Predictor-corrector Interior Point Method 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 new wide-neighborhood predictor-corrector interior-point method for semidefinite optimization

A new wide-neighborhood predictor-corrector interior-point method for semidefinite optimization

A Novel Low Complexity Faster-Than-Nyquist Signaling Detector Based on the Primal-Dual Predictor-Corrector Interior Point Method

Faster-than-Nyquist signaling (FTNS) is a promising non-orthogonal physical layer transmission technique to improve the spectral efficiency of next-generation communication systems but at the expense of increased transciever complexity to remove the artificially introduced intersymbol-interference (ISI). In this paper, we investigate the detection problem of FTNS and propose a low-complexity sub-optimal detection algorithm, stemming in operations research, that is based on convex relaxation, primal-dual predictor-corrector interior point method, and quantization. The proposed convex quadratic relax-and-quantize sequence estimation (CQRAQSE) algorithm is suitable for low rates FTNS and demonstrates an excellent trade-off between the detection performance and computational effort.

An iteration predictor–corrector interior-point method with a new one-norm neighbourhood for symmetric cone optimization

In this paper, we propose a predictor–corrector interior-point method for symmetric cone optimization. The proposed algorithm is based on a new one-norm neighbourhood, which is an even wider neighbourhood than a given negative infinity neighbourhood. The convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the xs and sx directions. We show that the algorithm has iteration complexity bound which is better than that of the usual wide neighbourhood algorithm . To our knowledge, these are the best complexity results obtained so far for the solution of symmetric cone optimization. We prove that beside the predictor steps, each corrector step also reduces the duality gap by a rate of . Finally, numerical experiments show that the proposed algorithm is efficient and reliable.

A New Predictor-corrector Infeasible Interior-point Algorithm for Linear Optimization in aWide Neighborhood

In this paper, we propose a Mizuno-Todd-Ye type predictor-corrector infeasible interior-point method for linear optimization based on a wide neighborhood of the central path. According to Ai-Zhang’s original idea, we use two directions of distinct and orthogonal corresponding to the negative and positive parts of the right side vector of the centering equation of the central path. In the predictor stage, the step size along the corresponded infeasible directions to the negative part is chosen. In the corrector stage by modifying the positive directions system a full-Newton step is removed. We show that, in addition to the predictor step, our method reduces the duality gap in the corrector step and this can be a prominent feature of our method. We prove that the iteration complexity of the new algorithm is 𝒪(n log ɛ−1), which coincides with the best known complexity result for infeasible interior-point methods, where ɛ > 0 is the required precision. Due to the positive direction new system, we improve the theoretical complexity bound for this kind of infeasible interior-point method [1] by a factor of n . Numerical results are also provided to demonstrate the performance of the proposed algorithm.

A wide neighborhood predictor-infeasible corrector interior-point algorithm for linear optimization

In this paper, we propose a theoretical framework of a predictor-corrector interior-point method for linear optimization based on the one-norm wide neighborhood of the central path, focusing on infeasible corrector steps. Here, we call the predictor-infeasible corrector algorithm. At each iteration, the proposed algorithm computes an infeasible corrector step in addition to the Ai-Zhang search directions and considers the Newton direction as a linear combination of these directions. We represent the complexity analysis of the algorithm and conclude that its iteration bound is $${\mathcal {O}}(n\log \varepsilon ^{-1})$$ . To our knowledge, this is the best complexity result up to now for infeasible interior-point methods based on these kinds of search directions. The complexity bound obtained here is the same as small neighborhood infeasible interior point algorithms.

Optimal Placement and Dispatch of LV-SVCs to Improve Distribution Circuit Performance

In this paper, a two-stage optimization framework is proposed for optimal placement and centralized real-time control of low-voltage static var compensators (LV-SVCs) in unbalanced distribution circuits to achieve feeder level benefits namely voltage regulation and energy loss minimization. The number, locations, and the optimal real-time reactive power injections from the LV-SVCs are determined from a proposed three-phase unbalanced ac optimal power flow (ACOPF). The ACOPF is formulated as a multi-objective optimization with operating constraints written in rectangular coordinates. The resulting nonlinear nonconvex problem is solved using the predictor-corrector primal-dual interior point method. The proposed approach is scalable for application to large distribution circuits, treats the constraints of physical space limitations effectively, and can take advantage of communication capabilities of LV-SVCs for centralized real-time control. The benefits of the proposed real-time control of LV-SVCs as compared to their operation with local autonomous voltage-based distributed control is demonstrated. The results show that, the proposed approach addresses the LV-SVC placement and control problem effectively to minimize energy losses while maintaining the voltage regulation.

