Predictor-corrector Interior Point Algorithm Research Articles

Large-step predictor-corrector interior point method for sufficient linear complementarity problems based on the algebraic equivalent transformation

We introduce a new predictor-corrector interior-point algorithm for solving P⁎(κ)-linear complementarity problems which works in a wide neighbourhood of the central path. We use the technique of algebraic equivalent transformation of the centering equations of the central path system. In this technique, we apply the function φ(t)=t in order to obtain the new search directions. We define the new wide neighbourhood Dφ. In this way, we obtain the first interior-point method, where not only the central path system is transformed, but the definition of the neighbourhood is also modified taking into consideration the algebraic equivalent transformation technique. This gives a new direction in the research of interior-point algorithms. We prove that the interior-point method has O((1+κ)nlog⁡((x0)Ts0ϵ)) iteration complexity. Furthermore, we show the efficiency of the proposed predictor-corrector algorithm by providing numerical results. To our best knowledge, this is the first predictor-corrector interior-point algorithm which works in the Dφ neighbourhood using φ(t)=t.

A predictor-corrector interior-point algorithm for $ P_{*}(\kappa) $-weighted linear complementarity problems

<abstract><p>In this paper, we present a predictor-corrector interior-point algorithm for $ P_{*}(\kappa) $-weighted linear complementarity problems. Based on the kernel function $ \varphi(t) = \sqrt{t} $, the search direction of the algorithm is obtained. By choosing appropriate parameters, we prove that the algorithm is feasible and convergent. It is shown that the proposed algorithm has polynomial iteration complexity. Numerical results illustrate the effectiveness of the algorithm.</p></abstract>

New Predictor–Corrector Algorithm for Symmetric Cone Horizontal Linear Complementarity Problems

We propose a new predictor–corrector interior-point algorithm for solving Cartesian symmetric cone horizontal linear complementarity problems, which is not based on a usual barrier function. We generalize the predictor–corrector algorithm introduced in Darvay et al. (SIAM J Optim 30:2628–2658, 2020) to horizontal linear complementarity problems on a Cartesian product of symmetric cones. We apply the algebraically equivalent transformation technique proposed by Darvay (Adv Model Optim 5:51–92, 2003), and we use the difference of the identity and the square root function to determine the new search directions. In each iteration, the proposed algorithm performs one predictor and one corrector step. We prove that the predictor–corrector interior-point algorithm has the same complexity bound as the best known interior-point methods for solving these types of problems. Furthermore, we provide a condition related to the proximity and update parameters for which the introduced predictor-corrector algorithm is well defined.

A Mehrotra-type second-order predictor–corrector algorithm for nonlinear complementarity problems over symmetric cones

This paper presents a Mehrotra-type predictor–corrector algorithm for monotone nonlinear complementarity problems over symmetric cones. The proposed algorithm is a second-order predictor–corrector interior point algorithm. Based on a one-norm wide neighbourhood, the iteration complexity of the algorithm is estimated and some numerical results are provided. The numerical results show that the algorithm is efficient and reliable.

Predictor-Corrector Interior-Point Algorithm for the General Linear Complementarity Problem

Abstract We study a predictor-corrector interior-point algorithm for solving general linear complementarity problems from the implementation point of view. We analyze the method proposed by Illés, Nagy and Terlaky [1] that extends the algorithm published by Potra and Liu [2] to general linear complementarity problems. A new method for determining the step size of the corrector direction is presented. Using the code implemented in the C++ programming language, we can solve large-scale problems based on sufficient matrices.

Predictor-corrector interior-point algorithm for [formula omitted]-linear complementarity problems based on a new type of algebraic equivalent transformation technique

We propose a new predictor-corrector (PC) interior-point algorithm (IPA) for solving linear complementarity problem (LCP) with P*(κ)-matrices. The introduced IPA uses a new type of algebraic equivalent transformation (AET) on the centering equations of the system defining the central path. The new technique was introduced by Darvay and Takács (2018) for linear optimization. The search direction discussed in this paper can be derived from positive-asymptotic kernel function using the function φ(t)=t2 in the new type of AET. We prove that the IPA has O((1+4κ)nlog3nμ04ϵ) iteration complexity, where κ is an upper bound of the handicap of the input matrix. To the best of our knowledge, this is the first PC IPA for P*(κ)-LCPs which is based on this search direction.

