Interior Point Method Research Articles

A barrier method for contact avoiding particles in Stokes flow

Rigid particles in a Stokesian fluid experience an increasingly strong lubrication resistance as particle gaps narrow. Numerically, resolving these lubrication forces comes at an intractably large cost, even for moderate system sizes. Hence, it can typically not be guaranteed that artificial particle collisions and overlaps do not occur in a dynamic simulation, independently of the choice of method to solve the Stokes equations. In this work, the potentially large set of non-overlap constraints, in terms of the Euclidean distance between boundary points on disjoint particles, are efficiently represented via a barrier energy. We solve for the minimum magnitudes of repelling contact forces and torques between any particle pair in contact to correct for overlaps by enforcing a zero barrier energy at the next time level, given a contact-free configuration at a previous instance in time. Robustness for the method is illustrated using a multiblob method to solve the mobility problem in Stokes flow, applied to suspensions of spheres, rods and boomerang shaped particles. Collision free configurations are obtained at all instances in time, and considerably larger time-steps can be taken than without the technique. The effect of the contact forces on the collective order of a set of rods in a background flow that naturally promote particle interactions is also illustrated.

Primal dual algorithm for solving the nonsmooth Twin SVM

In this paper, we propose an improved version of Twin SVM using a non-smooth optimization method. Twin SVM generally consists in determining two non-parallel planes by alternately solving two constrained optimization models. Solving this problem using the classical Lagrangian method has many limitations, notably: its only limited to handle Gaussian noise, generally exaggerates the influence of outliers and cannot handle unbalanced data, this due to the differentiability of the model. To circumvent these issues, we transform two-constraint optimization models using the penalty method into an unconstrained non-smooth optimization one. The non-smoothness nature of the problem has many advantages, but it requires special treatment, which is why we use the primal dual method to solve it, since it is the most appropriate and it is robust in terms of stability, convergence and speed (Lyaqini, Nachaoui and Hadri, 2022). To demonstrate the effectiveness of the proposed approach, several experiments were carried out on numerous UCI benchmarks, medical image and HandPD datasets. These experiments demonstrated the effectiveness and applicability of the proposed approach, with satisfactory results compared to the state of the art.

A regularized interior point method for sparse optimal transport on graphs

In this work, the authors address the Optimal Transport (OT) problem on graphs using a proximal stabilized Interior Point Method (IPM). In particular, strongly leveraging on the induced primal–dual regularization, the authors propose to solve large scale OT problems on sparse graphs using a bespoke IPM algorithm able to suitably exploit primal–dual regularization in order to enforce scalability. Indeed, the authors prove that the introduction of the regularization allows to use sparsified versions of the normal Newton equations to inexpensively generate IPM search directions. A detailed theoretical analysis is carried out showing the polynomial convergence of the inner algorithm in the proposed computational framework. Moreover, the presented numerical results showcase the efficiency and robustness of the proposed approach when compared to network simplex solvers.

Interior Point Methods in Optimal Control Problems of Affine Systems: Convergence Results and Solving Algorithms

Interior Point Methods in Optimal Control Problems of Affine Systems: Convergence Results and Solving Algorithms

A SSSC optimal configuration method to enhance available transfer capability considering multi‐wind farm access

AbstractIn the context of electricity market, the heavy load of some inter‐provincial and intra‐provincial lines limits available transfer capability (ATC). As a series compensation device, static synchronous series compensator (SSSC) can reallocate the power flow of adjacent lines. In order to maximize the inter‐regional ATC, a two‐stage optimal SSSC allocation method considering the uncertainty and correlation of multi‐wind farm is proposed. Firstly, the initial wind power scenarios are generated based on latin hypercube sampling (LHS) and reduced by the improved iterative self‐organizing data analysis techniques algorithm (I‐ISODATA). Then, a multi‐objective optimization model is constructed with the maximum expected power flow transfer entropy (PFTE) and minimum expected SSSC investment cost in all typical wind power scenarios, and non‐dominated sorting genetic algorithm‐II (NSGA‐II) method is used to determine the location of SSSC. In addition, based on the reduced wind power scenarios, the ATC calculation model containing SSSC is constructed, and the modern interior point method is used to determine the control parameters and configuration capacity of SSSC. Finally, based on the improved IEEE39 and IEEE118 node systems, case study proves that the proposed model can determine the optimal SSSC configuration scheme and significantly improves the ATC.

