Primal-dual Interior Point Method Research Articles

A numerical study of an infeasible interior-point algorithm for convex quadratic semi-definite optimization

The focus of this research is to apply primal-dual interior-point pathfollowing methods, specifically those derived from Newton’s method for solving convex quadratic semidefinite optimization (CQSDO) problems. In this paper, we present a numerical study of an infeasible primal-dual interior-point method for tackling this class of optimization problems. Unlike the feasible interior-point algorithms, the proposed algorithm can be start with any initial positive definite matrix and does not require the strictly feasible initial points. Under certain conditions, the Newton system is well defined and its Jacobian is nonsingular at the solution. For computing an iteration throughout the algorithm, a Newton direction and a step-size are determined. Here, our search direction is based on Alizadeh-Haeberly-Overton (AHO) symmetrization. However, for the step size along this direction an efficient procedure is suggested. Preliminary numerical results demonstrate the efficiency of our algorithm.

Establishment method of IGBT thermal network model based on electrothermal coupling simulation and parameter identification

Abstract To swiftly and precisely assess the IGBT’s thermal safety during the circuit design phase, this paper proposes a method of establishing an IGBT thermal network model by electrothermal coupling simulation and parameter identification. This model integrates electrothermal coupling simulation techniques and parameter identification methods. Firstly, the IGBT’s finite element model is established by COMSOL, aiming to acquire the temperature distribution patterns of the IGBT and its heat sink under various operating conditions. Then, the objective function of parameter identification is constructed. The IGBT thermal network model’s parameters are extracted by combining the electrothermal coupling simulation results with the primal-dual interior point method. Then the device’s junction-ambient thermal network model is obtained. The proposed method’s efficacy is validated through the comparison of the results and calculation time of the thermal network model and the electrothermal coupling simulation.

A primal-dual interior-point method with full-Newton step for semidefinite optimization

A primal-dual interior-point method with full-Newton step for semidefinite optimization

Proximal-stabilized semidefinite programming

AbstractA regularized version of the primal-dual Interior Point Method (IPM) for the solution of Semidefinite Programming Problems (SDPs) is presented in this paper. Leveraging on the proximal point method, a novel Proximal Stabilized Interior Point Method for SDP (PS-SDP-IPM) is introduced. The method is strongly supported by theoretical results concerning its convergence: the worst-case complexity result is established for the inner regularized infeasible inexact IPM solver. The new method demonstrates an increased robustness when dealing with problems characterized by ill-conditioning or linear dependence of the constraints without requiring any kind of pre-processing. Extensive numerical experience is reported to illustrate advantages of the proposed method when compared to the state-of-the-art solver.

A robust force identification method based on L1+2-norm approximation theory

Obtaining the real force of the structure is the basic condition for efficient vibration reduction design. However, the uneasiness and instability of the inverse problem make the force identification still challenging. On the basis of L2-norm regularization, we propose a new method for force identification based on L1+2-norm approximation theory. By introducing the L1-norm penalty operator, a joint norm approximation optimization model is constructed to limit the non-zero force and punish the force value which approaching to zero, so that it can adapt to the identification of various types of forces. Then the optimal force solution is obtained by using the Primal-dual interior point method. Finally, the effectiveness and accuracy of the proposed method are verified by numerical simulation and experiment of asymmetric plate and double-layer plate structures. Different from previous studies, the L1+2-norm method in this work has high robustness, considers the characteristics of sparse regularization and L2-norm regularization, and can adapt to the identification of continuous forces and sparse forces at the same time, which has potential engineering application value.

Solving Max-cut Problem with a mixed penalization method for Semidefinite Programming

This paper is an attempt to exploit the opportunity of Semi Definite Programming (SDP), which is an area of convex and conic optimization. Indeed, Numerous NP-hard problems can be solveby using this approach. Hence, we intend to investigate the strength of SDP to model and provide tight relaxations of combinatorial and quadratic problems in order to present a new polynomial time algorithm for solving this robust model. This algorithm was firstly use to solve the nonlinear programs, the reason for which we search to extend it to the SDP programs. Actually, this algorithm designs the combination of two penalization methods. The first one is a primal-dual interior point (PDIM) method while the second one is a primal-dual exterior point (PDEM) method. Unlike the first method, which converges globally, the second one, also called the primal dual nonlinear rescaling method, has local super linear/quadratic convergence. Therefore, it seems appropriate to use a mixed algorithm based on the interior-exterior point method (IEPM). In fact, this resolution starts from the interior method, and at a certain level of execution, it proceeds to exterior method. Hence, a convergence evaluation function is use to know the level of permutation. Through evaluation, it has been approve that our approach is use to solve some instances of max-cut problem. This problem is a central graph theory model that occurs in many real problems and it is one of many NP-hard problems, which has attracted many researchers over the years. Then, we have used a semi definite programming solver SDPA (Semi Definite Programming Algorithm) that is modify to include the exterior point method subroutine. From the computational performance, we conclude that as the problem size increases, interior-exterior point algorithm gets relatively faster. The numerical results obtained are promising.

Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a new type of kernel functions

Kernel functions are essential for designing and analyzing interior-point methods (IPMs). They are used to determine search directions and reduce the computational complexity of the interior point method. Currently, IPM based on kernel functions is one of the most effective methods for solving LO [1,20], second-order cone optimization (SOCO) [2], and symmetric optimization (SO) and is a very active research area in mathematicalprogramming. This paper presents a large-update primal-dual IPM for SDO based on a new bi-parameterized hyperbolic kernel function. Then we proved that the proposed large-update IPM has the same complexity bound as the best-known IPMs for solving these problems. Taking advantage of the favorable characteristics of the kernel function, we can deduce that the iteration bound for the large update method is $\mathcal{O}\left( \sqrt{n}\log n\log\dfrac{n}{\varepsilon }\right) $ when a takes a special value utilizing the favorable properties of the kernel function. These theoretical results play an essential role in the design and analysis of IPMs for CQSCO [7] and the Cartesian$\ P_{\ast }\left( \kappa \right) $-SCLCP [8]. The proximity function has never been used. In order to validate the efficacy of our algorithm and verify the effectiveness of our algorithm, examples aregiven to illustrate the applicability of our main results, and we compare our numerical results with some alternatives presented in the literature.

Solving clustered low-rank semidefinite programs arising from polynomial optimization

We study a primal-dual interior point method specialized to clustered low-rank semidefinite programs requiring high precision numerics, which arise from certain multivariate polynomial (matrix) programs through sums-of-squares characterizations and sampling. We consider the interplay of sampling and symmetry reduction as well as a greedy method to obtain numerically good bases and sample points. We apply this to the computation of three-point bounds for the kissing number problem, for which we show a significant speedup. This allows for the computation of improved kissing number bounds in dimensions 11 through 23. The approach performs well for problems with bad numerical conditioning, which we show through new computations for the binary sphere packing problem.

Detection of indentation damage in carbon fiber/epoxy composites via EIT during the application of bending loads

Electrical impedance tomography (EIT) is a method of spatially mapping the conductivity distribution of a domain and has been studied as a potential embedded sensing or nondestructive evaluation (NDE) tool. An often touted advantage of EIT is that it can be used in-situ; that is, because the method only requires the application of unobtrusive electrodes, it can conceivably be used while the component or structure is in operation. This material-as-the-sensor philosophy strongly aligns with key components of the NDE 4.0 vision such as the realization of intelligent cyber–physical systems (CPS) and digital twins. To date, however, the claim of in-situ sensing via EIT has not been significantly substantiated. This is problematic because operational loads induce strains that often change the conductivity of the material. Establishing that EIT can detect damage-induced conductivity changes through the presence of unrelated strain-induced conductivity changes is therefore important. To that end, we herein study the application of EIT for detecting indentation damage in a carbon fiber/epoxy composite as the composite is loaded in a four-point bend. It was found that the bending load changes the contact impedance of the electrodes, which resulted in poor EIT images when solving the EIT inverse problem with the ℓ2-norm on the error term. Using the ℓ1-norm on the error term, solved via the primal–dual interior point method (PDIPM), significantly improved image quality. Image quality was even further improved through the use of a mixed prior for regularization, and EIT images were compared to thermography with good agreement. These results show that EIT can indeed detect damage through the presence of an applied load, but care must be taken to account for factors such as outlier data arising from electrode degradation and changing contact impedance. Use of the ℓ1-norm on the error term is therefore highly recommended for in-situ imaging via EIT.

On Implementing a Two-Step Interior Point Method for Solving Linear Programs

A new two-step interior point method for solving linear programs is presented. The technique uses a convex combination of the auxiliary and central points to compute the search direction. To update the central point, we find the best value for step size such that the feasibility condition is held. Since we use the information from the previous iteration to find the search direction, the inverse of the system is evaluated only once every iteration. A detailed empirical evaluation is performed on NETLIB instances, which compares two variants of the approach to the primal-dual log barrier interior point method. Results show that the proposed method is faster. The method reduces the number of iterations and CPU time(s) by 27% and 18%, respectively, on NETLIB instances tested compared to the classical interior point algorithm.

