Interior Point Algorithm Research Articles

Hybridization of the swarming and interior point algorithms to solve the Rabinovich–Fabrikant system

In this study, a trustworthy swarming computing procedure is demonstrated for solving the nonlinear dynamics of the Rabinovich–Fabrikant system. The nonlinear system’s dynamic depends upon the three differential equations. The computational stochastic structure based on the artificial neural networks (ANNs) along with the optimization of global search swarming particle swarm optimization (PSO) and local interior point (IP) algorithms, i.e., ANNs-PSOIP is presented to solve the Rabinovich–Fabrikant system. An objective function based on the differential form of the model is optimized through the local and global search methods. The correctness of the ANNs-PSOIP scheme is observed through the performances of achieved and source solutions, while the negligible absolute error that is around 10−05–10−07 also represent the worth of the ANNs-PSOIP algorithm. Furthermore, the consistency of the ANNs-PSOIP scheme is examined by applying different statistical procedures to solve the Rabinovich–Fabrikant system.

Primal–dual interior-point algorithm for symmetric model predictive control

This paper presents a primal–dual interior-point (pdip) optimization algorithm for solving extreme-scale model predictive control (mpc) problems with linear dynamics, polytopic constraints, and quadratic/linear costs which are all invariant under the symmetric-group. We show that exploiting symmetry can reduce the computational and memory burden of extreme-scale or fast-paced applications of mpc. Our algorithm transforms the original inputs, states, and constraints of the mpc problem into a symmetric domain. The premise of our algorithm is that the numerical linear algebra used to solve the optimization problem has lower computational and memory complexity in the transformed domain. We demonstrate our algorithm for a heating, ventilation, and air-conditioning (hvac) numerical example. We show that, for our largest hvac control problem, our symmetry exploiting the pdip algorithm reduces the computation-time from minutes to seconds in comparison with the baseline pdip algorithm. Furthermore, we show that the presented symmetry exploiting pdip algorithm outperforms a state-of-the-art symmetry exploiting optimization algorithm.

Elastoplastic and limit analysis of reinforced concrete with an equilibrium-based finite element formulation

This paper introduces a 3D equilibrium-based finite element formulation for reinforced concrete. The stress unknowns in the proposed formulation strongly satisfy the equilibrium equations throughout the entire volume, with the exception of the mesh interfaces where the rebars intersect. The concrete stress tensor is interpolated using tetrahedral piece-wise linear elements with statically admissible discontinuities, while the rebars are considered as 1D elements embedded into the concrete, intersecting the triangular mesh. Thus, the global equilibrium is ensured by writing for each face of the mesh the traction continuity equation, while including the rebar stress contribution for the intersected triangular faces. The elastic equilibrium formulation is then written into an equivalent optimization problem, extended to the elastoplastic case by simply adding semi-definite matrix constraints on the concrete stress tensor, corresponding to a Rankine or a truncated Mohr–Coulomb criterion. As for the rebars, they are considered to obey a 1D perfectly elastoplastic behavior. The present formulation is also developed for limit analysis, to directly obtain an estimate of the lower bound of the collapse load. The resulting semi-definite programming optimization problems are solved using an in–house interior point algorithm.

Interior-point algorithm for symmetric cone horizontal linear complementarity problems based on a new class of algebraically equivalent transformations

AbstractWe generalize a primal-dual interior-point algorithm (IPA) proposed recently in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022) to $$P_*(\kappa )$$ P ∗ ( κ ) -horizontal linear complementarity problems (LCPs) over Cartesian product of symmetric cones. The algorithm is based on the algebraic equivalent transformation (AET) technique with a new class of AET functions. The new class is a modification of the class of AET functions proposed in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022) where only two conditions are used as opposed to three used in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022). Furthermore, the algorithm is a feasible algorithm that uses full Nesterov-Todd steps, hence, no calculation of step-size is necessary. Nevertheless, we prove that the proposed IPA has the iteration bound that matches the best known iteration bound for IPAs solving these types of problems.