A Distributed Block-Split Gibbs Sampler with Hypergraph Structure for High-Dimensional Inverse Problems

Sampling-based algorithms are classical approaches to perform Bayesian inference in inverse problems. They provide estimators with the associated credibility intervals to quantify the uncertainty on the estimators. Although these methods hardly scale to high dimensional problems, they have recently been paired with optimization techniques, such as proximal and splitting approaches, to address this issue. Such approaches pave the way to distributed samplers, splitting computations to make inference more scalable and faster. We introduce a distributed Split Gibbs sampler (SGS) to efficiently solve such problems involving distributions with multiple smooth and non-smooth functions composed with linear operators. The proposed approach leverages a recent approximate augmentation technique reminiscent of primal-dual optimization methods. It is further combined with a block-coordinate approach to split the primal and dual variables into blocks, leading to a distributed block-coordinate SGS. The resulting algorithm exploits the hypergraph structure of the involved linear operators to efficiently distribute the variables over multiple workers under controlled communication costs. It accommodates several distributed architectures, such as the Single Program Multiple Data and client-server architectures. Experiments on a large image deblurring problem show the performance of the proposed approach to produce high quality estimates with credibility intervals in a small amount of time. Supplementary material to reproduce the experiments is available online.

Contraception Use by Title X Clients and Clients of Other Providers, 2015–2019

IntroductionTitle X clinics provide access to a wide range of contraceptive options for individuals of all income levels and documentation statuses. As Title X continues to face political uncertainties, it is important to provide up-to-date information about its clients’ use of contraception. This study used recent nationally representative data to compare contraception received by Title X clients with that received by clients of other providers. MethodsThis article draws on 2015–2017 and 2017–2019 waves of the National Survey of Family Growth. The sample was restricted to 15- to 44-year-old women needing contraception. Logistic regressions estimated associations between receiving services at Title X clinics versus at other providers (including private) and use of a range of contraceptive options, as well as number of months’ supply for those using oral contraceptives. ResultsIn 2015–2017, Title X was associated with using any contraception (adjusted odds ratio [AOR], 4.11; p = .004). In both waves, Title X clients were more likely to use long-acting reversible contraceptives (AOR, 1.78 in 2015–2017 [p = .023] and AOR, 2.59 in 2017–2019 [p = .003]) and hormonal methods other than oral contraceptives (AOR, 2.31 in 2015–2017 [p = .007] and AOR, 3.04 in 2017–2019 [p = .001]). In both waves, Title X clients using oral contraceptives were also more likely than non-Title X clients to receive more than a 3-month supply (AOR, 3.54 in 2015–2017 [p = .008] and AOR, 2.61 in 2017–2019 [p = .043]). Title X was not associated in either wave with use of barrier or time-based methods, such as periodic abstinence or withdrawal. ConclusionsPatterns of contraceptive use by Title X clients compared with those of clients of other providers indicate that the Title X program has allowed access to a wide range of contraceptive methods. Ongoing research is necessary to see whether these patterns change over time.

Proximal Alternating Direction Method of Multipliers with Convex Combination Proximal Centers

Proximal alternating direction method of multipliers (PADMM) is a classical primal-dual splitting method for solving separable convex optimization problems with linear equality constraints, which have numerous applications in, e.g., signal and image processing, machine learning, and statistics. In this paper, we propose a new variant of PADMM, called PADMC, whose proximal centers are constructed by convex combinations of the iterates. PADMC is able to take advantage of problem structures and preserves the desirable properties of the classical PADMM. We establish iterate convergence as well as [Formula: see text] ergodic and [Formula: see text] nonergodic sublinear convergence rate results measured by function residual and feasibility violation, where [Formula: see text] denotes the iteration number. Moreover, we propose two fast variants of PADMC, one achieves faster [Formula: see text] ergodic convergence rate when one of the component functions is strongly convex, and the other ensures faster [Formula: see text] nonergodic convergence rate measured by constraint violation. Finally, preliminary numerical results on the LASSO and the elastic-net regularization problems are presented to demonstrate the performance of the proposed methods.

