Minimization Problem Research Articles

Isogeometric Methodfor Least Squares Problem

In the framework of this work, we used the isogeometric method to solve the least squares problem in one dimension, on a curve of R d , d = 1, 2, including a semicircle. For this purpose, we presented the isogeometric method and the tools necessary for the description of this method, namely, the b splines basis, the parameterization of the R d , d = 1, 2 curve. We formulated the least squares problem which is a minimization problem. This problem was solved by using the Discontinuous Galerkin (DG) and the b splines basis as the approximation basis. The numerical method was validated by evaluating the error. For this purpose, an inverse inequality was therefore used.

Fixed-Ratio Approximation Algorithm for the Minimum Cost Cover of a Digraph by Bounded Number of Cycles

We consider polynomial time approximation for the minimum cost cycle cover problem of an edge-weighted digraph, where feasible covers are restricted to have at most k disjoint cycles. In the literature this problem is referred to as Min-k-SCCP. The problem is closely related to classic Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP) and has many important applications in algorithms design and operations research. Unlike its unconstrained variant, the Min-k-SCCP is strongly NP-hard even on undirected graphs and remains intractable in very specific settings. For any metric, the problem can be approximated in polynomial time within ratio 2, while in fixed-dimensional Euclidean spaces it admits Polynomial Time Approximation Schemes (PTAS). In the same time, approximation of the more general asymmetric Min-k-SCCP still remains weakly studied. In this paper, we propose the first fixed-ratio approximation algorithm for this problem, which extends the recent breakthrough Svensson-Tarnawski-Vegh and Traub-Vygen results for the Asymmetric Traveling Salesman Problem.

Optimizing the Agricultural Internet of Things (IoT) with Edge Computing and Low-Altitude Platform Stations

Using low-altitude platform stations (LAPSs) in the agricultural Internet of Things (IoT) enables the efficient and precise monitoring of vast and hard-to-reach areas, thereby enhancing crop management. By integrating edge computing servers into LAPSs, data can be processed directly at the edge in real time, significantly reducing latency and dependency on remote cloud servers. Motivated by these advancements, this paper explores the application of LAPSs and edge computing in the agricultural IoT. First, we introduce an LAPS-aided edge computing architecture for the agricultural IoT, in which each task is segmented into several interdependent subtasks for processing. Next, we formulate a total task processing delay minimization problem, taking into account constraints related to task dependency and priority, as well as equipment energy consumption. Then, by treating the task dependencies as directed acyclic graphs, a heuristic task processing algorithm with priority selection is developed to solve the formulated problem. Finally, the numerical results show that the proposed edge computing scheme outperforms state-of-the-art works and the local computing scheme in terms of the total task processing delay.

Algorithms for optimizing systems with multiple extremum functionals

The problem of minimizing (maximizing) multiple extremum functionals (infinite-dimensional optimization) is considered. This problem cannot be solved by conventional gradient methods. New gradient methods with adaptive relaxation of steps in the vicinity of local extrema are proposed. The efficiency of the proposed methods is demonstrated by the example of optimizing the shape of a hydraulic gun nozzle with respect to the objective functional, which is the average force of the hydraulic pulse jet momentum acting on an obstacle. Two local maxima are found, the second of which is global; in the second maximum, the average force of the jet momentum is three times higher than in the first maximum. The corresponding nozzle shape is optimal. Conventional gradient methods have not found any maximum; i.e., they were unable to solve the problem.

Noncontace Monitoring of Respiration and Heartbeat Based on Two-Wave Model Using a Millimeter-Wave MIMO FM-CW Radar

This paper deals with the non-contact measurement of heartbeat and respiration using a millimeter-wave multiple-input–multiple-output (MIMO) frequency-modulated continuous-wave (FM-CW) radar. Monitoring heartbeat and respiration is useful for detecting cardiac diseases and understanding stress levels. Contact sensors are not suitable for these sorts of long-term measurements due to the discomfort and skin irritation they cause. Therefore, the use of non-contact sensors, such as radars, is desirable. In this study, we obtained heartbeat and respiration information from phase data measured using a millimeter-wave MIMO FM-CW radar. We propose a two-wave model based on a Fourier series expansion and extract respiration and heartbeat information as a minimization problem. This model makes it possible to produce respiration and heartbeat waveforms. The produced heartbeat waveform can be used for estimating the interbeat interval (IBI). Experiments were conducted to confirm the usefulness of the proposed method. Moreover, the estimated results were compared with the contact sensor’s results. The results for both types of sensors were in good agreement.