Optimized choice of parameters in interior-point methods for linear programming

In this work, we propose a predictor–corrector interior point method for linear programming in a primal–dual context, where the next iterate is chosen by the minimization of a polynomial merit function of three variables: the first is the steplength, the second defines the central path and the third models the weight of a corrector direction. The merit function minimization is performed by restricting it to constraints defined by a neighborhood of the central path that allows wide steps. In this framework, we combine different directions, such as the predictor, the corrector and the centering directions, with the aim of producing a better one. The proposed method generalizes most of predictor–corrector interior point methods, depending on the choice of the variables described above. Convergence analysis of the method is carried out, considering an initial point that has a good practical performance, which results in Q-linear convergence of the iterates with polynomial complexity. Numerical experiments using the Netlib test set are made, which show that this approach is competitive when compared to well established solvers, such as PCx.

An Efficient Parallel Sequential Approach for Transient Stability Emergency Control of Large-Scale Power System

Generator-tripping and load-shedding are important emergency control measures to maintain the transient stability of the power system. In this paper, emergency control is modeled as a large-scale optimal control problem. An efficient parallel sequential approach, which consists of preprocessing layer, simulation layer, and optimization layer, is proposed to solve the problem. In the preprocessing layer, power system partitioning is performed to construct a Jacobian matrix of double-layered bordered block diagonal (BBD) structure, control variable selection and initialization are designed to reduce computation cost and improve convergence. In the simulation layer, the differential algebraic equations are solved in parallel with the sensitivity analysis by reusing the LU-factorizations. Based on the BBD structure of Jacobian matrix, a parallel block algorithm is proposed to further accelerate computation speed. In the optimization layer, the resulting nonlinear programming problem is efficiently solved by predictor–corrector interior point method. The efficiency and practicality of the proposed approach are verified by numerical experiments on three test systems.

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 primal-dual predictor-corrector interior-point method for symmetric cone optimization

We present a primal-dual predictor-corrector interior-point method for symmetric cone optimization. The proposed algorithm is based on the Nesterov-Todd search directions and a wide neighborhood, which is an even wider neighborhood than a given negative infinity neighborhood. At each iteration, the method computes two corrector directions in addition to the Ai and Zhang directions (SIAM J. Optim. 16, 400–417, 2005), in order to improve performance. Moreover, we derive the complexity bound of the wide neighborhood predictor-corrector interior-point method for symmetric cone optimization that coincides with the currently best known theoretical complexity bounds for the short step algorithm. Finally, some numerical experiments are provided to reveal the effectiveness of the proposed method.

An Efficient Optimal Control Method for Open-Loop Transient Stability Emergency Control

With the expansion of modern power systems, the stability issues become more and more prominent. Transient stability emergency control is usually designed in open-loop schemes and applies proper actions to avoid system collapse when transient stability cannot be guaranteed in serious contingencies. Taking transient stability and economic efficiency of power system into consideration, the emergency control problem can be modeled as an optimal control problem, which is computational expensive. In this paper, an optimal control method with constraint aggregation is proposed to reduce computational complexity. The yield nonlinear problem is a fairly small-scale optimization problem which can be efficiently solved by predictor–corrector interior point method. The adjoint sensitivity analysis (ASA) is employed to evaluate the first-order derivative while Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is used to obtain the second-order derivative. Besides, very dishonest Newton (VDHN) method and reusage of LU factorization results are explored to accelerate the forward and backward integration phase of ASA, respectively. The proposed approach is tested on four cases with different scales, and shows its potential in computational efficiency.

Ip2go: An Interior Point Source Code Generator for Solving Linear-Quadratic Optimal Control Problems

Model predictive control applications have broadened from industrial plants with slow system dynamics to the implementation of control algorithms on low-cost hardware. Ip2go has been developed with the aim of making model predictive control readily accessible on embedded systems. It generates efficient, problem-tailored solvers for optimal control problems. The code generator implements Mehrotra’s predictor-corrector interior-point method with Riccati recursions. It can be applied to quadratic programming problems with linear discrete-time system dynamics and polytopic inequality constraints. Both hard and soft constraints can be implemented. In this paper the features of ip2go, the underlying algorithm, the benefits of code generation and the generation process are set out. The comparison of ip2go-generated solvers to state-of-the-art solvers shows that ip2go can compete in terms of efficiency and code-size. The open and modular design allows all users to amend the generator. The Matlab and based and LGPL licensed C-code generator ip2go is available under https://github.com/fabiankrank/ip2go.

A primal–dual predictor–corrector interior-point method for symmetric cone programming with O(√r log ϵ−1) iteration complexity

ABSTRACTIn this paper,we propose a new predictor–corrector interior-point method for symmetric cone programming. This algorithm is based on a wide neighbourhood and the Nesterov–Todd direction. We prove that, besides the predictor steps, each corrector step also reduces the duality gap by a rate of , where r is the rank of the associated Euclidean Jordan algebras. In particular, the complexity bound is , where is a given tolerance. To our knowledge, this is the best complexity result obtained so far for interior-point methods with a wide neighbourhood over symmetric cones. The numerical results show that the proposed algorithm is effective.

A variation on the interior point method for linear programming using the continued iteration

In this paper, we present a proposal for a variation of the predictor–corrector interior point method with multiple centrality corrections. The new method uses the continued iteration to compute a new search direction for the predictor corrector method. The purpose of incorporating the continued iteration is to reduce the overall computational cost required to solve a linear programming problem. The computational results constitute evidence of the improvement obtained with the use of this technique combined with the interior point method.

