Interior-point methods for linear optimization based on a kernel function with a trigonometric barrier term

In this paper, we present a new barrier function for primal–dual interior-point methods in linear optimization. The proposed kernel function has a trigonometric barrier term. It is shown that in the interior-point methods based on this function for large-update methods, the iteration bound is improved significantly. For small-update interior-point methods, the iteration bound is the best currently known bound for primal–dual interior-point methods.     

基于指数型核函数的线性规划原始对偶内点算法

给出线性规划原始对偶内点算法的一个单变量指数型核函数.首先研究了这个指数型核函数的性质以及其对应的障碍函数.其次,基于这个指数型核函数,设计了求解线性规划问题的原始对偶内点算法,得到了目前小步算法最好的理论迭代界.最后,通过数值算例比较了基于指数型核函数的原始对偶内点算法和基于对数型核函数的原始对偶内点算法的计算效果.     

Complexity analysis of an interior point algorithm for the semidefinite optimization based on a kernel function with a double barrier term

In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization(SDO) problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 q2 1, the algorithm has O((q1 + 1) nq1+1/2(q1-q2)logn/ε)and O((q1 + 1)3q1-2q2+1/2(q1-q2)n~1/2 logn/ε) complexity results for large- and small-update methods, respectively.     

A large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function

In this paper, we present a large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function. The proposed function is strongly convex. It is not self-regular function and also the usual logarithmic function. The goal of this paper is to investigate such a kernel function and show that the algorithm has favorable complexity bound in terms of the elegant analytic properties of the kernel function. The complexity bound is shown to be $O\left( {\sqrt n \left( {\log n} \right)^2 \log \frac{n} {\varepsilon }} \right)$ . This bound is better than that by the classical primal-dual interior-point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization fields. Some computational results have been provided.     

A full-Newton step infeasible interior-point method for linear optimization based on a trigonometric kernel function

An infeasible interior-point method (IIPM) for solving linear optimization problems based on a kernel function with trigonometric barrier term is analysed. In each iteration, the algorithm involves a feasibility step and several centring steps. The centring step is based on classical Newton’s direction, while we used a kernel function with trigonometric barrier term in the algorithm to induce the feasibility step. The complexity result coincides with the best-known iteration bound for IIPMs. To our knowledge, this is the first full-Newton step IIPM based on a kernel function with trigonometric barrier term.     

A new primal-dual path-following interior-point algorithm for semidefinite optimization

In this paper we present a new primal-dual path-following interior-point algorithm for semidefinite optimization. The algorithm is based on a new technique for finding the search direction and the strategy of the central path. At each iteration, we use only full Nesterov-Todd step. Moreover, we obtain the currently best known iteration bound for the algorithm with small-update method, namely, , which is as good as the linear analogue.     

A wide neighborhood interior-point algorithm for linear optimization based on a specific kernel function

This paper presents an interior point algorithm for solving linear optimization problems in a wide neighborhood of the central path introduced by Ai and Zhang (SIAM J Optim 16:400–417, 2005). In each iteration, the algorithm computes the new search directions by using a specific kernel function. The convergence of the algorithm is shown and it is proved that the algorithm has the same iteration bound as the best short-step algorithms. We demonstrate the computational efficiency of the proposed algorithm by testing some Netlib problems in standard form. To best our knowledge, this is the first wide neighborhood path-following interior-point method with the same complexity as the best small neighborhood path-following interior-point methods that uses the kernel function.

     

一个新的求解半正定规划问题的原始对偶内点算法

选择合适的核函数对设计求解线性规划与半正定规划的原始对偶内点算法以及复杂性分析都十分重要.Bai等针对线性规划提出三种核函数,并给出求解线性规划的大步迭代复杂界,但未给出数值算例验证算法的实际效果(Bai Y Q,Xie W,Zhang J.New parameterized kernel functions for linear optimization.J Global Optim,2012.DOI 10.1007/s10898-012-9934-z).基于这三种核函数设计了新的求解半正定规划问题的原始对内点算法.进一步分析了算法关于大步方法的计算复杂性界,同时通过数值算例验证了算法的有效性和核函数所带参数对计算复杂性的影响.     

