficiency of cloud servers. In this article, we aim to minimize the energy consumption of cloud servers under the soft-error reliability constraint by configuring the size and speed of servers. Specifically, we first derive the expected reliability based energy consumption of cloud servers to formulate the reliability-constrained energy minimization problem. We then leverage the reinforcement learning technique to obtain an optimal server configuration solution that maximizes system energy efficiency while maintaining the system reliability constraint. Finally, we perform extensive simulation experiments to analyze the relationship between system energy consumption and server configuration under varying arrival rates and execution requirements of service requests. Comparative experiments are also performed to validate the efficacy of the proposed learning-based server configuration scheme. Results show that compared to a benchmark method, the energy saved by the proposed scheme can reach up to 31.5%.

DMUST360: Distortion‐aware multicast‐unicast superposed transmission for multiuser 360‐degree video

Abstract360‐degree video has gained popularity since it is associated with immersive user experience of virtual reality content. However, providing high‐quality 360‐degree video for a large number of users is difficult, especially in limited bandwidth conditions. Nonorthogonal multiple access technology‐enabled multicast offers a possible solution, but it also introduces some challenges, including in terms of satisfying users' individual requirements, ensuring video quality and so forth. In order to address these issues, we introduce a novel distortion‐aware multicast‐unicast superposed transmission scheme for multiuser 360‐degree video, which combines the advantages of multicast, unicast, and nonorthogonal multiple access technologies with consideration of the field of view of multiple users. To improve video quality, a distortion minimization problem is formulated. Based on matching theory, we proposed a multicast‐unicast coalition game algorithm to solve the problem efficiently. A series of trace‐driven simulations are carried out, which have shown the effectiveness of the proposed solution in terms of video quality and throughput.

Unified Framework for Faster Clustering via Joint Schatten p-Norm Factorization With Optimal Mean.

To enhance the effectiveness and efficiency of subspace clustering in visual tasks, this work introduces a novel approach that automatically eliminates the optimal mean, which is embedded in the subspace clustering framework of low-rank representation (LRR) methods, along with the computationally factored formulation of Schatten p -norm. By addressing the issues related to meaningful computations involved in some LRR methods and overcoming biased estimation of the low-rank solver, we propose faster nonconvex subspace clustering methods through joint Schatten p -norm factorization with optimal mean (JS p NFOM), forming a unified framework for enhancing performance while reducing time consumption. The proposed approach employs tractable and scalable factor techniques, which effectively address the disadvantages of higher computational complexity, particularly when dealing with large-scale coefficient matrices. The resulting nonconvex minimization problems are reformulated and further iteratively optimized by multivariate weighting algorithms, eliminating the need for singular value decomposition (SVD) computations in the developed iteration procedures. Moreover, each subproblem can be guaranteed to obtain the closed-form solver, respectively. The theoretical analyses of convergence properties and computational complexity further support the applicability of the proposed methods in real-world scenarios. Finally, comprehensive experimental results demonstrate the effectiveness and efficiency of the proposed nonconvex clustering approaches compared to existing state-of-the-art methods on several publicly available databases. The demonstrated improvements highlight the practical significance of our work in subspace clustering tasks for visual data analysis. The source code for the proposed algorithms is publicly accessible at https://github.com/ZhangHengMin/TRANSUFFC.

Prediction-correction pedestrian flow by means of minimum flow problem

We study a new variant of mathematical prediction-correction model for crowd motion. The prediction phase is handled by a transport equation where the vector field is computed via an eikonal equation [Formula: see text], with a positive continuous function [Formula: see text] connected to the speed of the spontaneous travel. The correction phase is handled by a new version of the minimum flow problem. This model is flexible and can take into account different types of interactions between the agents, from gradient flow in Wassersetin space to granular type dynamics like in sandpile. Furthermore, different boundary conditions can be used, such as non-homogeneous Dirichlet (e.g. outings with different exit-cost penalty) and Neumann boundary conditions (e.g. entrances with different rates). Combining finite volume method for the transport equation and Chambolle–Pock’s primal dual algorithm for the eikonal equation and minimum flow problem, we present numerical simulations to demonstrate the behavior in different scenarios.