Interior-point Method For Linear Optimization Research Articles

A Geodesic Interior-Point Method for Linear Optimization over Symmetric Cones

A Geodesic Interior-Point Method for Linear Optimization over Symmetric Cones

A new search direction for full-Newton step infeasible interior-point method in linear optimization

In this work, we investigate a full Newton step infeasible interior-point method for linear optimization based on a new search direction which is obtained from an algebraic equivalent transformation of the central path system. Furthermore, we prove that the proposed method obtains an ε-optimal solution to the original problem in polynomial time.

Complexity analysis of primal-dual interior-point methods for linear optimization based on a new efficient Bi-parameterized kernel function with a trigonometric barrier term

In this paper we are generalizing the efficient kernel function with trigonometric barrier term given by (M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545). Using an elegant and simple analysis and under some easy to check conditions, we explore the best complexity result for the large update primal-dual interior point methods for linear optimization. This complexity estimate improves results obtained in (X. Li and M. Zhang, Oper. Res. Lett. 43 (2015) 471–475; M.R. Peyghami and S.F. Hafshejani, Numer. Algo. 67 (2014) 33–48; M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545). Our comparative numerical experiments on some test problems consolidate and confirm our theoretical results according to which the new kernel function has promising applications compared to the kernel function given by (M. Bouafia and A. Yassine, Optim. Eng. 21 (2020) 651–672). Moreover, the comparative numerical study that we have established favors our new kernel function better than other best trigonometric kernel functions (M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545; M. Bouafia and A. Yassine, Optim. Eng. 21 (2020) 651–672).

A second-order corrector wide neighborhood infeasible interior-point method for linear optimization based on a specific kernel function

In this paper, we present a second-order corrector infeasibleinterior-point method for linear optimization in a largeneighborhood of the central path. The innovation of our method is tocalculate the predictor directions using a specific kernel functioninstead of the logarithmic barrier function. We decompose thepredictor direction induced by the kernel function to two orthogonaldirections of the corresponding to the negative and positivecomponent of the right-hand side vector of the centering equation.The method then considers the new point as a linear combination ofthese directions along with a second-order corrector direction. Theconvergence analysis of the proposed method is investigated and itis proved that the complexity bound is Ο(n5/4 log e-1).

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 interior-point algorithm based on the trigonometric kernel function

In this paper, a wide neighborhood path-following interior point algorithm for linear optimization (LO) is proposed that uses a trigonometric kernel function to get search directions. The method treats the Newton direction as the sum of two other directions, according to the negative and positive parts of the right-hand-side based on the kernel function. By choosing different and appropriate step sizes for these two directions, the iterates stop in the Ai–Zhang’s wide neighborhood. By an elegant analysis, we show that the method enjoys the low iteration bound of $${\mathcal {O}}(\sqrt{n}\log \frac{(x^0)^Ts^0}{\varepsilon })$$ , where n is the dimension of the problem and $$(x^0, s^0)$$ the initial interior point with $$\varepsilon $$ the required precision. In our knowledge, this result is the first instance of a wide neighborhood interior point method for LO which involving the trigonometric kernel function.

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.

A new wide neighborhood primal-dual second-order corrector algorithm for linear optimization

We propose a new large-step primal-dual second-order corrector interior-point method for linear optimization. At each iteration, our method uses the new wide neighborhood introduced by Darvay and Takacs (Cent Eur J Oper Res 26(3):551–563, 2018. https://doi.org/10.1007/s10100-018-0524-0 ). In this paper we would like to improve the directions proposed by Darvay and Takacs by adding a second-order corrector direction. The corrector step is multiplied by the square of the step length in the expression of the new iterate. To our best knowledge, this is the first primal-dual second-order corrector interior-point algorithm based on Darvay–Takacs’s new wide neighborhood, which has the same complexity as the best short-step algorithms for linear optimization. Finally, numerical experiments show that the proposed algorithm is efficient and reliable.