An predictor–corrector interior-point algorithm for semidefinite optimization based on a wide neighbourhood

In this paper, we propose a new predictor–corrector interior-point algorithm for semidefinite optimization based on a wide neighbourhood of the central path. We show that, in addition to the predictor step, each corrector step decreases the duality gap as well. We also prove that the iteration complexity of the proposed algorithm coincides with the best iteration bound for small neighbourhood algorithms that use the Nesterov–Todd direction. Finally, some numerical results are provided as well.

A Mehrotra type predictor-corrector interior-point algorithm for linear programming

In this paper, we analyze a feasible predictor-corrector linear programming variant of Mehrotra's algorithm. The analysis is done in the negative infinity neighborhood of the central path. We demonstrate the theoretical efficiency of this algorithm by showing its polynomial complexity. The complexity result establishes an improvement of factor $ n^3 $ in the theoretical complexity of an earlier presented variant in [2], which is a huge improvement. We examine the performance of our algorithm by comparing its implementation results to solve some NETLIB problems with the algorithm presented in [2].

A long-step feasible predictor–corrector interior-point algorithm for symmetric cone optimization

ABSTRACTIn this paper, we present a feasible predictor–corrector interior-point method for symmetric cone optimization problem in the large neighbourhood of the central path. The method is generalization of Ai-Zhang's predictor–corrector algorithm to the symmetric cone optimization problem. Starting with a feasible point in given large neighbourhood of the central path, the algorithm still terminates in at most iterations. This matches the best known iteration bound that is usually achieved by short-step methods, thereby, closing the complexity gap between long- and short-step interior-point methods for symmetric cone optimization. The preliminary numerical results on a selected set of NETLIB problems show advantage of the method in comparison with the version of the algorithm that is not based on the predictor–corrector scheme.

A Wide-Neighborhood Predictor-Corrector Interior-Point Algorithm for Linear Complementarity Problems

In this paper, a wide-neighborhood predictor-corrector feasible interior-point algorithm for linear complementarity problems is proposed. The algorithm is based on using the classical affine scaling direction as a part in a corrector step, not in a predictor step. The convergence analysis of the algorithm is shown, and it is proved that the algorithm has the polynomial complexity $$O\left(\sqrt{n}\log \varepsilon ^{-1}\right)$$ which coincides with the best known iteration bound for this class of mathematical problems. The numerical results indicate the efficiency of the algorithm.

A predictor-corrector interior-point algorithm for monotone variational inequality problems

Mehrolra’s recent suggestion of a predictor-corrector variant of primal-dual interior-point method for linear programming is currently the interior-point method of choice for linear programming. In this work the authors give a predictor-corrector interior-point algorithm for monotone variational inequality problems. The algorithm was proved to be equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require the initial iteration to be feasible. Numerical results of experiments are presented.

A Wide Neighborhood Second-order Predictor-corrector Interior-point Algorithm for Semidefinite Optimization with Modified Corrector Directions

A Wide Neighborhood Second-order Predictor-corrector Interior-point Algorithm for Semidefinite Optimization with Modified Corrector Directions

Power System State and Transmission Line Conductor Temperature Estimation

This paper presents a state estimation technique for three-phase power systems where not only bus voltage phasors but also the temperature of transmission line conductors is considered as states. Transmission line admittance parameters depending on line conductor and ambient temperature are approximated from the pre-computed data based on polynomial interpolations. Weather environment and heat-balance equations have been also included in the measurement functions in order to estimate the temperature of conductors. The technique for segmenting transmission lines is also applied for handling temperature variations along the lines. The estimation problem is then formulated as a constrained nonlinear optimization based on the weighted least-squares criterion. A solution is obtained by applying a predictor–corrector interior point algorithm. Simulation results on some three-phase power systems indicate that the proposed method yields estimations with a better accuracy.

A Mehrotra predictor-corrector interior-point algorithm for semidefinite optimization

This paper proposes a second-order Mehrotra-type predictor-corrector feasible interior-point algorithm for semidefinite optimization problems. In each iteration, the algorithm computes the Newton search directions through a new form of combination of the predictor and corrector directions. Using the Ai-Zhang wide neighborhood for linear complementarity problems, it is shown that the complexity bound of the algorithm is $O(\sqrt{n}\log \varepsilon^{-1})$ for the Nesterov-Todd search direction and $O({n}\log \varepsilon^{-1})$ for the Helmberg-Kojima-Monteiro search directions.

