Interior-point Algorithm For Linear Optimization Research Articles

Large-step algorithm for linear complementarity problem with new search direction

Recently, [Darvay Zs, Takács PR: Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood. Cent Eur J Oper Res. 2018;26(3):551–563] introduced a new large-step interior-point algorithm (IPA) for linear optimization (LO). The purpose of this paper is to extend this method to the class of P ∗ ( κ ) -linear complementarity problems (LCPs). First, we propose a new wide neighbourhood of the central path for LCPs, which differs from the one presented by [Ai W, Zhang S: An O ( n L ) iteration primal–dual path-following method, based on wide neighbourhoods and large updates, for monotone LCP. SIAM J Optim. 2005;16(2):400–417]. Our method uses new search directions obtained by an algebraically equivalent transformation of the system which defines the central path. We present a large-step algorithm, but we demonstrate that it has the same iteration complexity as the best short-step algorithms. Finally, we give some numerical results to prove the efficiency of the new method.

A long-step second-order corrector interior point algorithm for linear optimization based on the algebraic equivalent transformation

In this paper, we present a new large-step primal-dual second-order corrector interior-point algorithm for linear optimization. This algorithm is based on the Nagy-Varga neighborhood of the central path, which uses the algebraic equivalent transformation strategy of Darvay et al. to determine the search directions. In addition, our method uses a second-order corrector direction in each iteration. The iterates remain in the proposed neighborhood by taking the largest possible step lengths along the search directions. We show that the new long-step algorithm is convergent and has the same complexity as the best short-step algorithms for linear optimization. To the best of our knowledge this is the first primal-dual second-order corrector interior-point algorithm that uses a different neighborhood from those available in the literature. Finally, numerical experiments show that the proposed algorithm is efficient and competitive.

Complexity analysis of an interior-point algorithm for linear optimization based on a new parametric kernel function with a double barrier term

<p style='text-indent:20px;'>Kernel functions play an important role in the complexity analysis of the interior point methods (IPMs) for linear optimization (LO). In this paper, an interior-point algorithm for LO based on a new parametric kernel function is proposed. By means of some simple analysis tools, we prove that the primal-dual interior-point algorithm for solving LO problems meets <inline-formula><tex-math id="M1">\begin{document}$ O\left(\sqrt{n} \log(n) \log(\frac{n}{\varepsilon}) \right) $\end{document}</tex-math></inline-formula>, iteration complexity bound for large-update methods with the special choice of its parameters.</p>

A class of new search directions for full-NT step feasible interior point method in semidefinite optimization

In this paper, based on Darvay et al.’s strategy for linear optimization (LO) (Z. Darvay and P.R. Takács, Optim. Lett. 12 (2018) 1099–1116.), we extend Kheirfam et al.’s feasible primal-dual path-following interior point algorithm for LO (B. Kheirfam and A. Nasrollahi, Asian-Eur. J. Math. 1 (2020) 2050014.) to semidefinite optimization (SDO) problems in order to define a class of new search directions. The algorithm uses only full Nesterov-Todd (NT) step at each iteration to find an ε-approximated solution to SDO. Polynomial complexity of the proposed algorithm is established which is as good as the LO analogue. Finally, we present some numerical results to prove the efficiency of the proposed algorithm.

Theoretical and numerical result for linear semidefinite programming based on a new kernel function

Kernel functions serve the central goal of creating new search directions for the primal-dual interiorpoint algorithm to solve linear optimization problems. A significantly improved primal-dual interior-point algorithm for linear optimization is presented based on a novel kernel function. We show a primal-dual interior-point technique for linear optimization based on a class of kernel functions that are eligible. This research presents a new efficient kernel function-based primal-dual IPM algorithm for semidefinite programming problems based on the Nesterov-Todd (NT) direction. With a new and simple technique, we propose a new kernel function to obtain an optimal solution of the perturbed problem (SDP)µ. We obtain the best-known complexity results,for smalland large-update namely O(pp+1/2p √nlog tr(X0S0)/ε) and O((pn)p+1/2p log trX0S0/ε) large update To prove the effectiveness of our proposed kernel function, we compare our numerical results with some alternatives presented by Touil et al. (2017).

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.

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.

A modified infeasible interior-point algorithm with full-Newton step for semidefinite optimization

ABSTRACTRecently, Mansouri et al. (J. Optim. Theory Appl. 166: 605-618, 2015) presented an improved infeasible interior-point algorithm for linear optimization. Their algorithm has the shortcoming that the proximity measure may be still large when the duality gap approaches to zero. In this paper, we propose an infeasible interior-point algorithm for semidefinite optimization with a modified search direction. This modification is an attempt to decrease the value of the proximity measure, which is important to determine whether or not to perform centreing steps in the classical infeasible interior-point algorithms. Some preliminary numerical results show the benefit of the proposed algorithm as well.