Unified Approach of Interior-Point Algorithms for P∗(κ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_*(\\kappa )$$\\end{document}-LCPs Using a New Class of Algebraically Equivalent Transformations

We propose new short-step interior-point algorithms (IPAs) for solving P∗(κ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_*(\\kappa )$$\\end{document}-linear complementarity problems (LCPs). In order to define the search directions, we use the algebraic equivalent transformation (AET) technique of the system describing the central path. A novelty of the paper is that we introduce a whole, new class of AET functions for which a unified complexity analysis of the IPAs is presented. This class of functions differs from the ones used in the literature for determining search directions, like the class of concave functions determined by Haddou, Migot and Omer, self-regular functions, eligible kernel and self-concordant functions. We prove that the IPAs using any member φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} of the new class of AET functions have polynomial iteration complexity in the size of the problem, in starting point’s duality gap, in the accuracy parameter and in the parameter κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}.

A Swarming Meyer Wavelet Computing Approach to Solve the Transport System of Goods

The motive of this work is to provide the numerical performances of the reactive transport model that carries trucks with goods on roads by exploiting the stochastic procedures based on the Meyer wavelet (MW) neural network. An objective function is constructed by using the differential model and its boundary conditions. The optimization of the objective function is performed through the hybridization of the global and local search procedures, i.e., swarming and interior point algorithms. Three different cases of the model have been obtained, and the exactness of the stochastic procedure is observed by using the comparison of the obtained and Adams solutions. The negligible absolute error enhances the exactness of the proposed MW neural networks along with the hybridization of the global and local search schemes. Moreover, statistical interpretations based on different operators, histograms, and boxplots are provided to validate the constancy of the designed stochastic structure.

A path-following interior-point algorithm for monotone LCP based on a modified Newton search direction

In this paper, we propose a short-step feasible full-Newton step path-following interior-point algorithm (IPA) for monotone linear complementarity problems (LCPs). The proposed IPA uses the technique of algebraic equivalent transformation (AET) induced by an univariate function to transform the centering equations which defines the central-path. By applying Newton’s method to the modified system of the central-path of LCP, a new Newton search direction is obtained. Under new appropriate defaults of the thresholdτwhich defines the size of the neighborhood of the central-path and ofθwhich determines the decrease in the barrier parameter, we prove that the IPA is well-defined and converges locally quadratically to a solution of the monotone LCPs. Moreover, we derive its iteration bound, namely,O(√nlogn\ϵ) which coincides with the best-known iteration bound for such algorithms. Finally, some numerical results are presented to show its efficiency.

Linear optimization over homogeneous matrix cones

A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone. Cones that are homogeneous and self-dual are called symmetric. Conic optimization problems over symmetric cones have been extensively studied, particularly in the literature on interior-point algorithms, and as the foundation of modelling tools for convex optimization. In this paper we consider the less well-studied conic optimization problems over cones that are homogeneous but not necessarily self-dual.We start with cones of positive semidefinite symmetric matrices with a given sparsity pattern. Homogeneous cones in this class are characterized by nested block-arrow sparsity patterns, a subset of the chordal sparsity patterns. Chordal sparsity guarantees that positive define matrices in the cone have zero-fill Cholesky factorizations. The stronger properties that make the cone homogeneous guarantee that the inverse Cholesky factors have the same zero-fill pattern. We describe transitive subsets of the cone automorphism groups, and important properties of the composition of log-det barriers with the automorphisms.Next, we consider extensions to linear slices of the positive semidefinite cone, and review conditions that make such cones homogeneous. An important example is the matrix norm cone, the epigraph of a quadratic-over-linear matrix function. The properties of homogeneous sparse matrix cones are shown to extend to this more general class of homogeneous matrix cones.We then give an overview of the algebraic theory of homogeneous cones due to Vinberg and Rothaus. A fundamental consequence of this theory is that every homogeneous cone admits a spectrahedral (linear matrix inequality) representation.We conclude by discussing the role of homogeneous structure in primal–dual symmetric interior-point methods, contrasting this with the well-developed algorithms for symmetric cones that exploit the strong properties of self-scaled barriers, and with symmetric primal–dual methods for general convex cones.