A hybrid algorithm for quadratically constrained quadratic optimization problems

Quadratically Constrained Quadratic Programs (QCQPs) are an important class of optimization problems with diverse real-world applications. In this work, we propose a variational quantum algorithm for general QCQPs. By encoding the variables in the amplitude of a quantum state, the requirement for the qubit number scales logarithmically with the dimension of the variables, which makes our algorithm suitable for current quantum devices. Using the primal-dual interior-point method in classical optimization, we can deal with general quadratic constraints. Our numerical experiments on typical QCQP problems, including Max-Cut and optimal power flow problems, demonstrate better performance of our hybrid algorithm over classical counterparts.

A convex cone programming based implicit material point method

For conventional Material Point Method (MPM), both explicit-based and implicit-based MPM have shortcomings: explicit MPM has high requirements on time steps, and implicit MPM has high requirements on convergence. To circumvent these limitations, this paper innovatively proposes a convex cone programming-based implicit MPM (CP-MPM) algorithm, which ensures excellent convergence of solving complex problems involving large deformation, regardless of the chosen time step. In the proposed CP-MPM, the governing equations are initially transformed into a stationary point of a multivariable functional, leveraging the generalized Hellinger-Reissner (HR) variational principle. This stationary point problem is subsequently reformulated as a min-max convex cone optimization problem, with constraints originating from elastoplastic constitutive equations. In addition, a novel particle-based adhesive-frictional contact algorithm is proposed to effectively tackle the interaction between MPM domain and rigid bodies. The contact inequality between material points and rigid bodies is transformed into convex cone constraints, which rigorously prevent material point penetration and facilitate the imposition of irregular boundary conditions. Both elastic and elastoplastic problems involving contact under static or dynamic loading are ultimately represented as standard second-order cone programming (SOCP) problems, which is effectively solved by employing the Primal-Dual Interior Point (PDIP) method. The robustness, accuracy and convergence of the proposed method are validated through a series of elastic and elastoplastic benchmark problems. All results demonstrate the CP-MPM is a very promising method for implicitly solving complex practices.

A primal–dual interior point method to implicitly update Gurson–Tvergaard–Needleman model

A primal–dual interior point method to implicitly update Gurson–Tvergaard–Needleman model

Local convergence of primal–dual interior point methods for nonlinear semidefinite optimization using the Monteiro–Tsuchiya family of search directions

Local convergence of primal–dual interior point methods for nonlinear semidefinite optimization using the Monteiro–Tsuchiya family of search directions

A projected-search interior-point method for nonlinearly constrained optimization

AbstractThis paper concerns the formulation and analysis of a new interior-point method for constrained optimization that combines a shifted primal-dual interior-point method with a projected-search method for bound-constrained optimization. The method involves the computation of an approximate Newton direction for a primal-dual penalty-barrier function that incorporates shifts on both the primal and dual variables. Shifts on the dual variables allow the method to be safely “warm started” from a good approximate solution and avoids the possibility of very large solutions of the associated path-following equations. The approximate Newton direction is used in conjunction with a new projected-search line-search algorithm that employs a flexible non-monotone quasi-Armijo line search for the minimization of each penalty-barrier function. Numerical results are presented for a large set of constrained optimization problems. For comparison purposes, results are also given for two primal-dual interior-point methods that do not use projection. The first is a method that shifts both the primal and dual variables. The second is a method that involves shifts on the primal variables only. The results show that the use of both primal and dual shifts in conjunction with projection gives a method that is more robust and requires significantly fewer iterations. In particular, the number of times that the search direction must be computed is substantially reduced. Results from a set of quadratic programming test problems indicate that the method is particularly well-suited to solving the quadratic programming subproblem in a sequential quadratic programming method for nonlinear optimization.

An efficient primal-dual interior point algorithm for linear optimization problems based on a new parameterized Kernel function with a logarithmic barrier term

In this paper, we present a primal-dual interior point method for linear optimization problems based on a new Kernel function with on a new parameterized logarithmic barrier term. We prove that the proposed kernel function belongs to the eligible class. We derive the complexity bounds for large and small-update methods respectively.

