Infeasible interior-point method for symmetric optimization using a positive-asymptotic barrier

We propose a new primal-dual infeasible interior-point method for symmetric optimization by using Euclidean Jordan algebras. Different kinds of interior-point methods can be obtained by using search directions based on kernel functions. Some search directions can be also determined by applying an algebraic equivalent transformation on the centering equation of the central path. Using this method we introduce a new search direction, which can not be derived from a usual kernel function. For this reason, we use the new notion of positive-asymptotic kernel function which induces the class of corresponding barriers. In general, the main iterations of the infeasible interior-point methods are composed of one feasibility and several centering steps. We prove that in our algorithm it is enough to take only one centering step in a main iteration in order to obtain a well-defined algorithm. Moreover, we conclude that the algorithm finds solution in polynomial time and has the same complexity as the currently best known infeasible interior-point methods. Finally, we give some numerical results.     

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.     

A new interior-point algorithm based on modified Nesterov–Todd direction for symmetric cone linear complementarity problem

We present a new primal-dual path-following interior-point algorithm for linear complementarity problem over symmetric cones. The algorithm is based on a reformulation of the central path for finding the search directions. For a full Nesterov–Todd step feasible interior-point algorithm based on the search directions, the complexity bound of the algorithm with the small update approach is the best available.     

A Polynomial Interior-Point Algorithm for Monotone Linear Complementarity Problems

In this paper, we propose an interior-point algorithm for monotone linear complementarity problems. 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-Newton steps. Moreover, it is proven that the number of iterations of the algorithm coincides with the well-known best iteration bound for monotone linear complementarity problems.     

A New Complexity Analysis for Full-Newton Step Infeasible Interior-Point Algorithm for Horizontal Linear Complementarity Problems

In this paper, we first present a full-Newton step feasible interior-point algorithm for solving horizontal linear complementarity problems. We prove that the full-Newton step to the central path is quadratically convergent. Then, we generalize an infeasible interior-point method for linear optimization to horizontal linear complementarity problems based on new search directions. This algorithm starts from strictly feasible iterates on the central path of a perturbed problem that is produced by a suitable perturbation in the horizontal linear complementarity problem. We use the so-called feasibility steps that find strictly feasible iterates for the next perturbed problem. By using centering steps for the new perturbed problem, we obtain a strictly feasible iterate close enough to the central path of the new perturbed problem. The complexity of the algorithm coincides with the best known iteration bound for infeasible interior-point methods.     

一个求解半正定规划问题的新原始一对偶内点算法

在原始对偶内点算法的设计和分析中,障碍函数对算法的搜索方法和复杂性起着重要的作用。本文由核函数来确定障碍函数,设计了一个求解半正定规划问题的原始。对偶内点算法。这个障碍函数即可以定义算法新的搜索方向,又度量迭代点与中心路径的距离,同时对算法的复杂性分析起着关键的作用。我们计算了算法的迭代界,得出了关于大步校正法和小步校正法的迭代界,它们分别是O（√n log n log n/c）和O（√n log n/ε）,这里n是半正定规划问题的维数。最后,我们根据一个算例,说明了算法的有效性以及对核函数的参数的敏感性。     

New Interior-Point Algorithm for Symmetric Optimization Based on a Positive-Asymptotic Barrier Function

Abstract

We define a new interior-point method (IPM), which is suitable for solving symmetric optimization (SO) problems. The proposed algorithm is based on a new search direction. In order to obtain this direction, we apply the method of algebraically equivalent transformation on the centering equation of the central path. We prove that the associated barrier cannot be derived from a usual kernel function. Therefore, we introduce a new notion, namely the concept of the positive-asymptotic kernel function. We conclude that this algorithm solves the problem in polynomial time and has the same complexity as the best known IPMs for SO.     

Complexity analysis of a full-Newton step interior-point method for linear optimization

This paper concerns a short-update primal-dual interior-point method for linear optimization based on a new search direction. We apply a vector-valued function generated by a univariate function on the nonlinear equation of the system which defines the central path. The common way to obtain the equivalent form of the central path is using the square root function. In this paper we consider a new function formed by the difference of the identity map and the square root function. We apply Newton’s method in order to get the new directions. In spite of the fact that the analysis is more difficult in this case, we prove that the complexity of the algorithm is identical with the one of the best known methods for linear optimization.     

A Polynomial Arc-Search Interior-Point Algorithm for Linear Programming

In this paper, ellipsoidal estimations are used to track the central path of linear programming. A higher-order interior-point algorithm is devised to search the optimizers along the ellipse. The algorithm is proved to be polynomial with the best complexity bound for all polynomial algorithms and better than the best known bound for higher-order algorithms.     

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 predictor-corrector interior-point algorithm for P ∗ ( κ ) -horizontal linear complementarity problem

We present a predictor-corrector path-following interior-point algorithm for \(P_*(\kappa )\) 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 \(P_*(\kappa )\) -horizontal linear complementarity problems.     

