Infeasible Interior-point Algorithm Research Articles

Simplified full Nesterov–Todd step infeasible interior-point algorithm for semidefinite optimization based on a kernel function

In this paper, we propose a new complexity analysis of the full Nesterov–Todd step infeasible interior-point algorithm for semidefinite optimization. With a specific feasibility step and the centering step induced by a well-known kernel function, the property of exponential convexity of the kernel function underlying the matrix barrier function is crucial in the analysis and enable us easily to estimate the proximity of iterates to center path. The analysis of the algorithm is simplified and the iteration bound obtained coincides with the currently best iteration bound for infeasible interior-point algorithm.

An Arc Search Infeasible Interior-Point Algorithm for Symmetric Optimization Using a New Wide Neighborhood

In this paper, we propose an infeasible interior-point algorithm for symmetric optimization problems using a new wide neighborhood and estimating the central path by an ellipse. In contrast of most interior-point algorithms for symmetric optimization which search an $\varepsilon$ -optimal solution of the problem in a small neighborhood of the central path, our algorithm searches for optimizers in a new wide neighborhood of the ellipsoidal approximation of central path. The convergence analysis of the algorithm is shown and it is proved that the iteration bound of the algorithm is $O ( r\log\varepsilon^{-1} ) $ which improves the complexity bound of the recent proposed algorithm by Liu et al. (J. Optim. Theory Appl., 2013, https://doi.org/10.1007/s10957-013-0303-y ) for symmetric optimization by the factor $r^{\frac{1}{2}}$ and matches the currently best-known iteration bound for infeasible interior-point methods.

Two computationally efficient polynomial-iteration infeasible interior-point algorithms for linear programming

Since the beginning of the development of interior-point methods, there exists a puzzling gap between the results in theory and the observations in numerical experience, i.e., algorithms with good polynomial bounds are not computationally efficient and algorithms demonstrated efficiency in computation do not have a good or any polynomial bound. Todd raised a question in 2002: “Can we find a theoretically and practically efficient way to reoptimize?” This paper is an effort to close the gap. We propose two arc-search infeasible interior-point algorithms with infeasible central path neighborhood wider than all existing infeasible interior-point algorithms that are proved to be convergent. We show that the first algorithm is polynomial and its simplified version has a complexity bound equal to the best known complexity bound for all (feasible or infeasible) interior-point algorithms. We demonstrate the computational efficiency of the proposed algorithms by testing all Netlib linear programming problems in standard form and comparing the numerical results to those obtained by Mehrotra’s predictor-corrector algorithm and a recently developed more efficient arc-search algorithm (the convergence of these two algorithms is unknown). We conclude that the newly proposed algorithms are not only polynomial but also computationally competitive compared to both Mehrotra’s predictor-corrector algorithm and the efficient arc-search algorithm.

New complexity analysis of full Nesterov-Todd step infeasible interior point method for second-order cone optimization

We present a full Nesterov-Todd (NT) step infeasible interior-point algorithm for second-order cone optimization based on a different way to calculate feasibility direction. In each iteration of the algorithm we use the largest possible barrier parameter value ?. Moreover, each main iteration of the algorithm consists of a feasibility step and a few centering steps. The feasibility step differs from the feasibility step of the other existing methods. We derive the complexity bound which coincides with the best known bound for infeasible interior point methods.

An infeasible interior point method for the monotone SDLCP based on a transformation of the central path

In this paper, we generalize Roos’s infeasible interior point algorithm for linear optimization (LO) (Roos in SIAM J Optim 25(1): 102–114, 2015) to the monotone semidefinite linear complementarity problem, based on Darvay et al.’s technique for LO (Darvay et al. in Period Math Hung 73(1): 27–42, 2016). The symmetrization of the search directions is based on the Nesterov–Todd scaling scheme. The algorithm uses only one full step as feasibility step at each iteration. The derived complexity bound coincides with the best obtained one for infeasible interior-point methods with small updates.

A full NT-step infeasible interior-point algorithm for semidefinite optimization

In this paper, a full Nesterov–Todd-step infeasible interior-point algorithm is presented for semidefinite optimization (SDO) problems. In contrast of some classical interior-point algorithms for SDO problems, this algorithm does not need to perform computationally expensive calculations for centering steps which are needed for classical interior-point methods. The convergence analysis of the algorithm is shown and it is also proved that the complexity bound of the algorithm coincides with the currently best iteration bound obtained by infeasible interior-point algorithms for this class of optimization problems.