Fast Algorithms for Maximizing the Minimum Eigenvalue in Fixed Dimension

In the minimum eigenvalue problem, we are given a collection of vectors in Rd, and the goal is to pick a subset B to maximize the minimum eigenvalue of the matrix ∑i∈Bvivi⊤. We give a O(nO(dlog⁡(d)/ϵ2))-time randomized algorithm that finds an assignment subject to a partition constraint whose minimum eigenvalue is at least (1−ϵ) times the optimum, with high probability. As a byproduct, we also get a simple algorithm for an algorithmic version of Kadison-Singer problem.

Implementation of spectral methods on Ising machines: toward flow simulations on quantum annealers

Abstract We investigate the possibility and current limitations of flow computations using quantum annealers by solving a fundamental flow problem on Ising machines. As a fundamental problem, we consider the one-dimensional advection-diffusion equation. We formulate it in a form suited to Ising machines (i.e., both classical and quantum annealers), perform extensive numerical tests on a classical annealer, and finally test it on an actual quantum annealer. To make it possible to process with an Ising machine, the problem is formulated as a minimization problem of the residual of the governing equation discretized using either the spectral method or the finite difference method. The resulting system equation is then converted to the Quadratic Unconstrained Binary Optimization (QUBO) form through the quantization of variables. It is found in numerical tests using a classical annealer that the spectral method requiring a smaller number of variables has a particular merit over
the finite difference method because the accuracy deteriorates with the increase of the number of variables. We also found that the computational error varies depending on the condition number of the coefficient matrix. In addition, we extended it to a two-dimensional problem and confirmed its fundamental applicability. From the numerical test using a quantum annealer, however, it turns out that the computation using a quantum annealer is still challenging due largely to the structural difference from the classical annealer, which leaves a number of issues toward its practical use.

Transmit Power Optimization in Multihop Amplify-and-Forward Relay Systems with Simultaneous Wireless Information and Power Transfer

In this paper, we study a multi-hop amplify-and-forward (AF) simultaneous wireless information and power transmission (SWIPT) relay system. Each relay node harvests power using a power split (PS) method from a portion of the received signal, amplifies the remaining received signal, and passes it to the next relay. Based on this system model and signal flow, we derived and solved the convex power minimization problem with the optimal PS ratio. In this case, it was found that using the optimal PS ratio consumed a lower amount of power than when using a fixed PS ratio (0.5). We then investigated the impact of processing cost on the AF-SWIPT system using decoding and forwarding SWIPT as benchmarks, and found that AF-SWIPT was superior.

Broadband photonic frequency measurement based on the frequency-to-carrier phase-shifted mapping and finite extinction ratio

A novel frequency measurement is proposed based on frequency-to-carrier phase-shifted mapping and finite extinction ratio. Due to the frequency dependence of the dispersion phase shift, carrier phase-shifted double-sideband modulation and dispersive effect are combined for effective mapping. Additionally, the finite extinction ratio is introduced to mitigate the influence of unideal parameters on measurement error. To achieve this, a dual-parallel Mach-Zehnder modulator and a dispersion compensating fibre are combined, and a power minimization problem is formulated to establish a mapping table. Theoretical analysis and simulation results show the frequency measurement range over 30 GHz with errors below 180 MHz. Furthermore, the measurement error is reduced below 40 MHz after considering the finite extinction ratio.

On a minimization problem of the maximum generalized eigenvalue: properties and algorithms

AbstractWe study properties and algorithms of a minimization problem of the maximum generalized eigenvalue of symmetric-matrix-valued affine functions, which is nonsmooth and quasiconvex, and has application to eigenfrequency optimization of truss structures. We derive an explicit formula of the Clarke subdifferential of the maximum generalized eigenvalue and prove the maximum generalized eigenvalue is a pseudoconvex function, which is a subclass of a quasiconvex function, under suitable assumptions. Then, we consider smoothing methods to solve the problem. We introduce a smooth approximation of the maximum generalized eigenvalue and prove the convergence rate of the smoothing projected gradient method to a global optimal solution in the considered problem. Also, some heuristic techniques to reduce the computational costs, acceleration and inexact smoothing, are proposed and evaluated by numerical experiments.

