Primal-dual Interior-point Algorithm Research Articles

An efficient primal-dual interior point algorithm for convex quadratic semidefinite optimization

An efficient primal-dual interior point algorithm for convex quadratic semidefinite optimization

IPRSDP: a primal-dual interior-point relaxation algorithm for semidefinite programming

IPRSDP: a primal-dual interior-point relaxation algorithm for semidefinite programming

Complexity of primal-dual interior-point algorithm for linear programming based on a new class of kernel functions

Complexity of primal-dual interior-point algorithm for linear programming based on a new class of kernel functions

A Full-Newton Step Feasible Interior-Point Algorithm for the Special Weighted Linear Complementarity Problems Based on a Kernel Function

In this paper, a new full-Newton step primal-dual interior-point algorithm for solving the special weighted linear complementarity problem is designed and analyzed. The algorithm employs a kernel function with a linear growth term to derive the search direction, and by introducing new technical results and selecting suitable parameters, we prove that the iteration bound of the algorithm is as good as best-known polynomial complexity of interior-point methods. Furthermore, numerical results illustrate the efficiency of the proposed method.

Second-order cone programming for frictional contact mechanics using interior point algorithm

We report experiments of an implementation of a primal–dual interior point algorithm for solving mechanical models of one-sided contact problems with Coulomb friction. The objective is to recover an optimal solution with high accuracy and as quickly as possible. These developments are part of the design of Siconos, an open-source software for modelling and simulating non-smooth dynamical systems. Currently, Siconos uses mainly first-order methods for the numerical solution of these systems. These methods are very robust, but suffer from a linear rate of convergence and are therefore too slow to find accurate solutions in a reasonable time. Our main objective is to apply second-order methods to speed up convergence. In this paper, we focus on solving a relaxed formulation of the initial mechanical model, corresponding to the optimality conditions of a convex quadratic minimization problem with second-order cone constraints. We will present in detail a primal–dual interior point algorithm for solving this type of problem. The main difficulty in implementing this algorithm arises from the fact that at each iteration of the algorithm, a change of variable, called a scaling, has to be performed in order to guarantee the non-singularity of the linear system to be solved, as well as to recover a symmetric system. Although this scaling strategy is very nice from a theoretical point of view, it leads to an enormous deterioration of the conditioning of the linear system when approaching the optimal solution and therefore to all the numerical difficulties that result from it. We will describe in detail the numerical algebra that we have developed in our implementation, in order to overcome these problems of numerical instability. We will also present the solution of the models resulting from the problems of rolling friction, for which the cone of constraints is no longer self-dual like the Lorentz cone.

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.

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.

IPRQP: a primal-dual interior-point relaxation algorithm for convex quadratic programming

IPRQP: a primal-dual interior-point relaxation algorithm for convex quadratic programming

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.

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.

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.

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.

An Efficient Primal-Dual Interior-Point Algorithm for Volt/VAR Optimization in Rectangular Voltage Coordinates

Security and reliability of electrical power supply has become indispensable to modern society, and the system operator is challenged to manage the increasingly complex modern power system in a manner that ensures the expected reliability and security of system operation. In this context, Volt/VAR optimization (VVO) plays a key role in the efficient delivery of power through the transmission system, contributing significantly to the security, reliability, quality and economy of system operation. This article presents the design and implementation of an efficient primal-dual interior-point algorithm for the solution of the VVO problem. The primal-dual interior-point method combines efficient constraint handling by means of logarithmic barrier functions, Lagrangian theory of optimization, and the Newton method to constitute one of the most efficient deterministic algorithms for large-scale nonlinear optimization. The developed algorithm also incorporates the efficient Newton-Raphson load flow computation, which ensures that the solution is feasible with respect to the power flow balance equations at each iteration of the VVO algorithm. Both the VVO and Newton-Raphson load flow problems are formulated in the rectangular coordinates of system voltages. This is a departure from most researchers, who make use of the polar formulation, and adds considerably to the efficiency of the developed algorithm. The efficiency and effectiveness of the developed algorithm has been demonstrated by means of case studies performed on the 6-bus and IEEE 14-bus, 30-bus and 118-bus test systems, which have been selected to analyse the computational efficiency and scalability of the algorithm as it is applied to systems of various sizes. The extensive analyses that have been conducted reveal the developed primal-dual interior-point algorithm’s effectiveness and efficiency, particularly in being able to successfully solve the VVO problem for systems of widely varying sizes without disproportionate increase in computational cost or deterioration in the quality of the results. The developed algorithm exhibits characteristics of fast convergence, high efficiency, and scalability to large-scale problems.

A class of new search directions for full-NT step feasible interior point method in semidefinite optimization

In this paper, based on Darvay et al.’s strategy for linear optimization (LO) (Z. Darvay and P.R. Takács, Optim. Lett. 12 (2018) 1099–1116.), we extend Kheirfam et al.’s feasible primal-dual path-following interior point algorithm for LO (B. Kheirfam and A. Nasrollahi, Asian-Eur. J. Math. 1 (2020) 2050014.) to semidefinite optimization (SDO) problems in order to define a class of new search directions. The algorithm uses only full Nesterov-Todd (NT) step at each iteration to find an ε-approximated solution to SDO. Polynomial complexity of the proposed algorithm is established which is as good as the LO analogue. Finally, we present some numerical results to prove the efficiency of the proposed algorithm.

