Primal-dual Interior Point Method Research Articles

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 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 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 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 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.

Extension of primal-dual interior point method based on a kernel function for linear fractional problem

Our aim in this work is to extend the primal-dual interior point method based on a kernel function for linear fractional problem. We apply the techniques of kernel function-based interior point methods to solve a standard linear fractional program. By relying on the method of Charnes and Cooper [3], we transform the standard linear fractional problem into a linear program. This transformation will allow us to define the associated linear program and solve it efficiently using an appropriate kernel function. To show the efficiency of our approach, we apply our algorithm on the standard linear fractional programming found in numerical tests in the paper of A. Bennani et al. [4], we introduce the linear programming associated with this problem. We give three interior point conditions on this example, which depend on the dimension of the problem. We give the optimal solution for each linear program and each linear fractional program. We also obtain, using the new algorithm, the optimal solutions for the previous two problems. Moreover, some numerical results are illustrated to show the effectiveness of the method.

A conic programming approach to the wrinkling of pneumatic membranes using convex potentials

This paper presents a novel conic programming approach for modeling the wrinkling of isotropic hyperelastic membranes subject to nonlinear loading and finite strains, employing convex potentials derived from tension field theory. The incompressible neo-Hookean strain energy is recast as a minimization problem over a set of cones, with a semidefinite constraint on the deformed surface metric. To address pneumatic membranes, a linearized volumetric potential is introduced, reestablishing convexity for follower forces, and Boyle's law is expressed as a minimization problem over the exponential cone.The primal–dual interior point method solves the total potential energy minimization problem, with convergence verified using the analytical solution of a sheared planar membrane. The proposed model, applied to examples of increasing complexity, reveals intriguing membrane behaviors rarely discussed in existing literature. The method is shown to be robust, even when handling highly wrinkled membranes, which are challenging to address using direct nonlinear equilibrium analysis or nonlinear interior point methods.Implementation using high-level automatic code generation tools (FEniCS) results in concise, extensible code. The findings point to several avenues for future research, such as exploring complex material models, mesh refinement, dynamics, and parametric design. Additionally, the linearization process implies potential applicability of the methods to nonlinear problems beyond membrane mechanics.

DC-based security constraints formulation: A perspective of primal-dual interior point method

The DC network security constraints have been extensively studied in numerous power system problems, such as optimal power flow (OPF), security-constrained economic dispatch (SCED), and security-constrained unit commitment (SCUC). Power transfer distribution factors (PTDFs) are widely applied in DC network constraints. However, the PTDF matrix is extremely dense, making it difficult to solve security-constraint optimization problems. This paper investigates the computational inefficiency of PTDF-based security constraints from the perspective of matrix sparsity of primal–dual interior point method (IPM), and proposes a matrix transformation to restore sparsity during IPM iterations. It turns out that the transformation method is equivalent to solving the original optimization problem expressed in pure voltage angle. The regular B-θ formulation is also a variant of the proposed transformation. Numerical studies show that sparsity rather than the size of variables and constraints is the key factor impacting the speed of solving convex quadratic problems, i.e., OPF and SCED problems. The proposed transformation significantly outperforms the PTDF-based approach, achieving a 100x reduction in non-zero elements of the coefficient matrix and a 40x speed increase in barrier processes in a 25,000-bus network.

Comparative Analysis of the Particle Swarm Optimization and Primal-Dual Interior-Point Algorithms for Transmission System Volt/VAR Optimization in Rectangular Voltage Coordinates

Optimal power flow (OPF) is one of the most widely studied problems in the field of operations research, as it applies to the optimal and efficient operation of the electric power system. Both the problem formulation and solution techniques have attracted significant research interest over the decades. A wide range of OPF problems have been formulated to cater for the various operational objectives of the power system and are mainly expressed either in polar or rectangular voltage coordinates. Many different solution techniques falling into the two main categories of classical/deterministic optimization and heuristic/non-deterministic optimization techniques have been explored in the literature. This study considers the Volt/VAR optimization (VVO) variant of the OPF problem formulated in rectangular voltage coordinates, which is something of a departure from the majority of the studies, which tend to use the polar coordinate formulation. The heuristic particle swarm optimization (PSO) and the classical primal-dual interior-point method (PDIPM) are applied to the solution of the VVO problem and a comparative analysis of the relative performance of the two algorithms for this problem is presented. Four case studies based on the 6-bus, IEEE 14-bus, 30-bus, and 118-bus test systems are presented. The comparative performance analysis reveals that the two algorithms have complementary strengths, when evaluated on the basis of the solution quality and computational efficiency. Particularly, the PSO algorithm achieves greater power loss minimization, whereas the PDIPM exhibits greater speed of convergence (and, thus, better computational efficiency) relative to the PSO algorithm, particularly for higher-dimensional problems. An additional distinguishing characteristic of the proposed solution is that it incorporates the Newton–Raphson load flow computation, also formulated in rectangular voltage coordinates, which adds to the efficiency and effectiveness of the presented solution method.