On Discrete Subproblems in Integer Optimal Control with Total Variation Regularization in Two Dimensions

We analyze integer linear programs that we obtain after discretizing two-dimensional subproblems arising from a trust-region algorithm for mixed integer optimal control problems with total variation regularization. We discuss NP-hardness of the discretized problems and the connection to graph-based problems. We show that the underlying polyhedron exhibits structural restrictions in its vertices with regard to which variables can attain fractional values at the same time. Based on this property, we derive cutting planes by employing a relation to shortest-path and minimum bisection problems. We propose a branching rule and a primal heuristic which improves previously found feasible points. We validate the proposed tools with a numerical benchmark in a standard integer programming solver. We observe a significant speedup for medium-sized problems. Our results give hints for scaling toward larger instances in the future. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [Grant MA 10080/2-1]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0680 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0680 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Rooftop Solar PV, Coal Plant Inflexibility and the Minimum Load Problem

Australia’s National Electricity Market (NEM) has amongst the highest take-up rates of rooftop solar PV in the world. As with California, this has produced a distinctive load shape termed the “duck curve.” The Queensland version is being principally driven by non-scheduled (i.e., uncontrolled) rooftop solar PV. When combined with inflexible coal plant, it leads to a “minimum load problem.” In this article, we examine the feasibility of dispatch with ever-expanding rooftop solar PV resources in the NEM’s Queensland region and minimal demand elasticity. We find episodes of intractable dispatch throughout the year with rising intensity in the winter and spring months. Furthermore, we find no ability to “export your way out of the problem” via larger interstate interconnectors because the same problem is emerging in the adjacent region at the same time. Resolution ultimately requires inflexible coal plant exit, and entry of flexible plant. JEL Classification: D25, D80, G32, L51, Q41

An inertial projective forward-backward-forward algorithm for constrained convex minimization problems and application to cardiovascular disease prediction

An inertial projective forward-backward-forward algorithm for constrained convex minimization problems and application to cardiovascular disease prediction

TOP2DFVT: An Efficient Matlab Implementation for Topology Optimization based on the Finite-Volume Theory.

The finite-volume theory has shown to be numerically efficient and stable for topology optimization of continuum elastic structures. The significant features of this numerical technique are the local satisfaction of equilibrium equations and the employment of compatibility conditions along edges in a surface-averaged sense. These are essential properties to adequately mitigate some numerical instabilities in the gradient version of topology optimization algorithms, such as checkerboard, mesh dependence, and local minima issues. Several computational tools have been proposed for topology optimization employing analysis domains discretized with essential features for finite-element approaches. However, this is the first contribution to offer a platform to generate optimized topologies by employing a Matlab code based on the finite-volume theory for compliance minimization problems. The Top2DFVT provides a platform to perform 2D topology optimization of structures in Matlab, from domain initialization for structured meshes to data post-processing. This contribution represents a significant advancement over earlier publications on topology optimization based on the finite-volume theory, which needed more efficient computational tools. Moreover, the Top2DFVT algorithm incorporates SIMP and RAMP material interpolation schemes alongside sensitivity and density filtering techniques, culminating in a notably enhanced optimization tool. The application of this algorithm to various illustrative cases confirms its efficacy and underscores its potential for advancing the field of structural optimization.

Solving convex optimization problems via a second order dynamical system with implicit Hessian damping and Tikhonov regularization

AbstractThis paper deals with a second order dynamical system with a Tikhonov regularization term in connection to the minimization problem of a convex Fréchet differentiable function. The fact that beside the asymptotically vanishing damping we also consider an implicit Hessian driven damping in the dynamical system under study allows us, via straightforward explicit discretization, to obtain inertial algorithms of gradient type. We show that the value of the objective function in a generated trajectory converges rapidly to the global minimum of the objective function and depending the Tikhonov regularization parameter the generated trajectory converges weakly to a minimizer of the objective function or the generated trajectory converges strongly to the element of minimal norm from the $$\mathop {\text {argmin}}$$ argmin set of the objective function. We also obtain the fast convergence of the velocities towards zero and some integral estimates. Our analysis reveals that the Tikhonov regularization parameter and the damping parameters are strongly correlated, there is a setting of the parameters that separates the cases when weak convergence of the trajectories to a minimizer and strong convergence of the trajectories to the minimal norm minimizer can be obtained.