An efficient multi parametric kernel function for large and small-update methods interior point algorithm for P*(κ)-horizontal linear complementarity problem

In this paper, we propose the first efficient multi parametric kernel function with logarithmic barrier term. A class of polynomial interior-point algorithms for P*(κ)-horizontal linear complementarity problem based on this kernel function, with parameters pi > 0 for all i ∈ 1, 2, , m, are presented. Then by using some simple analysis tools, we present a primal-dual interior point method (IPM) for P*(κ)-horizontal linear complementarity problems based on this kernel function. At the same time, we derive the complexity bounds small and large-update methods, respectively. In particular, if we take many different values of the parameters, we obtain the best known iteration bounds for the algorithms with large- and small-update methods are derived, namely, O((1 + 2κ)√n(log n)log n/ϵ) and O((1 + 2κ)√n log n/ϵ) respectively. We illustrate the performance of the proposed kernel function by some numerical results that are derived by applying our algorithm.

Pure-displacement formulation and bulk modulus propagation for topology optimization with pressure loads

This work proposes a pure-displacement formulation for topology optimization with pressure loads and the use of a continuous boundary propagation technique. This technique allows controlling the non-solid (fluid or void) constitutive behavior by disseminating the desired bulk modulus to holes in the solid domain. The novelty of the proposed approach is to enable the distribution of solid, fluid, and void simultaneously with one design variable field. A density-based topology optimization approach is considered, and a gradient-based interior point algorithm is used. The linear elasticity equilibrium equations are discretized by the finite element method and solved with a sparse direct solver. The sensitivities are calculated by the discrete adjoint method with automatic differentiation. The optimization problems encompass compliance minimization (in 2D and 3D) and a compliant mechanism design to maximize the output displacement. Three examples evaluate the proposed formulation: the internally-pressurized lid, the piston head, and the inverter-compliant mechanism. The results show that different designs are obtained by considering different constitutive behavior for the non-solid phase.

A Primal-Dual Interior-Point Algorithm Based on a Kernel Function with a New Barrier Term

In this paper, we propose a path-following interior-point method (IPM) for solving linear optimization (LO) problems based on a new kernel function (KF). The latter differs from other KFs in having an exponential-hyperbolic barrier term that belongs to the hyperbolic type, recently developed by I. Touil and W. Chikouche \cite{filomat2021,acta2022}. The complexity analysis for large-update primal-dual IPMs based on this KF yields an $\mathcal{O}\left( \sqrt{n}\log^2n\log \frac{n}{\epsilon }\right)$ iteration bound which improves the classical iteration bound. For small-update methods, the proposed algorithm enjoys the favorable iteration bound, namely, $\mathcal{O}\left( \sqrt{n}\log \frac{n}{\epsilon }\right)$. We back up these results with some preliminary numerical tests which show that our algorithm outperformed other algorithms with better theoretical convergence complexity. To our knowledge, this is the first feasible primal-dual interior-point algorithm based on an exponential-hyperbolic KF.

A full-Newton step infeasible interior-point algorithm based on a kernel function with a new barrier term

A full-Newton step infeasible interior-point algorithm based on a kernel function with a new barrier term

A Cooperative Recharging-Transmission Strategy in Powered UAV-Aided Terahertz Downlink Networks

Considering that unmanned aerial vehicles (UAVs) with the advantages of flexible deployment and high maneuverability have the limited energy to provide long-term seamless coverage, a cooperative recharging-transmission strategy is studied in this paper for the powered UAV-aided terahertz (THz) downlink networks, where a UAV-to-user THz sub-band association scheme is proposed for the THz frequency transmission to eliminate interference. Based on this, the minimum requirements of the wireless charging window and THz-transmitting window are derived. A minimization problem of minimum timeslot requirement is formulated to jointly optimize UAV placement and UAV-to-user THz sub-band association. The optimization problem is non-convex, and we propose an alternating optimization solution via iteratively solving placement subproblem and association subproblem. Specifically, we resort to the coalescent method of interior point and gradient descent algorithms to optimize UAV placement. The UAV-to-user THz sub-band association is solved as a knapsack problem by the transformation of association matrix. Simulation results validate the solution and demonstrate that the proposed strategy can achieve quasi-optimal performance compared with other contrasting strategies.

