Minimization Problem Research Articles

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.

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

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.

Approximation algorithms for minimum ply covering of points with unit squares and unit disks

Given a set P of points and a set U of geometric objects in the Euclidean plane, a minimum ply cover of P with U is a subset of U that covers P and minimizes the number of objects that share a common intersection, called the minimum ply cover number of P with U. Biedl et al. (2021) [9] showed that for both unit squares and unit disks, determining the minimum ply cover number for a set of points is NP-hard. They gave polynomial-time 2-approximation algorithms for the special case when the minimum ply cover number is constant, and asked whether there exists polynomial-time O(1)-approximation algorithms for these problems. In this paper, we settle the question posed by Biedl et al. by providing polynomial-time O(1)-approximation algorithms for the minimum ply cover problem for both unit squares and unit disks.

Secure Unmanned Aerial Vehicle Communication in Dual-Function Radar Communication System by Exploiting Constructive Interference

In contrast from traditional unmanned aerial vehicle communication via unlicensed spectrum, connecting unmanned aerial vehicles with cellular networks can extend their communication coverage and improve the quality of their service. In addition, the emerging dual-functional radar communication paradigm in cellular systems can better meet the requirements of location-sensitive tasks such as reconnaissance and cargo delivery. Based on the above considerations, in this paper, we study the simultaneous communication and target sensing issue in cellular-connected unmanned aerial vehicle systems. Specifically, we consider a two-cell coordinated system with two base stations, cellular unmanned aerial vehicles, and potential aerial targets. In such systems, the communication security issue of cellular unmanned aerial vehicles regarding eavesdropping on their target is inevitable since the main beam of the transmit waveform needs to point to the direction of the target for achieving a sufficient detection performance. Aiming at protecting the privacy of cellular transmission as well as performing target sensing, we exploit the physical layer security technique with the aid of constructive interference-based precoding. A transmit power minimization problem is formulated with constraints on secure and reliable cellular transmission and a sufficient radar signal-to-interference-plus-noise ratio. By specially designing the transmit beamforming vectors at the base stations, the received signals at the cellular users are located in the decision regions of the transmitted symbols while the targets can only receive wrong symbols. We also compare the performance of the proposed scheme with that of the traditional one without constructive interference. The simulation results show that the proposed constructive interference-based strategy can meet the requirements of simultaneous target sensing and secure communication, and also save transmit power compared with the traditional scheme.

An iterative approach for addressing monotone inclusion and fixed point problems with generalized demimetric mappings

Throughout this study, we present a new algorithm for finding the common solution of a finite family of monotone inclusion and the fixed point problem of a finite family of generalized demimetric operators in the context of a real Hilbert space and show its strong convergence. Moreover, we utilize our result to solve the minimization problem.

Controllability and minimum energy control problem of two-dimensional continuous-time fractional linear systems

The effectiveness of this paper lies in a new formulation and solution of the minimum energy control problem for the fractional two-dimensional systems described by the Fornasini–Marchesini first models. The problem of controllability and minimum energy control of a general fractional two-dimensional Fornasini–Marchesini model is newly considered by the authors in this work. Necessary and sufficient controllability conditions are then established using the Gramian matrix. To carry out this study, a new test was described and used in order to give the existence conditions of the solution to the minimum energy control problem and the procedures for calculating an input minimising the given performance index. The usefulness and the precision of the proposed procedure is demonstrated by a numerical examples, Darboux equation and metal rolling process model as a practical application.

Research on path planning based on improved artificial potential field method

Aiming at the unreachable target and local minimum problems existing in the traditional artificial potential field method in the path planning of mobile robots, an improved artificial potential field method is proposed. Firstly, when there are obstacles near the target point, the robot is difficult to reach the target point due to the large repulsive force. A safety distance factor is introduced in the potential field, and the parameter is optimized, so that the robot can maintain a suitable distance from the obstacle and reach the target point smoothly. Secondly, in order to solve the local minimum problem, the local minimum judgment condition is introduced, and when the condition is triggered, the local minimum area is bypassed, so that the robot can reach the target point smoothly. The simulation results show that the improved algorithm runs in the environment of different numbers of obstacles and has strong robustness. The proposed algorithm can enable the robot to bypass the local minimum area in the U-shaped obstacle environment, and successfully solves the local minimum problem in the path planning of mobile robots.