A Study on the Normalized Delineation of Airspace Sectors Based on Flight Conflict Dynamics

The study of airspace sector demarcation can help controllers to deal with complex air situations, provide reference for air traffic control services, reduce the workload of controllers, and ensure safe and efficient airspace operation. Based on complex network theory, this paper proposes a sector division method based on a mean shift clustering algorithm and a Voronoi diagram. Firstly, a flight conflict network is constructed based on the structural characteristics of the airborne flight situation, combined with the three-dimensional velocity barrier method and the aircraft protected area model. Secondly, six network topology indexes, such as the total node degree, average point strength and network density, are selected to construct the flight-situation-assessment index system, and the busy and idle time segments of the airspace are divided according to the comprehensive network indexes; finally, based on the historical flight data, the airborne aircraft in the busy and idle time segments are clustered using the mean shift clustering algorithm and the Voronoi diagram method, respectively, so as to obtain a sector division scheme. The simulation results show that this method can equalize the network topology indexes and provide a reference for the scientific allocation of controllers’ energy.

Nonlinear Forward-Backward Splitting with Momentum Correction

The nonlinear, or warped, resolvent recently explored by Giselsson and Bùi-Combettes has been used to model a large set of existing and new monotone inclusion algorithms. To establish convergent algorithms based on these resolvents, corrective projection steps are utilized in both works. We present a different way of ensuring convergence by means of a nonlinear momentum term, which in many cases leads to cheaper per-iteration cost. The expressiveness of our method is demonstrated by deriving a wide range of special cases. These cases cover and expand on the forward-reflected-backward method of Malitsky-Tam, the primal-dual methods of Vũ-Condat and Chambolle-Pock, and the forward-reflected-Douglas-Rachford method of Ryu-Vũ. A new primal-dual method that uses an extra resolvent step is also presented as well as a general approach for adding momentum to any special case of our nonlinear forward-backward method, in particular all the algorithms listed above.

Cooperative Beamforming for RIS-Aided Cell-Free Massive MIMO Networks

The combination of cell-free massive multiple-input multiple-output (CF-mMIMO) and reconfigurable intelligent surface (RIS) is envisioned as a promising paradigm to improve network capacity and enhance coverage capability. However, to reap full benefits of RIS-aided CF-mMIMO, the main challenge is to efficiently design cooperative beamforming (CBF) at base stations (BSs), RISs, and users. Firstly, we investigate the fractional programing to convert the weighted sum-rate (WSR) maximization problem into a tractable optimization problem. Then, the alternating optimization framework is employed to decompose the transformed problem into a sequence of subproblems, i.e., hybrid BF (HBF) at BSs, passive BF at RISs, and combining at users. In particular, the alternating direction method of multipliers algorithm is utilized to solve the HBF subproblem at BSs. Concretely, the analog BF design with unit-modulus constraints is solved by the manifold optimization (MO) while we obtain a closed-form solution to the digital BF design that is essentially a convex least-square problem. Additionally, the passive BF at RISs and the analog combining at users are designed by primal-dual subgradient and MO methods. Moreover, considering heavy communication costs in conventional CF-mMIMO systems, we propose a partially-connected CF-mMIMO (P-CF-mMIMO) framework to decrease the number of connections among BSs and users. To better compromise WSR performance and network costs, we formulate the BS selection problem in the P-CF-mMIMO system as a binary integer quadratic programming (BIQP) problem, and develop a relaxed linear approximation algorithm to handle this BIQP problem. Finally, numerical results demonstrate superiorities of our proposed algorithms over baseline counterparts.

Queue-aware computation offloading for UAV-assisted edge computing in wind farm routine inspection

Integration of unmanned aerial vehicles (UAVs) and edge computing into the wind farm routine inspection provides a promising approach to enhancing inspection effectiveness and decreasing operation maintenance costs. In light of the finite battery power and computational capacity of UAVs, a dynamic queue-aware UAV-assisted edge computing inspection wind farm framework is investigated with the goal of minimizing the long-term energy consumption of UAVs. The Lyapunov optimization theory is utilized to decouple the long-term stochastic optimization problem into four short-term deterministic subproblems, including the task splitting, the UAV-side computing resource allocation, the task offloading, and the edge server-side computing resource allocation. Furthermore, a Lyapunov optimization-based dynamic queue-aware computation offloading algorithm (LODQCO) is presented to optimize task offloading and resource allocation jointly. The optimal UAV-side computing resource is determined by a closed form formula, and then the optimal task offloading decision is tackled by applying the classical interior point method. Finally, the edge server-side computing resource is addressed via a linear optimization CPLEX solver. Based on simulation results, LODQCO is superior to the benchmark algorithms with respect to the energy consumption, queue backlogs, and queuing delays.