Influence of soft band on seismic slope stability by finite-element limit-analysis modelling

Soft bands may widely exist in sites, and their presence complicates the analysis of slope stability, particularly when the limit analysis method is adopted. Note that the use of the limit analysis method is capable of providing rigorous lower and upper bound solutions and hence is widely used in soil slope stability analyses. To account for soft band effects, finite-element lower-bound (FELB) and finite-element upper-bound (FEUB) approaches are introduced in this study. Finite element discretization aids in the construction of a statically allowable stress field and kinematically admissible velocity field, which are prerequisites for limit analysis. In combination with a modified pseudodynamic approach, seismic slope stability analysis is evaluated through linear programming. Lower and upper bound solutions of the slope safety factor (FoS) are optimized with an interior point algorithm, which aids in better estimating the true FoS of seismic slope stability. The nonassociated flow rule is also discussed herein. After validating the upper and lower linear programming models by limit equilibrium and FEM, the soft band’s effects on seismic slope stability are discussed, and a rotational-translational failure mechanism is revealed by critical velocity fields. A less than 5.1% difference between lower- and upper-bound solutions is favourable for providing a reliable reference for slope design.

Particle swarm optimisation strategies for IOL formula constant optimisation.

To investigate particle swarm optimisation (PSO) as a modern purely data driven non-linear iterative strategy for lens formula constant optimisation in intraocular lens power calculation. A PSO algorithm was implemented for optimising the root mean squared formula prediction error (rmsPE, defined as achieved refraction minus predicted refraction) for the Castrop formula in a dataset of N= 888 cataractous eyes with implantation of the Hoya Vivinex hydrophobic acrylic aspheric lens. The hyperparameters were set to inertia: 0.8, accelerations c1=c2=0.1. The algorithm was initialised with NP = 100 particles having random positions and velocities within the box constraints of the constant triplet parameter space C=0.25 to 0.45, H=-0.25 to 0.25 and R=-0.25 to 0.25. The performance of the algorithm was compared to classical gradient-based Trust-Region-Reflective and Interior-Point algorithms. The PSO algorithm showed fast and stable convergence after 37 iterations. The rmsPE reduced systematically to 0.3440 diopters (D). With further iterations the scatter of the particle positions in the swarm decreased but without further reduction of rmsPE. The final constant triplet was C/H/R=0.2982/0.2497/0.1435. The Trust-Region-Reflective/Interior-Point algorithms showed convergence after 27/17 iterations, respectively, resulting in formula constant triplets C/H/R=0.2982/0.2496/0.1436 and 0.2982/0.2495/0.1436, both with the same rmsPE as the PSO algorithm (rmsPE=0.3440 D). The PSO appears to be a powerful adaptive nonlinear iteration algorithm for formula constant optimisation even in formulae with more than 1 constant. It acts independently of an analytical or numerical gradient and is in general able to search for the best solution even with multiple local minima of the target function.

An Algebraic-Based Primal–Dual Interior-Point Algorithm for Rotated Quadratic Cone Optimization

In rotated quadratic cone programming problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of rotated quadratic cones. In this paper, we introduce the rotated quadratic cone programming problems as a “self-made” class of optimization problems. Based on our own Euclidean Jordan algebra, we present a glimpse of the duality theory associated with these problems and develop a special-purpose primal–dual interior-point algorithm for solving them. The efficiency of the proposed algorithm is shown by providing some numerical examples.