A new extension of the core inverse

There are a number of extensions of the core inverse represented by the products of known generalized inverses, like the core-EP inverse, DMP inverse, GC inverse and so on. However, none of them is an inner inverse of the matrix, and especially for an arbitrary nilpotent matrix, such inverses are always null. Our goal is to use a new technic to introduce an extension of the core inverse that preserves the interesting property of being an inner inverse of the matrix which need not necessarily be the null matrix of a nilpotent matrix. We define the extended core inverse for square complex matrices combining the sum and the difference of three known generalized inverses. Various properties and representations of the extended core inverse are developed. The extended dual core inverse is investigated too. We apply the extended core inverse to solve some systems of linear equations and one minimization problem. A significant normal equation related to least-squares solutions can be solved using the extended core inverse.

Bias and multiscale correction methods for variational state estimation

Data assimilation performance can be significantly impacted by biased noise in observations, altering the signal magnitude and introducing fast oscillations or discontinuities when the system lacks smoothness. To mitigate these issues, this paper employs variational state estimation using the so-called parametrized-background data-weak method. This approach relies on a background manifold parametrized by a set of constraints, enabling the state estimation by solving a minimization problem on a reduced-order background model, subject to constraints imposed by the input measurements. The proposed formulation incorporates a novel bias correction mechanism and a manifold decomposition that handles rapid oscillations by treating them as slow-decaying modes based on a two-scale splitting of the classical reconstruction algorithm. The method is validated in different examples, including the assimilation of biased synthetic data, discontinuous signals, and Doppler ultrasound data obtained from experimental measurements.

FreeTO - Freeform 3D topology optimization using a structured mesh with smooth boundaries in Matlab

Topology optimization has revolutionized the design of structures for various applications, particularly with the advancement of additive manufacturing. However, existing open-source codes for topology optimization have limitations, such as restricted domain initialization and lack of a CAD output after optimization. A novel open-source Matlab code, FreeTO, is presented, and it addresses these limitations by enabling the initialization of 3D arbitrary geometries and providing an STL file post-optimization. FreeTO utilizes a structured mesh and a smooth-edge (boundary) algorithm to generate smooth topological boundaries. The code is demonstrated through six practical design cases, showcasing its effectiveness in compliance minimization, compliant mechanisms, and self-supporting problems. FreeTO offers a user-friendly, all-in-one topology optimization package, making it an invaluable tool for educators, researchers, and practitioners. Future developments will focus on eliminating a few geometrical deviations in the optimized topologies, incorporating speedups, and extending the code to apply to more applications.

On the atout ticket learning problem for neural networks and its application in securing federated learning exchanges

Artificial Neural Networks (ANNs) have become the backbone of many real-world applications, including distributed applications relying on Federated Learning (FL). However, several vulnerabilities/attacks have emerged in recent years, affecting the benefits of using ANNs in FL, such as reconstruction attacks and membership inference attacks. These attacks can have severe impacts on both the societal and professional levels. For instance, inferring the presence of a patient’s private health record in a medical study or a clinic database violates the patient’s privacy and can have legal or ethical consequences. Therefore, protecting the data and model from malicious attacks in FL systems is important. This paper introduces the Atout Ticket Learning (ATL) problem. This new problem consists of identifying sensitive parameters (atout tickets) of a neural network model, which, if modified, will increase the model’s loss by at least a given threshold ϵ. First, we formulate ATL as an ℓ0-norm minimization problem, and we derive a lower bound on the number of atout tickets needed to achieve a model degradation of ϵ. Second, we design the Atout Ticket Protocol (ATP) as an effective solution for privacy-preserving in FL systems using atout tickets, along with the benefit of noise perturbations and secure aggregation techniques. Finally, we experiment ATP against FL reconstruction attacks using new selection strategies, namely Inverting Gradients, Deep Leakage, and Improved Deep Leakage. The results show that ATP is highly robust against these attacks.

Local minima in Newton's aerodynamic problem and inequalities between norms of partial derivatives

The problem considered first by Newton (1687) [1] consists in finding a surface of the minimal frontal resistance in a parallel flow of non-interacting point particles. The standard formulation assumes that the surface is convex with a given convex base Ω and a bounded altitude. Newton found the solution for surfaces of revolution. Without this assumption the problem is still unsolved, although many important results have been obtained in the last decades. We consider the problem to characterize the domains Ω for which the flat surface gives a local minimum. We show that this problem can be reduced to an inequality between L2-norms of partial derivatives for bivariate concave functions on a convex domain that vanish on the boundary. Can the ratio between those norms be arbitrarily large? The answer depends on the geometry of the domain. A complete criterion is derived, which also solves the local minimality problem.