An arc-search infeasible interior-point algorithm for horizontal linear complementarity problem in the neighbourhood of the central path

ABSTRACTAn arc-search infeasible interior point algorithm is proposed for solving a horizontal linear complementarity problem. The algorithm produces a sequence of iterates in the negative infinity neighbourhood of the central path and searches the solutions along the ellipses that approximate the whole central path. We study the theoretical convergence properties and also establish polynomial complexity bound for the proposed algorithm. Moreover, our numerical results suggest that the proposed algorithm is very efficient and competitive.

配电网中分布式源-荷协同运行多目标优化的机会约束二阶锥规划模型

在双馈风电机组和光伏发电站等多种类型的分布式电源以及电动汽车接入配电网系统的情况下，考虑配电网电源出力和负荷的随机性，建立了含不确定性DG与EV的配电网优化模型。该模型以高置信水平的最小经济运行成本和最小有功网络损耗为目标，同时加入节点电压期望值最大偏差为惩罚函数，用全牛顿步不可行内点法进行求解。对IEEE33节点系统进行仿真计算并对比其他规划及锥规划的其他算法，证明了本文所提出的优化模型计算法的有效性。 In the case of distributed power supply (DG) and electric vehicle (EV) connected to distribution network system such as double-fed wind turbine and photovoltaic power station, considering the randomness of power distribution and load of distribution network, the optimization model of distribution network with DG and EV is established. In the model, the minimum economic operating cost and the minimum active network loss are taken as goals, and the maximum deviation of the expected value of the node voltage is added as the penalty function. Full Newton step infeasible interior point algorithm is used to solve the model. The simulation of the IEEE 33-node system is carried out and compared with other algorithms of other programming, which proves the validity of the optimization model proposed in this paper.

Modified polynomial-time full-NT-step infeasible interior-point algorithm for symmetric optimisation

Modified polynomial-time full-NT-step infeasible interior-point algorithm for symmetric optimisation

Modified polynomial-time full-NT-step infeasible interior-point algorithm for symmetric optimisation

In this paper, we improve the full Nesterov-Todd (NT)-step infeasible-interior-point algorithm for symmetric optimisation of Gu et al. (2011). Our proposed algorithm differs from Gu et al.'s algorithm such that it has the advantage that no centring steps are needed. In each main iteration of Gu et al.'s algorithm, one feasibility and a few centring steps are needed to get feasible iterates for a pair of perturbed problems, close enough to its central path. In this paper, we perform only one full-NT-step in each main iteration of the algorithm. Despite eliminating the centring steps, the complexity bound of our algorithm matches the best obtained one for symmetric optimisation.

An Arc-search Interior Point Method in the 𝒩−∞ Neighborhood for Symmetric Optimization

In this paper, we propose an arc-search infeasible interior point algorithm for symmetric optimization using the negative infinity neighborhood of the central path. The algorithm searches the optimizers along the ellipses that approximate the entire central path. The convergence of the algorithm is shown for the set of commutative scaling class, which includes some of the most interesting choice of scalings such as xs, sx and the Nesterov-Todd scalings.

CurveLP-A MATLAB implementation of an infeasible interior-point algorithm for linear programming

Mehrotra's algorithm has been the most successful infeasible interior-point algorithm for linear programming since 1990. Most popular interior-point software packages for linear programming are based on Mehrotra's algorithm. This paper proposes an alternative algorithm, arc-search infeasible interior-point algorithm. We will demonstrate, by testing Netlib problems and comparing the test results obtained by arc-search infeasible interior-point algorithm and Mehrotra's algorithm, that the proposed arc-search infeasible interior-point algorithm is a more efficient algorithm than Mehrotra's algorithm.

An $$\ell _{2}$$ ℓ 2 -neighborhood infeasible interior-point algorithm for linear complementarity problems

In this paper, we propose an infeasible interior-point algorithm for linear complementarity problems. In every iteration, the algorithm constructs an ellipse and searches an \(\varepsilon \)-approximate solution of the problem along the ellipsoidal approximation of the central path. The theoretical iteration-complexity of the algorithm is derived and the algorithm is proved to be polynomial with the complexity bound \(O\left(n\log \varepsilon ^{-1}\right)\) which coincides with the best known iteration bound for infeasible interior-point methods.