Primal-dual interior-point methods for PDE-constrained optimization

This paper provides a detailed analysis of a primal-dual interior-point method for PDE-constrained optimization. Considered are optimal control problems with control constraints in L ^p . It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier L ^∞-setting is analyzed, but also a more involved L ^q -analysis, q < ∞, is presented. In L ^∞, the set of feasible controls contains interior points and the Fréchet differentiability of the perturbed optimality system can be shown. In the L ^q -setting, which is highly relevant for PDE-constrained optimization, these nice properties are no longer available. Nevertheless, a convergence analysis is developed using refined techniques. In parti- cular, two-norm techniques and a smoothing step are required. The L ^q -analysis with smoothing step yields global linear and local superlinear convergence, whereas the L ^∞-analysis without smoothing step yields only global linear convergence.     

A new second-order corrector interior-point algorithm for semidefinite programming

In this paper, we propose a second-order corrector interior-point algorithm for semidefinite programming (SDP). This algorithm is based on the wide neighborhood. The complexity bound is O(?nL){O(\sqrt{n}L)} for the Nesterov-Todd direction, which coincides with the best known complexity results for SDP. To our best knowledge, this is the first wide neighborhood second-order corrector algorithm with the same complexity as small neighborhood interior-point methods for SDP. Some numerical results are provided as well.     

Complexity analysis of an interior-point algorithm for linear optimization based on a new proximity function

In this paper, we focus on a family of modified Chebyshev methods and study the semilocal convergence for these methods. Different from the results in reference (Hernández and Salanova, J. Comput. Appl. Math. 126:131–143, 2000), the Hölder continuity of the second derivative is replaced by its generalized continuity condition, and the latter is weaker than the former. Using the recurrence relations, we establish the semilocal convergence of these methods and prove a convergence theorem to show the existence-uniqueness of the solution. The R-order of these methods is also analyzed. Especially, when the second derivative of the operator is Hölder continuous, the R-order of these methods is at least 3 + 2p, which is higher than the one of Chebyshev method considered in reference (Hernández and Salanova, J. Comput. Appl. Math. 126:131–143, 2000) under the same condition. Finally, we give some numerical results to show our approach.     

Simplified infeasible interior-point algorithm for linear optimization based on a simple function

Based on a similar kernel function, we present an infeasible version of the interior-point algorithm for linear optimization introduced by Wang et al. (2016). The property of exponential convexity is still important to simplify the analysis of the algorithm. The iteration bound coincides with the currently best iteration bound for infeasible interior-point algorithms.     

Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood

Central European Journal of Operations Research - The interior-point algorithms can be classified in multiple ways. One of these takes into consideration the length of the step. In this way, we can...     

半定规划的一个新的宽邻域非可行内点算法

基于一种新的宽邻域,提出一个求解半定规划的新的非可行内点算法．在适当的假设条件下,证明了该算法具有较好的迭代复杂界O（√nL）,优于目前此类算法的最好的复杂性O（n√nL）,等同于可行内点算法．     

An interior-point algorithm for semidefinite least-squares problems

Applications of Mathematics - We propose a feasible primal-dual path-following interior-point algorithm for semidefinite least squares problems (SDLS). At each iteration, the algorithm uses only...     

A full-NT-step infeasible interior-point algorithm for SDP based on kernel functions

This paper proposes an infeasible interior-point algorithm with full Nesterov-Todd (NT) steps for semidefinite programming (SDP). The main iteration consists of a feasibility step and several centrality steps. First we present a full NT step infeasible interior-point algorithm based on the classic logarithmical barrier function. After that a specific kernel function is introduced. The feasibility step is induced by this kernel function instead of the classic logarithmical barrier function. This kernel function has a finite value on the boundary. The result of polynomial complexity, O(nlogn/ε), coincides with the best known one for infeasible interior-point methods.     

关于半定规划的一种宽邻域不可行内点算法的注记

针对半定规划的宽邻域不可行内点算法, 将牛顿法和预估校正法进行结合, 构造出适当的迭代方向, 提出一个修正的半定规划宽邻域不可行内点算法, 并在适当的假设条件下, 证明了该算法具有O(\sqrt{n}L)的迭代复杂界．最后利用Matlab编程, 给出了基于KM方向和NT方向的数值实验结果．     

A primal-dual interior-point algorithm for symmetric optimization based on a new method for finding search directions

We introduce an interior-point method for symmetric optimization based on a new method for determining search directions. In order to accomplish this, we use a new equivalent algebraic transformation on the centring equation of the system which characterizes the central path. In this way, we obtain a new class of directions. We analyse a special case of this class, which leads to the new interior-point algorithm mentioned before. Another way to find the search directions is using barriers derived from kernel functions. We show that in our case the corresponding direction cannot be deduced from a usual kernel function. In spite of this fact, we prove the polynomial complexity of the proposed algorithm.