A New Primal–Dual Predictor–Corrector Interior-Point Method for Linear Programming Based on a Wide Neighbourhood

In this paper, we propose a new predictor---corrector interior-point algorithm for linear programming based on a wide neighbourhood. In each iteration, the algorithm computes the Ai-Zhang's predictor direction (SIAM J. Optim. 16(2):400---417, 2005) and a new corrector direction, in an attempt to improve its performance. We drive that the duality gap reduces in both predictor and corrector steps. Moreover, we also prove that the complexity of the algorithm coincides with the best iteration bound for small neighbourhood algorithms. Finally, some numerical experiments are provided which reveal capability and effectiveness of the proposed method.

A New Wide Neighborhood Primal-Dual Predictor-Corrector Interior-Point Method for Linear Programming

ABSTRACTIn this article, we present a new predictor-corrector interior-point method, based on a wide neighborhood of the central path, for linear programming problems. In the corrector step ofthe proposed method, we derive the step size and corrector directions that guarantee that each iteration lies in a wide neighborhood of the central path. We also prove that the algorithm has iteration complexity, which coincides with the best bound derived for linear programming problems.

Predictor-corrector interior point method for contact analysis models with multi-point constraints

Predictor-corrector interior point method for contact analysis models with multi-point constraints

A new predictor–corrector method for optimal power flow

A predictor–corrector interior-point method is developed in order to deal with the AC active and reactive optimal power flow problem. Voltage rectangular coordinates are adopted instead of polar ones, since they allow nonlinear corrections for the primal and dual feasibility conditions and not only for the complementary constraints as in the traditional nonlinear programming methods. A new heuristic is proposed to handle voltage magnitude constraints. Computational experiments for IEEE test systems and a real Brazilian system are presented and show the advantages of the proposed approach.

Computing a hybrid preconditioner approach to solve the linear systems arising from interior point methods for linear programming using the conjugate gradient method

(ProQuest: ... denotes formulae omitted.)1. IntroductionThe development of sophisticated software to solve linear optimization problems by interior point methods has started since the early works on this subject. There are three main research lines aimed at improving the efficiency of such methods for solving large-scale problems: reduction of the total number of iterations, techniques to obtain a fast iteration and specific methods for particular classes of problems.This work addresses the second one. Iterative methods are used to solve the linear systems of equations which are the most expensive step at each iteration of interior point methods. Since such systems are very ill-conditioned near a solution, the design of specially tailored preconditioners is an important implementation issue. On the other hand, since the early linear systems do not present the same features, it is advisable to adopt hybrid preconditioners that begin as a generic preconditioner and adapt during the course of the iteration, becoming ever more specialized as convergence takes place (Bocanegra et al., 2007).During the initial iterations a controlled Cholesky factorization is adopted (Campos & Birkett, 1998). Its major advantage is the control parameter that allows the preconditioner to vary all way from a diagonal preconditioner to the full Cholesky factorization, if desired. At the onset of convergence, a splitting preconditioner is used (Oliveira & Sorensen, 2005). Its major advantage is its excellent behavior near a solution of the linear program. However, this desirable feature has a price: the preconditioner could be very expensive to compute. A careful implementation must be performed in order to achieve competitive numerical results regarding both: speed and robustness. An effective implementation of the splitting preconditioner depends crucially upon finding a suitable set of linearly independent columns to form a nonsingular matrix, to be factored, from the constraint matrix.There are several techniques for finding such a set of columns such as the delayed update form for the LU factorization, the symbolic dependent columns, the symbolic independent columns, the combination of symbolic dependent and independent columns and strongly connected components. Some are well known and already applied in other contexts (Coleman & Pothen, 1987; Duff & Reid, 1986; El-Bakry et al., 1994). Others were developed to compute the splitting preconditioner (Oliveira, 1997; Oliveira & Sorensen, 2005). Among the techniques used is the study of the nonzero structure of the constraint matrix to speed up the numerical factorization, such as using key columns, symbolically dependent and independent columns, finding strongly connected components (Oliveira & Sorensen, 2005). Other implementation issues, include ways for changing preconditioners, are also discussed in (Ghidini et al., 2012; Velazco et al., 2010).The choice of the controlled factorization is justified due to the possibility of computing an inexpensive preconditioner in the initial interior point iterations and, as the linear systems become more ill conditioned, the controlled preconditioner can be improved with just the change of a parameter value. Numerical experiments illustrating the effectiveness of such strategies in order to solve large scale linear programming problems are presented in Bocanegra et al. (2007) and Velazco et al. (2010).This work is organized as follows: Section 2 presents the predictor-corrector interior point method, defines its search directions and explains how to solve the resulting linear systems of equations. The controlled Cholesky factorization and splitting preconditioners are discussed in this section. Sections 3 and 4 study several techniques in order to achieve an efficient implementation of the splitting preconditioner. In Section 5 the numerical experiments are shown and discussed. Conclusions follow in Section 6. …