Higher-order interior point methods for convex nonlinear programming

This paper extends the concept of higher-order search directions within interior point methods to convex nonlinear programming. This includes the mathematical framework needed to compute the higher-order derivatives. The paper also highlights some special cases where the computation of these higher-order derivatives is simplified and a dimensional lifting procedure for transforming a large number of general nonlinear problems into one of these more efficient forms. The paper further describes the algorithmic development required to employ these higher-order search directions in a practical algorithm. Computational results are presented for a large number of test problems, highlighting higher-order methods’ strong potential for decreasing iteration count and their case-by-case potential for decreasing CPU time.

A rate-independent crystal plasticity algorithm based on the interior point method

A robust rate-independent crystal plasticity algorithm is proposed where the amount of plastic slip required to satisfy the multi-surface yield conditions are determined using a flow rule that stems from the maximization of plastic dissipation. The stress update procedure, which is based on the backward Euler time integration, is treated as an optimization problem and the interior point method is utilized in obtaining the stress. As opposed to common return mapping algorithms, such as the closest point projection method, the yield conditions are not violated during the iterative update of the slip amounts which helps to avoid convergence issues for both the local consistency and the global equilibrium solutions. An efficient implementation of this method for large deformations is given together with an algorithmic tangent modulus that makes it attractive to be used in both Finite Element simulations as well as analytical homogenization algorithms.

Distributed continuous-time proximal algorithm for nonsmooth resource allocation problem with coupled constraints

This paper studies the distributed resource allocation problem with nonsmooth local cost functions subject to the coupled equality and inequality constraints. In particular, each local cost function is expressed as the sum of a differentiable function and two nonsmooth functions. By using the operator splitting and primal–dual method, a continuous-time distributed proximal algorithm is developed, which can be applied to more general local cost functions that are convex but not necessarily smooth. In addition, the proposed algorithm is fully distributed in the sense that the gain parameter can be determined locally and does not require any global information of the network. By applying Lyapunov stability analysis and convex optimization theory, it is shown that the decision variables of all the agents converge to an optimal solution. Finally, a simulation example is carried out to demonstrate the effectiveness of the proposed algorithm.

A Fractional-Order Fidelity-Based Total Generalized Variation Model for Image Deblurring

Image deblurring is a fundamental image processing task, and research for efficient image deblurring methods is still a great challenge. Most of the currently existing methods are focused on TV-based models and regularization term construction; little efforts are paid to model proposal and correlated algorithms for the fidelity term in fractional-order derivative space. In this paper, we propose a novel fractional-order variational model for image deblurring, which can efficiently address three different blur kernels. The objective functional contains a fractional-order gradient fidelity term and a total generalized variation (TGV) regularization term, and it highlights the ability to preserve details and eliminate the staircase effect. To solve the problem efficiently, we provide two numerical algorithms based on the Chambolle-Pock primal-dual method (PD) and the alternating direction method of multipliers (ADMM). A series of experiments show that the proposed method achieves a good balance between detail preservation and deblurring compared with several existing advanced models.

Fast Trajectory Tracking Control Algorithm for Autonomous Vehicles Based on the Alternating Direction Multiplier Method (ADMM) to the Receding Optimization of Model Predictive Control (MPC).

In order to improve the real-time performance of the trajectory tracking of autonomous vehicles, this paper applies the alternating direction multiplier method (ADMM) to the receding optimization of model predictive control (MPC), which improves the computational speed of the algorithm. Based on the vehicle dynamics model, the output equation of the autonomous vehicle trajectory tracking control system is constructed, and the auxiliary variable and the dual variable are introduced. The quadratic programming problem transformed from the MPC and the vehicle dynamics constraints are rewritten into the solution of the ADMM form, and a decreasing penalty factor is used during the solution process. The simulation verification is carried out through the joint simulation platform of Simulink and Carsim. The results show that, compared with the active set method (ASM) and the interior point method (IPM), the algorithm proposed in this paper can not only improve the accuracy of trajectory tracking, but also exhibits good real-time performance in different prediction time domains and control time domains. When the prediction time domain increases, the calculation time shows no significant difference. This verifies the effectiveness of the ADMM in improving the real-time performance of MPC.

