A NEW FRAMEWORK OF PRIMAL-DUAL INFEASIBLE INTERIOR-POINT METHOD FOR LINEAR PROGRAMMING

On the basis of the formulations of the logarithmic barrier function and the idea of following the path of minimizers for the logarithmic barrier family of problems the so called "centralpath" for linear programming, we propose a new framework of primal-dual infeasible interiorpoint method for linear programming problems. Without the strict convexity of the logarithmic barrier function, we get the following results: (a) if the homotopy parameterμcan not reach to zero,then the feasible set of these programming problems is empty; (b) if the strictly feasible set is nonempty and the solution set is bounded, then for any initial point x, we can obtain a solution of the problems by this method; (c) if the strictly feasible set is nonempty and the solution set is unbounded, then for any initial point x, we can obtain a (?)-solution; and(d) if the strictly feasible set is nonempty and the solution set is empty, then we can get the curve x(μ), which towards to the generalized solutions.

A Primal-Dual Interior-Point Method for Optimal Grasping Manipulation of Multi-fingered Hand-Arm Robots

In this paper,we consider an optimization problem of the grasping manipulation of multi-fingered hand-arm robots.We first formulate an optimization model for the problem,based on the dynamic equations of the object and the friction constraints.Then,we reformulate the model as a convex quadratic programming over circular cones.Moreover,we propose a primal-dual interior-point algorithm based on the kernel function to solve this convex quadratic programming over circular cones.We derive both the convergence of the algorithm and the iteration bounds for largeand small-update methods,respectively.Finally,we carry out the numerical tests of 180◦and 90◦manipulations of the hand-arm robot to demonstrate the effectiveness of the proposed algorithm.

Kernel Function-Based Primal-Dual Interior-Point Methods for Symmetric Cones Optimization

In this paper, we present a large-update primal-dual interior-point method for symmetric cone optimization（SCO） based on a new kernel function, which determines both search directions and the proximity measure between the iterate and the center path. The kernel function is neither a self-regular function nor the usual logarithmic kernel function. Besides, by using Euclidean Jordan algebraic techniques, we achieve the favorable iteration complexity O（ √r（1/2）（log r）^2 log（r/ ε））, which is as good as the convex quadratic semi-definite optimization analogue.

A class of polynomial primal-dual interior-point algorithms for semidefinite optimization

In the present paper we present a class of polynomial primal-dual interior-point algorithms for semidefmite optimization based on a kernel function. This kernel function is not a so-called self-regular function due to its growth term increasing linearly. Some new analysis tools were developed which can be used to deal with complexity ＂analysis of the algorithms which use analogous strategy in [5] to design the search directions for the Newton system. The complexity bounds for the algorithms with large- and small-update methodswere obtained, namely,O（qn^（p＋q/q（P＋1）log n/ε and O（q^2√n）log n/ε,respectlvely.

A new primal-dual interior-point algorithm for convex quadratic optimization

In this paper, a new primal-dual interior-point algorithm for convex quadratic optimization （CQO） based on a kernel function is presented. The proposed function has some properties that are easy for checking. These properties enable us to improve the polynomial complexity bound of a large-update interior-point method （IPM） to O（√n log nlog n/e）, which is the currently best known polynomial complexity bound for the algorithm with the large-update method. Numerical tests were conducted to investigate the behavior of the algorithm with different parameters p, q and θ, where p is the growth degree parameter, q is the barrier degree of the kernel function and θ is the barrier update parameter.

Primal-Dual Interior-Point Algorithms with Dynamic Step-Size Based on Kernel Functions for Linear Programming

In this paper, primal-dual interior-point algorithm with dynamic step size is implemented for linear programming （LP） problems. The algorithms are based on a few kernel functions, including both serf-regular functions and non-serf-regular ones. The dynamic step size is compared with fixed step size for the algorithms in inner iteration of Newton step. Numerical tests show that the algorithms with dynaraic step size are more efficient than those with fixed step size.

A new primal-dual path-following interior-point algorithm for linearly constrained convex optimization

In this paper, a primal-dual path-following interior-point algorithm for linearly constrained convex optimization(LCCO) is presented.The algorithm is based on a new technique for finding a class of search directions and the strategy of the central path.At each iteration, only full-Newton steps are used.Finally, the favorable polynomial complexity bound for the algorithm with the small-update method is deserved, namely, O(√n log n /ε).

Weighted Variational Minimization Model for Wavelet Domain Inpainting with Primal-Dual Method

To preserve the edges and details of the image,a new variational model for wavelet domain inpainting was proposed which contained a non-convex regularizer. The non-convex regularizer can utilize the local information of image and perform better than those usual convex ones. In addition, to solve the non-convex minimization problem,an iterative reweighted method and a primaldual method were designed. The numerical experiments show that the new model not only gets better visual effects but also obtains higher signal to noise ratio than the recent method.

A New Infeasible Interior-point Method for Linear Complementarity Problem Based on Full Newton Step

This paper proposes an infeasible interior-point algorithm with full-Newton step for linear complementarity problem,which is an extension of Roos about linear optimization. The main iteration of the algorithm consists of a feasibility step and several centrality steps. At last,we prove that the algorithm has O(nlog n/ε) polynomial complexity,which coincides with the best known one for the infeasible interior-point algorithm at present.