A superlinearly convergent wide-neighborhood predictor–corrector interior-point algorithm for linear programming

In this paper we propose a new predictor-corrector algorithm with superlinear convergence in a wide neighborhood for linear programming problems. We let the centering parameter in a predictor step is chosen adaptively, which is different from other algorithms in the same wide neighborhood. The choice is a key for the local convergence of the new algorithm. In addition, we use the classical affine scaling direction as a part in a corrector step, not in a predictor step, which contributes to the complexity result. We prove that the new algorithm has a polynomial complexity of \(O(\sqrt{n}L)\), and the duality gap sequence is superlinearly convergent to zero, under the assumption that the iterate points sequence is convergent. Finally, numerical tests indicate its effectiveness.

Solving the logit-based stochastic user equilibrium problem with elastic demand based on the extended traffic network model

This paper proposes a novel extended traffic network model to solve the logit-based stochastic user equilibrium (SUE) problem with elastic demand. In this model, an extended traffic network is established by properly adding dummy nodes and links to the original traffic network. Based on the extended traffic network, the logit-based SUE problem with elastic demand is transformed to the SUE problem with fixed demand. Such problem is then further converted to a linearly constrained convex programming and addressed by a predictor–corrector interior point algorithm with polynomial complexity. A numerical example is provided to compare the proposed model with the method of successive averages (MSA). The numerical results indicate that the proposed model is more efficient and has a better convergence than the MSA.

Polynomial convergence of Mehrotra-type predictor–corrector algorithm for the Cartesian P∗(κ)-LCP over symmetric cones

We propose a new infeasible variant of Mehrotra-type predictor–corrector interior-point algorithms which can be regarded as an extension of Salahi et al. (SIAM J. Optim. 18: 1377-1397, 2007) for linear programming to the Cartesian linear complementarity problems over symmetric cone. That algorithm incorporates a safeguard in Mehrotra’s original algorithm, which allows us to prove polynomial iteration complexity. In our algorithm, the safeguard strategy is implemented by bounding the central parameter, which is different from the algorithm of Salahi et al. We modify the maximum step size in the affine scaling step and the Newton system in the corrector step, and extend the algorithm to symmetric cones using the machinery of Euclidean Jordan algebras. We show that the iteration-complexity bound of the proposed algorithm is , where is the rank of the associated Euclidean Jordan algebras, is the handicap of the problem, is the condition number of matrix and is the required precision. Some numerical tests are reported to illustrate our theoretical results.

A predictor-corrector interior-point algorithm for P ∗ ( κ ) $P_{*}(\kappa )$ -horizontal linear complementarity problem

We present a predictor-corrector path-following interior-point algorithm for P ? ( ? ) $P_*(\kappa )$ horizontal linear complementarity problem based on new search directions. In each iteration, the algorithm performs two kinds of steps: a predictor (damped Newton) step and a corrector (full Newton) step. The full Newton-step is generated from an algebraic reformulation of the centering equation, which defines the central path and seeks directions in a small neighborhood of the central path. While the damped Newton step is used to move in the direction of optimal solution and reduce the duality gap. We derive the complexity for the algorithm, which coincides with the best known iteration bound for P ? ( ? ) $P_*(\kappa )$ -horizontal linear complementarity problems.

New complexity analysis of a Mehrotra-type predictor–corrector algorithm for semidefinite programming

In this paper, we propose a new Mehrotra-type predictor–corrector interior-point algorithm for semidefinite programming. This algorithm is an extension of the variant of Mehrotra-type algorithm that was proposed by Salahi et al. [On Mehrotra-type predictor–corrector algorithms, SIAM J. Optim. 18 (2007), pp. 1377–1397] for linear programming problems. We modify the step sizes lightly in the predictor step of Koulaei and Terlaky [On the complexity analysis of a Mehrotra-type primal–dual feasible algorithm for semidefinite optimization, Optim. Methods Softw. 25 (2010), pp. 467–485]. In such a way, we obtain O(nlog(Tr(X 0 S 0)/ϵ)) iteration complexity of the algorithm, where (X 0, y 0, S 0) is the initial feasible point and ϵ is the required precision.

Power system with multi-type FACTS devices states estimation based on predictor–corrector interior point algorithm

This paper addresses a state estimation problem of power systems incorporating various flexible alternating current transmission system (FACTS) devices. Static var compensators, thyristor controlled series compensators, and unified power flow controllers are considered since they represent various kinds of FACTS devices. By considering constraints of system buses and devices, the estimation problem can be formulated as a nonlinear optimization with constraints. An algorithm based on predictor–corrector interior point is applied to solve the problem. The IEEE 14- and 118-bus systems modified by incorporating FACTS devices are used as test systems to verify and demonstrate the effectiveness of the proposed algorithm. Numerical results indicate that the proposed algorithm performs better than the primal dual interior point based method even the presence of bad measurement data.