Newton and interior-point methods for (constrained) nonconvex–nonconcave minmax optimization with stability and instability guarantees

We address the problem of finding a local solution to a nonconvex–nonconcave minmax optimization using Newton type methods, including primal-dual interior-point ones. The first step in our approach is to analyze the local convergence properties of Newton’s method in nonconvex minimization. It is well established that Newton’s method iterations are attracted to any point with a zero gradient, irrespective of it being a local minimum. From a dynamical system standpoint, this occurs because every point for which the gradient is zero is a locally asymptotically stable equilibrium point. We show that by adding a multiple of the identity such that the Hessian matrix is always positive definite, we can ensure that every non-local-minimum equilibrium point becomes unstable (meaning that the iterations are no longer attracted to such points), while local minima remain locally asymptotically stable. Building on this foundation, we develop Newton-type algorithms for minmax optimization, conceptualized as a sequence of local quadratic approximations for the minmax problem. Using a local quadratic approximation serves as a surrogate for guiding the modified Newton’s method toward a solution. For these local quadratic approximations to be well-defined, it is necessary to modify the Hessian matrix by adding a diagonal matrix. We demonstrate that, for an appropriate choice of this diagonal matrix, we can guarantee the instability of every non-local-minmax equilibrium point while maintaining stability for local minmax points. Using numerical examples, we illustrate the importance of guaranteeing the instability property. While our results are about local convergence, the numerical examples also indicate that our algorithm enjoys good global convergence properties.

Dynamic Surface Properties of α-Lactalbumin Fibril Dispersions.

The dynamic surface properties of aqueous dispersions of α-lactalbumin (ALA) amyloid fibrils differ noticeably from the properties of the fibril dispersions of other globular proteins. As a result, the protocol of the application of ALA fibrils to form stable foams and emulsions has to be deviate from that of other protein fibrils. Unlike the fibrils of β-lactoglobulin and lysozyme, ALA fibrils can be easily purified from hydrolyzed peptides and native protein molecules. The application of the oscillating barrier method shows that the dynamic surface elasticity of ALA fibril dispersions exceeds the surface elasticity of native protein solutions at pH 2. ALA fibrils proved to be stable at this pH, but the stability breaks at higher pH levels when the fibrils start to release small peptides of high surface activity. As a result, the dynamic surface properties of ALA coincide with those of native protein solutions. The ionic strength strongly influences the adsorption kinetics of both fibril dispersions and native protein solutions but have almost no impact on the structure of the adsorption layers.

In-Network Computation for Large-Scale Federated Learning Over Wireless Edge Networks

Most conventional Federated Learning (FL) models are using a star network topology where all users aggregate their local models at a single server (e.g., a cloud server). That causes significant overhead in terms of both communications and computing at the server, delaying the training process, especially for large scale FL systems with straggling nodes. This paper proposes a novel edge network architecture that enables decentralizing the model aggregation process at the server, thereby significantly reducing the training delay for the whole FL network. Specifically, we design a highly-effective in-network computation framework (INC) consisting of a user scheduling mechanism, an in-network aggregation process (INA) which is designed for both primal- and primal-dual methods in distributed machine learning problems, and a network routing algorithm with theoretical performance bounds. The in-network aggregation process, which is implemented at edge nodes and cloud node, can adapt two typical methods to allow edge networks to effectively solve the distributed machine learning problems. Under the proposed INA, we then formulate a joint routing and resource optimization problem, aiming to minimize the aggregation latency. The problem turns out to be NP-hard, and thus we propose a polynomial time routing algorithm which can achieve near optimal performance with a theoretical bound. Simulation results showed that the proposed algorithm can achieve more than 99<inline-formula><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> of the optimal solution and reduce the FL training latency, up to 5.6 times w.r.t other baselines. The proposed INC framework can not only help reduce the FL training latency but also significantly decrease cloud&#x2019;s traffic and computing overhead. By embedding the computing/aggregation tasks at the edge nodes and leveraging the multi-layer edge-network architecture, the INC framework can liberate FL from the star topology to enable large-scale FL.