An interior-point method for linear optimization based on a trigonometric kernel function

An interior-point method for linear optimization based on a trigonometric kernel function

A primal-dual interior-point method for linear optimization based on a new parameterized kernel function

A primal-dual interior-point method for linear optimization based on a new parameterized kernel function

An extension for identifying search directions for interior-point methods in linear optimization

In this paper, based on the transformation [Formula: see text] introduced by Darvay and Takács [New method for determining search directions for interior-point algorithms in linear optimization, Optim. Lett. 12(5) (2018) 1099–1116], we present a full-Newton step interior-point method for linear optimization. They consider the case [Formula: see text]. Here, we extend this to the case [Formula: see text] to obtain our search directions. We show that the iterates lie in the neighborhood of the local quadratic convergence of the proximity measure. Finally, the polynomial complexity of the proposed algorithm is proved.

Primal–dual interior-point method for linear optimization based on a kernel function with trigonometric growth term

In this paper, we propose a large-update primal–dual interior-point algorithm for linear optimization problems based on a new kernel function with a trigonometric growth term. By simple analysis, we prove that in the large neighbourhood of the central path, the worst case iteration complexity of the new algorithm is bounded above by , which matches the currently best known iteration bound for large-update methods. Moreover, we show that, most of the so far proposed kernel functions can be rewritten as a kernel function with trigonometric growth term. Finally, numerical experiments on some test problems confirm that the new kernel function is well promising in practice in comparison with some existing kernel functions in the literature.

A generalization of IPMs for linear optimization to convex quadratic SDO based on a trigonometric kernel function

In this paper, we generalize primal-dual interior-point methods for linear optimization to convex quadratic semidefinite optimization, which is a wide class of optimization problems that contains linear optimization, convex quadratic optimization, second-order cone optimization and semidefinite optimization as special cases. Based on the Nesterov and Todd scaling scheme, we establish the currently best known complexity bounds of large- and small-update interior-point methods for convex quadratic semidefinite optimization, namely, and , respectively, which are as good as the linear optimization analogue.

Primal-dual entropy-based interior-point algorithms for linear optimization

We propose a family of search directions based on primal-dual entropy in the context of interior-point methods for linear optimization. We show that by using entropy-based search directions in the predictor step of a predictor-corrector algorithm together with a homogeneous self-dual embedding, we can achieve the current best iteration complexity bound for linear optimization. Then, we focus on some wide neighborhood algorithms and show that in our family of entropy-based search directions, we can find the best search direction and step size combination by performing a plane search at each iteration. For this purpose, we propose a heuristic plane search algorithm as well as an exact one. Finally, we perform computational experiments to study the performance of entropy-based search directions in wide neighborhoods of the central path, with and without utilizing the plane search algorithms.

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.

Infeasible Interior-Point Methods for Linear Optimization Based on Large Neighborhood

In this paper, we design a class of infeasible interior-point methods for linear optimization based on large neighborhood. The algorithm is inspired by a full-Newton step infeasible algorithm with a linear convergence rate in problem dimension that was recently proposed by the second author. Unfortunately, despite its good numerical behavior, the theoretical convergence rate of our algorithm is worse up to square root of problem dimension.

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.

An Improved and Simplified Full-Newton Step $O(n)$ Infeasible Interior-Point Method for Linear Optimization

We present an improved version of an infeasible interior-point method for linear optimization published in 2006. In the earlier version each iteration consisted of one so-called feasibility step and a few---at most three---centering steps. In this paper each iteration consists of only a feasibility step, whereas the iteration bound improves the earlier bound by a factor $2\sqrt{2}$. The improvements are due to a new lemma that gives a much tighter upper bound for the proximity after the feasibility step.

Complexity Analysis of Primal-Dual Interior-Point Methods for Linear Optimization Based on a New Parametric Kernel Function with a Trigonometric Barrier Term

We introduce a new parametric kernel function, which is a combination of the classic kernel function and a trigonometric barrier term, and present various properties of this new kernel function. A class of large- and small-update primal-dual interior-point methods for linear optimization based on this parametric kernel function is proposed. By utilizing the feature of the parametric kernel function, we derive the iteration bounds for large-update methods,O(n2/3log⁡(n/ε)), and small-update methods,O(nlog⁡(n/ε)). These results match the currently best known iteration bounds for large- and small-update methods based on the trigonometric kernel functions.

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.