A duality and free boundary approach to adverse selection

Adverse selection is a version of the principal-agent problem that includes monopolist nonlinear pricing, where a monopolist with known costs seeks a profit-maximizing price menu facing a population of potential consumers whose preferences are known only in the aggregate. For multidimensional spaces of agents and products, Rochet and Choné (Econometrica, 1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences combine bilinearity in the product and agent parameters with a quasilinear sensitivity to prices. We characterize solutions to this problem by identifying a dual minimization problem. This duality allows us to reduce the solution of the square example of Rochet–Choné to a novel free boundary problem, giving the first analytical description of an overlooked market segment.

Explicit A Posteriori Error Representation for Variational Problems and Application to TV-Minimization

AbstractIn this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant vector fields as well as a discrete integration-by-parts formula between the Crouzeix–Raviart and the Raviart–Thomas element, all convex duality relations are transferred to a discrete level, making the explicit a posteriori error representation –initially based on continuous arguments only– practicable from a numerical point of view. In addition, we provide a generalized Marini formula that determines a discrete primal solution in terms of a given discrete dual solution. We benchmark all these concepts via the Rudin–Osher–Fatemi model. This leads to an adaptive algorithm that yields a (quasi-optimal) linear convergence rate.

Leader-follower control and APF for Multi-USV coordination and obstacle avoidance

To overcome the limitations of a single USV in task execution, this paper investigates the issues of cooperative motion and obstacle avoidance for USV swarm. The leader-follower control approach is chosen as the strategy for coordinating the formation of multiple USVs. Kinematic and kinetic controllers are designed using sliding mode control, with a saturation function replacing the sign function to avoid chattering. The validity of the kinematic and kinetic controllers is demonstrated using the Lyapunov stability theorem. To tackle the problem of goal reachability when obstacles are nearby in the artificial potential field method for obstacle avoidance, the repulsive potential field function is modified by including the distance factor between the USV and the target point. To address the problem of local minimum, the simulated annealing algorithm is incorporated into the traditional artificial potential field method. Multi-condition simulation experiments are conducted using MATLAB. The experimental results demonstrate that multiple USVs can form a formation in a short time and maintain the formation while navigating, exhibiting high robustness. Furthermore, they are able to effectively navigate through complex marine environments, successfully avoiding obstacles and ultimately reaching their target destination.

Approach for multi-valued integer programming in multi-material topology optimization: Random discrete steepest descent (RDSD) algorithm

The present study models the multi-material topology optimization problems as the multi-valued integer programming (MVIP) or named as combinatorial optimization. By extending classical convex analysis and convex programming to discrete point-set functions, the discrete convex analysis and discrete steepest descent (DSD) algorithm are introduced. To overcome combinatorial complexity of the DSD algorithm, we employ the sequential approximate integer programming (SAIP) to explicitly and linearly approximate the implicit objective and constraint functions. Considering the multiple potential changed directions for multi-valued design variables, the random discrete steepest descent (RDSD) algorithm is proposed, where a random strategy is implemented to select a definitive direction of change. To analytically calculate multi-material discrete variable sensitivities, topological derivatives with material contrast is applied. In all, the MVIP is finally transferred as the linear 0–1 programming that can be efficiently solved by the canonical relaxation algorithm (CRA). Explicit nonlinear examples demonstrate that the RDSD algorithm owns nearly three orders of magnitude improvement compared with the commercial software (GUROBI). The proposed approach, without using any continuous variable relaxation and interpolation penalization schemes, successfully solves the minimum compliance problem, strength-related problem, and frequency-related optimization problems. Given the algorithm efficiency, mathematical generality and merits over other algorithms, the proposed RDSD algorithm is meaningful for other structural and topology optimization problems involving multi-valued discrete design variables.

A Lagrange barrier approach for the minimum concave cost supply problem via a logarithmic descent direction algorithm

The minimisation of concave costs in the supply chain presents a challenging non-deterministic polynomial (NP) optimisation problem, widely applicable in industrial and management engineering. To approximate solutions to this problem, we propose a logarithmic descent direction algorithm (LDDA) that utilises the Lagrange logarithmic barrier function. As the barrier variable decreases from a high positive value to zero, the algorithm is capable of tracking the minimal track of the logarithmic barrier function, thereby obtaining top-quality solutions. The Lagrange function is utilised to handle linear equality constraints, whilst the logarithmic barrier function compels the solution towards the global or near-global optimum. Within this concave cost supply model, a logarithmic descent direction is constructed, and an iterative optimisation process for the algorithm is proposed. A corresponding Lyapunov function naturally emerges from this descent direction, thus ensuring convergence of the proposed algorithm. Numerical results demonstrate the effectiveness of the algorithm.