A full-Newton step infeasible interior-point algorithm for monotone LCP based on a locally-kernel function

We propose a new full-Newton step infeasible interior-point algorithm for monotone linear complementarity problems based on a simple locally-kernel function. The algorithm uses the simple locally-kernel function to determine the search directions and define the neighborhood of central path. Two types of full-Newton steps are used, feasibility step and centering step. The algorithm starts from strictly feasible iterates of a perturbed problem, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed problem. By using centering steps for the new perturbed problem, we obtain strictly feasible iterates close enough to the central path of the new perturbed problem. The procedure is repeated until an ?-approximate solution is found. We analyze the algorithm and obtain the complexity bound, which coincides with the best-known result for monotone linear complementarity problems.     

A New Full Nesterov–Todd Step Primal–Dual Path-Following Interior-Point Algorithm for Symmetric Optimization

In this paper, we generalize a primal–dual path-following interior-point algorithm for linear optimization to symmetric optimization by using Euclidean Jordan algebras. The proposed algorithm is based on a new technique for finding the search directions and the strategy of the central path. At each iteration, we use only full Nesterov–Todd steps. Moreover, we derive the currently best known iteration bound for the small-update method. This unifies the analysis for linear, second-order cone, and semidefinite optimizations.     

A modified layered-step interior-point algorithm for linear programming

The layered-step interior-point algorithm was introduced by Vavasis and Ye. The algorithm accelerates the path following interior-point algorithm and its arithmetic complexity depends only on the coefficient matrixA. The main drawback of the algorithm is the use of an unknown big constant in computing the search direction and to initiate the algorithm. We propose a modified layered-step interior-point algorithm which does not use the big constant in computing the search direction. The constant is required only for initialization when a well-centered feasible solution is not available, and it is not required if an upper bound on the norm of a primal—dual optimal solution is known in advance. The complexity of the simplified algorithm is the same as that of Vavasis and Ye. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research supported in part by ONR contract N00014-94-C-0007 and the Grant-in-Aid for Scientific Research (C) 08680478 and the Grant-in-Aid for Encouragement of Young Scientists (A) 08780227 of the Ministry of Science, Education and Culture of Japan. This research was partially done while S. Mizuno and T. Tsuchiya were visiting IBM Almaden Research Center in the summer of 1995.     

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.     

Quadratic convergence of potential-reduction methods for degenerate problems

Global and local convergence properties of a primal-dual interior-point pure potential-reduction algorithm for linear programming problems is analyzed. This algorithm is a primal-dual variant of the Iri-Imai method and uses modified Newton search directions to minimize the Tanabe-Todd-Ye (TTY) potential function. A polynomial time complexity for the method is demonstrated. Furthermore, this method is shown to have a unique accumulation point even for degenerate problems and to have Q-quadratic convergence to this point by an appropriate choice of the step-sizes. This is, to the best of our knowledge, the first superlinear convergence result on degenerate linear programs for primal-dual interior-point algorithms that do not follow the central path. Received: February 12, 1998 / Accepted: March 3, 2000?Published online January 17, 2001     

A corrector-predictor arc search interior-point algorithm for symmetric optimization

In this paper, a corrector-predictor interior-point algorithm is proposed for symmetric 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 iterates 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 algorithm is shown and it is proved that the algorithm has the complexity bound O$\left(\sqrt{r}L\right)$ for the well-known Nesterov-Todd search direction and O(rL) for the xs and sx search directions.     

Weighted-path-following interior-point algorithm to monotone mixed linear complementarity problem

In this paper we propose a weighted-path-following interior-point algorithm to monotone mixed linear complementarity problem. 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, we only use full-Newton step. Finally, the currently best known iteration bound for the algorithm with a small-update method, namely, O(√nlog n/ε) is derived, which is as good as the bound for the linear optimization analogue.     

An Interior-Point Algorithm for Nonconvex Nonlinear Programming

The paper describes an interior-point algorithm for nonconvex nonlinear programming which is a direct extension of interior-point methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.     

A new infeasible interior-point method based on Darvay’s technique for symmetric optimization

We present a full Nesterov and Todd step primal-dual infeasible interior-point algorithm for symmetric optimization based on Darvay’s technique by using Euclidean Jordan algebras. The search directions are obtained by an equivalent algebraic transformation of the centering equation. The algorithm decreases the duality gap and the feasibility residuals at the same rate. During this algorithm we construct strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. Each main iteration of the algorithm consists of a feasibility step and some centering steps. The starting point in the first iteration of the algorithm depends on a positive number ξ and it is strictly feasible for a perturbed pair. The feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close to the central path of the new perturbed pair. The algorithm finds an ?-optimal solution or detects infeasibility of the given problem. Moreover, we derive the currently best known iteration bound for infeasible interior-point methods.