A stable implicit nodal integration-based particle finite element method (N-PFEM) for modelling saturated soil dynamics

In this study, we present a novel nodal integration-based particle finite element method (N-PFEM) designed for the dynamic analysis of saturated soils. Our approach incorporates the nodal integration technique into a generalised Hellinger-Reissner (HR) variational principle, creating an implicit PFEM formulation. To mitigate the volumetric locking issue in low-order elements, we employ a node-based strain smoothing technique. By discretising field variables at the centre of smoothing cells, we achieve nodal integration over cells, eliminating the need for sophisticated mapping operations after re-meshing in the PFEM. We express the discretised governing equations as a min-max optimisation problem, which is further reformulated as a standard second-order cone programming (SOCP) problem. Stresses, pore water pressure, and displacements are simultaneously determined using the advanced primal-dual interior point method. Consequently, our numerical model offers improved accuracy for stresses and pore water pressure compared to the displacement-based PFEM formulation. Numerical experiments demonstrate that the N-PFEM efficiently captures both transient and long-term hydro-mechanical behaviour of saturated soils with high accuracy, obviating the need for stabilisation or regularisation techniques commonly employed in other nodal integration-based PFEM approaches. This work holds significant implications for the development of robust and accurate numerical tools for studying saturated soil dynamics.

A nonsmooth modified symplectic integration scheme for frictional contact dynamics of rigid–flexible multibody systems

Constructing an efficient high-order nonsmooth time-stepping integration scheme for rigid–flexible multibody systems with frictional contacts and impacts is a challenging task. This paper presents a nonsmooth modified symplectic integration scheme that can automatically achieve second-order convergence if possible. This scheme is obtained in three main steps. Firstly, the position coordinates are used to interpolate midpoint approximated integrals, which include velocity jumps implicitly; Secondly, a two-timestep central difference scheme is formulated to approximate the measure differential inclusions; Thirdly, composing two half-condensed two-timestep schemes leads to a one-timestep scheme, which retains the structure of a second-order scheme. Furthermore, while bilateral constraints are satisfied at the position level, unilateral constraints at the velocity level are subject to constraint drifts. To alleviate this issue, a classified cone complementarity problem is formulated to restrain penetration from deteriorating and sticking contacts from fictitious slipping. After time discretization, a set of nonlinear equations with cone complementarity conditions at each timestep is solved by a two-layer iterative computational strategy, where the outer layer is Newton iteration and the inner layer is a primal–dual interior point method. Several numerical examples demonstrate the high accuracy attained by the proposed method. Nonsmooth dynamics of a rigid–flexible robot involving high-speed and high-friction impacts can be simulated faster than real-time.

A Primal-Dual Box-Constrained QP Pressure Poisson Solver With Topology-Aware Geometry-Inspired Aggregation AMG.

We propose a new barrier-based box-constrained convex QP solver based on a primal-dual interior point method to efficiently solve large-scale pressure Poisson problems with non-negative pressure constraints, which commonly arise in liquid animation. The performance of prior active-set-based approaches is limited by the need to repeatedly update the active set. Our solver eliminates this issue by entirely avoiding the use of an active set, which in turn makes the inner problems of our Newton iteration process fully unconstrained. For efficiency, exploiting the solution uniqueness of convex QPs and the fact that the pressure constraints are simple box constraints, we aggressively update solution candidates without performing any step selection procedure (such as line search) and instead directly clamp candidates back to the bounds wherever constraint violations occur. Additionally, to accelerate the inner linear solves, we present a topology-aware geometry-inspired aggregation algebraic multigrid preconditioner and describe in detail several key performance optimizations that we incorporate. We demonstrate the efficacy of our solver in various practical scenarios and show that it often surpasses various alternatives in terms of speed and memory usage.

Fast Quasi-Optimal Power Flow of Flexible DC Traction Power Systems

This paper proposes a quasi-optimal power flow (OPF) algorithm for flexible DC traction power systems (TPSs). Near-optimal OPF solutions can be solved with high computational efficiency by quasi-OPF. Unlike conventional OPF algorithms, quasi-OPF does not utilize mathematical optimization algorithms but adopts a new methodology. First, we adopt a new modeling method and successfully reveal the physical meaning of OPF solutions in flexible DC TPSs. Then, by converting the physical meaning of OPF solutions into mathematical expressions, a simple mapping from the power flow solution to the near-optimal OPF solution is obtained, and the quasi-OPF algorithm is designed based on power flow and this mapping. Since calculating power flow is computationally cheap and the mapping is based on simple arithmetic, the quasi-OPF algorithm can solve OPF with much less execution time, achieving subsecond level calculation and a speed-up of 57 times compared to the primal-dual interior point method. The effectiveness is verified by mathematical proofs and a case study with Beijing Metro Line 13. This study provides insight into the physical meaning of OPF solutions and is a powerful tool for flexible DC TPSs to analyze the effects of coordinated control, design real-time coordinated control strategies, and solve operational problems in planning.