An infeasible interior-point algorithm for monotone linear complementarity problem based on a specific kernel function

In this paper, a full-Newton step infeasible kernel-based interior-point algorithm for solving monotone linear complementarity problems is proposed. In each iteration, the algorithm computes the new feasibility search directions by using a specific kernel function with the trigonometric barrier term and obtains the centering search directions using the classical kernel function. The algorithm takes only full-Newton steps and therefore no line-searches are needed for generating the new iterations. The convergence of the algorithm is shown and it is proved that the iteration bound of the algorithm coincides with the currently best iteration bound for monotone linear complementarity problem.

An infeasible full-NT step interior point algorithm for CQSCO

In this paper, we propose a full Nesterov-Todd (NT) step infeasible interior-point algorithm for convex quadratic symmetric cone optimization based on Euclidean Jordan algebra. The algorithm uses only one feasibility step in each main iteration. The complexity result coincides with the best-known iteration bound for infeasible interior-point methods.

An infeasible interior-point algorithm for linear optimization over Cartesian symmetric cones

We investigate the effect of making equivalent transformation for the central path from \(x\diamond s=\mu e\) to \(x\diamond s=\mu v\), and we obtain a new class of search directions in both version of feasible and infeasible interior point algorithms for linear optimization over Cartesian symmetric cones. We propose a new modified infeasible interior-point algorithm based on elimination of the centering steps. Moreover, the proposed algorithm finds an \(\varepsilon \)-optimal solution of the underlying problem in polynomial time complexity and its iteration bound coincides with the best known iteration bound for this class of problems.

A modified infeasible interior-point algorithm for P*(κ)-HLCP over symmetric cones

An improved version of infeasible interior-point algorithm for [Formula: see text] horizontal linear complementarity problem over symmetric cones is presented. In the earlier version (optimization, doi: 10.1080/02331934.2015.1062011) each iteration of the algorithm consisted of one so-called feasibility step and some centering steps. The main advantage of the modified version is that it uses only one feasibility step in each iteration and the centering steps not to be required. Furthermore, giving a complexity analysis of the algorithm, we derive the currently best-known iteration bound for infeasible interior-point methods.

A Path-Following Full Newton-Step Infeasible Interior-Point Algorithm for $$P_*(\kappa )$$ P ∗ ( κ ) -HLCPs Based on a Kernel Function

In this paper, we present a path-following infeasible interior-point method for \(P_*(\kappa )\) horizontal linear complementarity problems (\(P_*(\kappa )\)-HLCPs). The algorithm is based on a simple kernel function for finding the search directions and defining the neighborhood of the central path. The algorithm follows the central path related to some perturbations of the original problem, using the so-called feasibility and centering steps, along with only full such steps. Therefore, it has the advantage that the calculation of the step sizes at each iteration is avoided. The complexity result shows that the full-Newton step infeasible interior-point algorithm based on the simple kernel function enjoys the best-known iteration complexity for \(P_*(\kappa )\)-HLCPs.

An infeasible interior-point method with improved centering steps for monotone linear complementarity problems

In this paper, we improve the infeasible full-Newton interior-point algorithm presented by Mansouri et al. [A full-Newton step [Formula: see text] infeasible interior-point algorithm for linear complementarity problems, Nonlinear Anal. Real World Appl. 12 (2011) 545–561] for monotone linear complementarity problems (MLCPs). In each iteration of Mansouri’s algorithm two types of full-Newton steps are used, one feasibility step and some ordinary (centering) steps. In this paper, we use a new search direction, and reduce the number of the centering steps, so that only one centering step is needed. We prove that the complexity of the algorithm is as good as the best-known complexity for infeasible interior-point methods for MLCPs.

A full Nesterov–Todd step infeasible-interior-point algorithm for Cartesian P*(κ) horizontal linear complementarity problems over symmetric cones

Euclidean Jordan algebra is a commonly used tool in designing interior-point algorithms for symmetric cone programs. In this paper, we present a full Nesterov–Todd (NT) step infeasible interior-point algorithm for horizontal linear complementarity problems over Cartesian product of symmetric cones. Since the algorithm uses only full-NT feasibility and centring steps, it has the advantage that no line searches are needed. The complexity result obtained here for symmetric cones using NT directions coincides with the best bound obtained for horizontal linear complementarity problems.