Numerical study of nonlinear flow and mass transfer through a channel filled with a porous medium using a hybrid heuristic paradigm

ABSTRACT The effects of mass transfer and chemical reactions on laminar viscous flow in a porous channel with moving or stationary walls are examined in this study. The mathematical model of the mass and flow transfer is in the form of a partial differential equation (PDE), a similarity transformation method used to convert the physical problem's governing partial differential equations (PDEs) into a group of nonlinear coupled ordinary differential equations (ODEs). The main accomplishment of the present study is the utilization of a precise and effective method based on artificial neural networks (ANNs) for the model of flow in a porous channel of a viscous fluid with a chemical reaction. To investigate the coupled nonlinear ordinary differential equations, a differential evolutionary (DE) is used for exploration to get optimal weights for the model, and then we use those optimal weights in interior point algorithms (IPA) for exploitation of the laminar viscous flow model in a porous channel with moving or stationary walls. Finally, the effects of Darcy number (Da), suction/injection parameter (α), Reynolds number (Re), and power-law index (m) influence are investigated in five different scenarios, each with four different cases. The outcomes of the hybrid of differential evolution and interior point algorithm (DE-IPA) are compared to the RK4 results. The statistical analysis (MAD, TIC, RMSE, and ENSE) is also performed to ensure that the proposed technique is stable and accurate.

Predictive Model of Humidity in Greenhouses through Fuzzy Inference Systems Applying Optimization Methods

Establishing the indoor and outdoor humidity values in a greenhouse allows us to describe the crop yield during its entire developmental cycle. This study seeks to develop a predictive model of indoor relative humidity values in a greenhouse with high accuracy and interpretability through the use of optimized fuzzy inference systems, in order to offer greenhouse users a clear and simple description of their behaviour. The three-phase methodology applied made use of descriptive statistics techniques, correlation analysis, and prototyping paradigm for the iterative and incremental development of the predictive model, validated through error measurement. The research resulted in six models which define the behaviour of humidity as a result of temperature, CO2, and soil moisture, with percentages of effectiveness above 90%. The implementation of a Mamdani-type fuzzy inference system, optimized by a hybrid method combining genetic and interior point algorithms, allowed to predict the relative humidity in greenhouses with high interpretability and precision, with an effectiveness percentage of 90.97% and MSE (mean square error) of 8.2e − 3.

AN INTERIOR-POINT ALGORITHM FOR SIMPLICIAL CONE CONSTRAINED CONVEX QUADRATIC OPTIMIZATION

In this paper, we are concerned with the numerical solution of simplicial cone constrained convex quadratic optimization (SCQO) problems. A reformulation of the K.K.T optimality conditions of SCQOs as an equivalent linear complementarity problem with $\mathcal{P}$-matrix ($\mathcal{P}$-LCP) is considered. Then, a feasible full-Newton step interior-point algorithm (IPA) is applied for solving SCQO via $\mathcal{P}$-LCP. For the completeness of the study, we prove that the proposed algorithm is well-defined and converges locally quadratic to an optimal of SCQOs. Moreover, we obtain the currently best well-known iteration bound for the algorithm with short-update method, namely,$ \mathcal{O}(\sqrt{n}\log\frac{n}{\epsilon })$. Finally, we present a various set of numerical results to show its efficiency.

Modeling and analysis of a variable stiffness nonlinear permanent magnetic spring

The magnetic spring is a practical element to isolate external vibration to instruments in ultra-precision manufacturing and ultra-precision testing. Accurate magnetic force and stiffness models are necessary to analyze and optimize the design of magnetic spring isolators. Thus, this study proposes an accurate magnetic force and stiffness model to design and analyze a variable stiffness nonlinear magnetic spring (VSNMS) considering the permanent magnet’s (PM’s) relative permeability, positions, sizes, and magnetic properties. Firstly, the VSNMS’s structure is designed and introduced, and its magnetic force and stiffness models are derived. Secondly, the established model is validated by comparison with the numerical simulation. Thirdly, the VSNMS prototype is fabricated, and then tested through an electronic universal testing machine. The model has the advantage of high accuracy, high efficiency, and ease of use compared with numerical simulations and experimental tests. Furthermore, parametric studies are conducted to learn the effect of PMs’ sizes, positions, and magnetic properties on VSNMS’s magnetic force and stiffness characteristics. Finally, the optimized structural size parameters of VSNMS are obtained using the derived model and the interior-point algorithm. As a result, the established model contributes to the analysis and optimal design of VSNMS, supporting the design of magnetic spring isolators in ultra-precision applications.