A polynomial-time weighted path-following interior-point algorithm for linear optimization

In this paper, a weighted short-step primal-dual path-following interior-point algorithm for solving linear optimization (LO) is presented. The algorithm uses at each interior-point iteration a full-Newton step, thus no need to use line search, and the strategy of the central-path to obtain an [Formula: see text]-approximated solution of LO. We show that the algorithm yields the iteration bound, namely, [Formula: see text]. This bound is currently the best iteration bound for LO. Finally, some numerical results are reported in order to analyze the efficiency of the proposed algorithm.

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

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 speak about large-step and short-step methods, that work in different neighbourhoods of the central path. The large-step algorithms work in a wide neighbourhood, while the short-step ones determine the new iterates that are in a smaller neighbourhood. In spite of the fact that the large-step algorithms are more efficient in practice, the theoretical complexity of the short-step ones is generally better. Ai and Zhang introduced a large-step interior-point method for linear complementarity problems using a wide neighbourhood of the central path, which has the same complexity as the best short-step methods. We present a new wide neighbourhood of the central path. We prove that the obtained large-step primal–dual interior-point method for linear programming has the same complexity as the best short-step algorithms.

New method for determining search directions for interior-point algorithms in linear optimization

We introduce an interior-point algorithm for linear optimization, which is based on a new technique for determining search directions. This method consists of a new type of transformation on the centering equations of the system which characterizes the central path. It can be shown that these new directions cannot be derived from usual kernel functions. Therefore, we extend the concept of the kernel functions, and we establish an equivalence between this approach and the proposed method for obtaining search directions. Moreover, we prove the polynomial complexity of the algorithm.

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.

A full-Newton step interior-point algorithm for linear optimization based on a finite barrier

In this paper, we propose a finite barrier kernel function for primal–dual interior-point algorithm in linear optimization with a full-Newton step. To our best knowledge, it is the first time that the property of exponential convexity is used for full-Newton step interior-point methods(IPMs). Moreover, the analysis is simplified and the complexity of the algorithm coincides with the currently best iteration bound for linear optimization problems.

An infeasible interior-point algorithm for linear optimization over Cartesian symmetric cones

We investigate the effect of making equivalent transformation for the central path from \(x\diamond s=\mu e\) to \(x\diamond s=\mu v\), and we obtain a new class of search directions in both version of feasible and infeasible interior point algorithms for linear optimization over Cartesian symmetric cones. We propose a new modified infeasible interior-point algorithm based on elimination of the centering steps. Moreover, the proposed algorithm finds an \(\varepsilon \)-optimal solution of the underlying problem in polynomial time complexity and its iteration bound coincides with the best known iteration bound for this class of problems.

Interior-point algorithm for linear optimization based on a new trigonometric kernel function

In this paper, we present a new primal–dual interior-point algorithm for linear optimization based on a trigonometric kernel function. By simple analysis, we derive the worst case complexity for a large-update primal–dual interior-point method based on this kernel function. This complexity estimate improves a result from El Ghami et al. (2012) and matches the one obtained in Reza Peza Peyghami et al. (2014).

A full-modified-Newton step infeasible interior-point algorithm for linear optimization

Based on an equivalent reformulation of the central path, we obtain a modified-Newton step for linear optimization. Using this step, we propose an infeasible interior-point algorithm. The algorithm uses only one full-modified-Newton step search in each iteration. The complexity bound of the algorithm is the best known for infeasible interior-point algorithm.

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

Kernel functions play an important role in the complexity analysis of the interior point methods for linear optimization. In this paper, we present a primal-dual interior point method for linear optimization based on a new kernel function consisting of a trigonometric function in its barrier term. By simple analysis, we show that the feasible primal-dual interior point methods based on the new proposed kernel function enjoys O n log n 2 log n 𝜖 $O\left (\sqrt {n}\left (\log {n}\right )^{2}\log \frac {n}{\epsilon }\right )$ worst case complexity result which improves the results obtained by El Ghami et al. (J Comput Appl Math 236:3613---3623, 2012) for the kernel functions with trigonometric barrier terms.

A Full Nesterov-Todd Step Feasible Weighted Primal-Dual Interior-Point Algorithm for Symmetric Optimization

In this paper a weighted short-step primal-dual interior-point algorithm for linear optimization over symmetric cones is proposed that uses new search directions. The algorithm uses at each interior-point iteration a full Nesterov-Todd step and the strategy of the central path to obtain a solution of symmetric optimization. We establish the iteration bound for the algorithm, which matches the currently best-known iteration bound for these methods, and prove that the algorithm is quadratically convergent.