Interior-Point Methods Applied to the Predispatch Problem of a Hydroelectric System with Scheduled Line Manipulations

Transmission line manipulations in a power system are necessary for the execution of preventative or corrective main- tenance in a network, thus ensuring the stability of the system. In this study, primal-dual interior-point methods are used to minimize costs and losses in the generation and transmission of the predispatch active power flow in a hydroelectric system with previously scheduled line manipulations for preventative maintenance, over a period of twenty-four hours. The matrix structure of this problem and the modification that it imposes on the system is also broached in this study. From the computational standpoint, the effort required to solve a problem with or without line manipulations is similar, and the reasons for this are also discussed in this study. Computational results sustain our findings.

Complexity analysis of interior-point algorithm based on a new kernel function for semidefinite optimization

Interior-point methods （IPMs） for linear optimization （LO） and semidefinite optimization （SDO） have become a hot area in mathematical programming in the last decades. In this paper, a new kernel function with simple algebraic expression is proposed. Based on this kernel function, a primal-dual interior-point methods （IPMs） for semidefinite optimization （SDO） is designed. And the iteration complexity of the algorithm as O（n^3/4 log n/ε） with large-updates is established. The resulting bound is better than the classical kernel function, with its iteration complexity O（n log n/ε） in large-updates case.

A PREDICTOR-CORRECTOR INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interiorpoint method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.

A Modified Full-NT-Step Infeasible Interior-Point Algorithm for SDP Based on a Specific Kernel Function

This paper proposes a new full Nesterov-Todd(NT) step infeasible interior-point algorithm for semidefinite programming. Our algorithm uses a specific kernel function, which is adopted by Liu and Sun, to deduce the feasibility step. By using the step, it is remarkable that in each iteration of the algorithm it needs only one full-NT step, and can obtain an iterate approximate to the central path. Moreover, it is proved that the iterative bound corresponds with the known optimal one for semidefinite optimization problems.

A CORRECTOR-PREDICTOR ARC SEARCH INTERIOR-POINT ALGORITHM FOR SYMMETRIC OPTIMIZATION

In this paper, a corrector-predictor interior-point algorithm is proposed for sym- metric optimization. The algorithm approximates the central path by an ellipse, follows the ellipsoidal approximation of the central-path step by step and generates a sequence of iter- ates in a wide neighborhood of the central-path. Using the machinery of Euclidean Jordan algebra and the commutative class of search directions, the convergence analysis of the algo- rithm is shown and it is proved that the algorithm has the complexity bound O （√τL） for the well-known Nesterov-Todd search direction and O （τL） for the xs and sx search directions.

Polynomial-time interior-point algorithm based on a local self-concordant finite barrier function

The choice of self-concordant functions is the key to efficient algorithms for linear and quadratic convex optimizations, which provide a method with polynomial-time iterations to solve linear and quadratic convex optimization problems. The parameters of a self-concordant barrier function can be used to compute the complexity bound of the proposed algorithm. In this paper, it is proved that the finite barrier function is a local self-concordant barrier function. By deriving the local values of parameters of this barrier function, the desired complexity bound of an interior-point algorithm based on this local self-concordant function for linear optimization problem is obtained. The bound matches the best known bound for small-update methods.

Interior-point algorithm based on general kernel function for monotone linear complementarity problem

A polynomial interior-point algorithm is presented for monotone linear complementarity problem （MLCP） based on a class of kernel functions with the general barrier term, which are called general kernel functions. Under the mild conditions for the barrier term, the complexity bound of algorithm in terms of such kernel function and its derivatives is obtained. The approach is actually an extension of the existing work which only used the specific kernel functions for the MLCP.

A New Full-NT-Step Infeasible Interior-Point Algorithm for SDP Based on a Specific Kernel Function

In this paper, we propose a new infeasible interior-point algorithm with full NesterovTodd (NT) steps for semidefinite programming (SDP). The main iteration consists of a feasibility step and several centrality steps. We used a specific kernel function to induce the feasibility step. The analysis is more simplified. The iteration bound coincides with the currently best known bound for infeasible interior-point methods.

A New Kernel Function Yielding the Best Known Iteration Bounds for Primal-Dual Interior-Point Algorithms

Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithm for solving linear optimization problems. In this paper we present a new kernel function which yields an algorithm with the best known complexity bound for both large- and small-update methods.

Primal-dual Interior-point Algorithms for Second-order Cone Optimization Based on a New Parametric Kernel Function

A class of polynomial primal-dual interior-point algorithms for second-order cone optimization based on a new parametric kernel function, with parameters p and q, is presented. Its growth term is between linear and quadratic. Some new tools for the analysis of the algorithms are proposed. The complexity bounds of O（√Nlog N log N/ε） for large-update methods and O（√Nlog N/ε） for smallupdate methods match the best known complexity bounds obtained for these methods. Numerical tests demonstrate the behavior of the algorithms for different results of the parameters p and q.

基于非线性优化算法的电力系统状态估计技术研究

海上油田电网的安全运行直接影响着我国海上油气生产企业的稳定运行,通过了解状态估计的功能和研究现状,结合海洋石油生产电网的实际情况,提出了一种基于非线性优化算法的状态估计算法。基于状态估计的基本原理和流程,包括目标函数、约束条件的建立以及原-对偶内点算法的求解,采用了状态估计中不良数据检测与识别的计算方法,通过软件模块程序实现状态估计结果的求解。结合实际算例对比分析,得出了在正常情况下、不检测不良数据情况下、系统分裂运行情况下的状态估计结果与系统状态,为实现电力系统状态估计提供了高可靠性、高可信的数据,为电力系统调度中心电网运行状态分析提供参考。