A NEW KERNEL FUNCTION GENERATING THE BEST COMPLEXITY ANALYSIS FOR MONOTONE SDLCP

In this article, we propose a new class of search directions based on new kernel function to solve the monotone semidefinite linear complementarity problem by primal-dual interior point algorithm. We show that this algorithm based on this function benefits from the best polynomial complexity, namely $ O(\sqrt{n}(\log n)^{2}\log\dfrac{n}{\epsilon})$. The implementation of the algorithm showed a great improvement concerning the time and the number of iterations.

Primal–dual interior-point algorithm for electricity cost minimization in a prosumer-based smart grid environment: A convex optimization approach

This study proposes a primal–dual interior-point algorithm, which is used by the prosumers to efficiently manage the operation of their generating and consuming devices, to maximize their own profits while contributing to the stability of the overall electricity grid. The energy management problem investigated in this study is formulated as a convex optimization problem and the primal–dual interior-point algorithm is leveraged to acquire the optimal solution which minimizes the energy cost for prosumers in a smart grid environment. The proposed algorithm considers the cost of energy generation by each available device or the charge of a utility company, as well as the predicted consumption and production levels of the facility, and develops an operating plan that maximizes the prosumer’s profitability. The cost function for energy generation is convex, as are the constraints that must be satisfied. The primal–dual interior-point algorithm is an efficient algorithm and has attracted attention in the optimization community. The proposed algorithm is evaluated on three prosumer profiles that have different generation and storage characteristics. A variable pricing structure for power supplied or bought by the utility company is considered in this study and a scenario in which prosumers are not allowed to sell to the utility company. It is evident from the simulation results of the MATLAB environment that scenario 1 generates the minimum cost operation profiles compared to scenario 2. The effectiveness and the robustness of the proposed algorithm are demonstrated in the optimal solution obtained in solving the energy management problem.

A wide neighbourhood primal-dual second-order corrector interior point algorithm for semidefinite optimization

In this paper, we proposed a new primal-dual second-order corrector interior-point algorithm for semidefinite optimization. The algorithm is based on Darvay–Takács neighbourhood of the central path. In the new algorithm, the search directions are determined by the Darvay–Takács's direction and a second-order corrector direction in each iteration. The iteration complexity bound is for the Nesterov–Todd scaling direction, which coincides with the best-known complexity results for semidefinite optimization. Finally, numerical experiments show that the proposed algorithm is promising.

A dynamic optimization method for power distribution network operation with high ratio photovoltaics

With the high ratio distributed photovoltaics (PVs) penetration to the distribution network (DN), the bearing capacity, integration capacity, consuming capacity and controlling capacity of DN for PVs are all facing great challenges. Enhancing the flexible regulation and operation optimization capabilities of DN have become key factors to satisfy these new challenges. For the operation optimization problem of high ratio PVs penetration, a novel quasi-equal curtailment ratio (QCR) constraint of PVs is proposed in the dynamic optimization method for DN operation. Firstly, the uncertainty of high ratio PVs and their active power deviations are discussed. And a dynamic optimization framework of PVs based on optimal power flow (OPF) has been proposed. Secondly, the optimization model for high ratio PVs is formulated with multi-objective of maximum PVs output, minimum voltage deviation and minimum line loss, under the equality and inequality constraints of power flow, PVs power generation limit, QCR of PVs and so on. Then the multi-objective optimization problem (MOP) is transformed into a single-objective optimization problem (SOP) by weighting coefficients, and solved by sequential quadratic programming (SQP) with trust-region (TR) searching algorithm. Finally, the proposed method is tested, verified and compared with the primal dual interior point (PDIP) and traditional SQP algorithms in IEEE 33-bus, PG&E 69-bus, 292-bus and 1180-bus test cases. The experimental results show the rapidity and robustness of the proposed method.

A Semidistributed Interior Point Algorithm for Optimal Coordination of Automated Vehicles at Intersections

In this article, we consider the optimal coordination of automated vehicles at intersections under fixed crossing orders. We formulate the problem using direct optimal control and exploit the structure to construct a semidistributed primal-dual interior-point algorithm to solve it by parallelizing most of the computations. Differently from standard distributed optimization algorithms, where the optimization problem is split, in our approach we split the linear algebra steps, such that the algorithm takes the same steps as a fully centralized one, while still performing computations in a distributed fashion. We analyze the communication requirements of the algorithm, and propose an approximation scheme which can significantly reduce the data exchange. We demonstrate the effectiveness of the algorithm in hard but realistic scenarios, which show that the approximation leads to reductions in communicated data of almost 99% of the exact formulation, at the expense of less than 1% suboptimality.

A new long-step interior point algorithm for linear programming based on the algebraic equivalent transformation

In this paper, we investigate a new primal-dual long-step interior point algorithm for linear optimization. Based on the step size, interior point algorithms can be divided into two main groups, short-step, and long-step methods. In practice, long-step variants perform better, but usually, a better theoretical complexity can be achieved for the short-step methods. One of the exceptions is the large-update algorithm of Ai and Zhang. The new wide neighborhood and the main characteristics of the presented algorithm are based on their approach. In addition, we use the algebraic equivalent transformation technique of Darvay to determine new modified search directions for our method. We show that the new long-step algorithm is convergent and has the best known iteration complexity of short-step variants. We present our numerical results and compare the performance of our algorithm with two previously introduced Ai-Zhang type interior point algorithms on a set of linear programming test problems from the Netlib library.