Three-dimensional plasticity-based topology optimization with smoothed finite element analysis

AbstractThis paper presents a novel plasticity-based formulation for three-dimensional (3D) topology optimization of continuum structures. The proposed formulation addresses the optimization problem by combining mixed rigid-plastic analysis with density-based topology optimization, resulting in a volume minimization approach. Unlike conventional stress-constrained topology optimization methods that rely on linear elastic structural analysis, our developed formulation focuses on enhancing the loading capacity of the designed structures based on the plastic limit theory, leading to more cost-effective designs. To improve computational efficiency, we employ the smoothed finite element technique in our proposed method, enabling the utilization of linear tetrahedral elements for 3D mesh refinement. Moreover, the final formulation of our developed method can be efficiently solved using the advanced primal–dual interior point method, eliminating the need for a separate nonlinear finite element structural analysis. Numerical examples are presented to demonstrate the effectiveness of the proposed approach in offering enhanced design possibilities for continuum structures.

Minimum path deviation rapid online path planning of unmanned aerial vehicles with avoidance zones constraints

This paper investigates a minimum path deviation online path planning problem for UAV with constrainted avoidance zones. The nonlinear and nonconvex of this problem make the path planning is very time consuming and not acceptable for using onboard. Three new state variables and two control variables are redefined to convert the nonconvex problem into a second-order cone convex programming problem, which can be solved in polynomial time by primal-dual interior-point method, theoretical proof demonstrates the equivalence of the relaxed convex problem and the original problem. We propose a sequential second-order cone programming algorithm in order to get the optimal solution of the convexified problem. Numerical simulations show the effectiveness and less time-consuming of the proposed algorithm, no initial guess required and efficient solving process indicate that the algorithm has potential future for online path planning applications.

Nodal integration-based particle finite element method (N-PFEM) for poro-elastoplastic modelling of saturated soils under large deformation

This paper presents the nodal integration-based particle finite element method (N-PFEM) for poro-elastoplastic analysis of saturated soils subject to large deformation, utilising the generalised Hellinger-Reissner variational principle to reformulate the governing equations for saturated soil dynamics into a min–max optimisation problem. With finite element discretisation and nodal integration over cells, the problem is transformed into a standard second-order cone programming problem, efficiently resolved using the advanced primal–dual interior point method. The N-PFEM method has several advantages, including the use of linear triangular elements without volumetric locking issues, the avoidance of regularisation techniques, and the elimination of tedious variable mapping after remeshing. The numerical model is validated for large deformation analysis of saturated soils with a series of benchmarks against available analytical and numerical solutions, with a case study of the deformation of an embankment considering stone column reinforcement also carried out. This N-PFEM framework offers an effective and efficient simulation approach for the evolutionary behaviour of saturated soils with large deformation in complex geotechnical configurations of practical relevance.

An Optimal Iterative Learning Control Approach for Linear Systems With Nonuniform Trial Lengths Under Input Constraints

In practical applications of iterative learning control (ILC), the repetitive process may end up early by accident during the performance improvement along the trial axis, which yields the nonuniform trial length problem. For such practical systems, input signals are usually constrained because of some certain physical limitations. This article proposes an optimal ILC algorithm for linear time-invariant multiple-input–multiple-output (MIMO) systems with nonuniform trial lengths under input constraints. The optimal ILC framework is specifically modified for the nonuniform trial length problem, where the primal–dual interior point method is introduced to deal with the input constraints. Hence, the constraint handling capability are improved compared with the conventional counterparts for nonuniform trial lengths. Also, the monotonic convergence property of the proposed optimal ILC algorithm is obtained in the sense of mathematical expectation. Finally, the effectiveness of the proposed algorithm is verified on the numerical simulation of a mobile robot.

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.

APPLICATION OF THE QUASI-NEWTON IN INTERIOR POINT METHODS FOR SOLVING THE PREDISPATCH PROBLEM

Brazilian energy matrix is essentially based on hydroelectricity with long transmission lines, allowing the exchange of energy produced in all regions of the country, the increased demand for energy and the search for lower costs, the application of more efficient and robust methods to minimize generation and transmission losses is necessary, since these are functions of generated and transmitted power, respectively. The purpose of this work is to implement primal dual interior point method for the predispatchof a hydroelectric system, with partial replacement of the Newton method with the quasi-Newton method in order to compute the system Jacobian matrix and reduce the computational costs of the iterations arising from approximations of the inverses of the Hessian matrix. This means that, in order to obtain a search direction, only a matrix vector product is necessary, which is much more efficient, for example, than the Newton method, in which a linear system has to be solved at each iteration. Computational results prove the efficiency of the approach used.

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.