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

Based on a new parametric kernel function, this paper presents a primal-dual large-update interior-point algorithm (IPM) for semi-definite optimization (SDO) problems. The new parametric function is neither self-regular function nor the usual logarithmic barrier function. It is strongly convex and possesses some novel analytic properties. We analyse this new parametric kernel function and show that the proposed algorithm has favorable complexity bound in terms of the analytic properties of the kernel function. Moreover, the complexity bound for our large-update IPM is shown to be O(\sqrt{n}(\log n)^2 \log\frac{n}{\epsilon}). Some numerical results are reported to illustrate the feasibility of the proposed algorithm.

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).

Exploring complexity of large update interior-point methods for [formula omitted] linear complementarity problem based on Kernel function

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.

In this paper, we present a primal-dual interior point method for linear optimization problems based on a new Kernel function with on a new parameterized logarithmic barrier term. We prove that the proposed kernel function belongs to the eligible class. We derive the complexity bounds for large and small-update methods respectively.

Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithm. In this paper, a new kernel function which its barrier term is integral type is proposed. We study the properties of the new kernel function, and give a primal-dual interior-point algorithm for solving linear optimization based on the new kernel function. Polynomial complexity of algorithm is analyzed. The iteration bounds both for large-update and for small-update methods are obtained, respectively. The iteration bound for small-update method is the best known complexity bound.

In this paper a primal-dual path-following interior-point algorithm for the monotone semidefinite linear complementarity problem is presented. The algorithm is based on Nesterov-Todd search directions and on a suitable proximity for tracing approximately the central-path. We provide an unified analysis for both long and small-update primal-dual algorithms. Finally, the iteration bounds for these algorithms are obtained.

The introduction of kernel function in primal-dual interior point methods represents not only a measure of the distance between the iterate and the central path, but also plays an important role in the amelioration of the computational complexity of an interior point algorithm. In this work, we present a primal-dual interior point method for linear optimization problems based on a new kernel function with an efficient logarithmic barrier term. We derive the complexity bounds for large and small-update methods respectively. We obtain the best known complexity bound for large-update given by Peng et al., which improves significantly the so far obtained complexity results based on a logarithmic kernel function given by El Ghami et al. The results obtained in this paper are the first with respect to the kernel function with a logarithmic barrier term.

A class of polynomial primal-dual interior-point algorithms for P * (κ ) linear complementarity problems (LCPs) are presented. We generalize Ghami et al.'s[A polynomial-time algorithm for linear optimization based on a new class of kernel functions(2008)] algorithm for linear optimization (LO) problem to P * (κ ) LCPs. Our analysis is based on a class of finite kernel functions which have the linear and quadratic growth terms. Since P *(κ ) LCP is a generalization of LO problem, we lose the orthogonality of the vectors dx and ds . So our analysis is different from the one in Ghami et al's algorithm. Despite this, the favorable complexity result is obtained, namely, $O((1 + 2\kappa)n^{\frac{1}{1+p}}\log n \log(n/\epsilon))$, which is better than the usual large-update primal-dual algorithm based on the classical logarithmic barrier function for P * (κ ) LCP.

Second-order cone optimization (SOCO) problems are crucial in various fields such as engineering, finance, and machine learning due to their ability to model complex convex optimization tasks efficiently. In this article, we propose an interior point algorithm tailored for SOCO, leveraging a novel kernel function to enhance optimization performance. Traditional interior point methods for SOCO often encounter challenges in scalability and efficiency, particularly with large-scale problems. These methods typically rely on barrier functions that may suffer from slow convergence rates, especially when dealing with highly ill-conditioned or degenerate cones. To address these limitations, our approach introduces a new kernel function that exploits the geometric properties of second-order cones, thereby improving convergence behavior and computational efficiency. The key innovation of our algorithm lies in the construction of the kernel function, which incorporates insights from the structure of second-order cones to efficiently capture the curvature information of the optimization problem. By exploiting this curvature information, our algorithm can effectively navigate the optimization landscape, leading to faster convergence and enhanced robustness compared to traditional methods. Furthermore, our algorithm is equipped with advanced techniques for handling various challenges encountered in SOCO, such as dealing with degenerate cones and exploiting problem structure to accelerate convergence. Additionally, we incorporate strategies for warm-starting and adaptive step-size selection to further improve efficiency, particularly in iterative optimization processes. Finally, our proposed interior point algorithm for second-order cone optimization, based on a new kernel function, derives the iteration bounds O (Nlog (N/∊)) for large update methods. Here denotes N the number of second-order cones in the problem formulation and ∊ the desired accuracy. These iteration bounds are currently the best known bounds for such methods.

This article studies a distributed model-predictive control (DMPC) strategy for a class of discrete-time linear systems subject to globally coupled constraints. To reduce the computational burden, the constraint tightening technique is adopted for enabling the early termination of the distributed optimization algorithm. Using the Lagrangian method, we convert the constrained optimization problem of the proposed DMPC to an unconstrained saddle-point seeking problem. Due to the presence of the global dual variable in the Lagrangian function, we propose a primal-dual algorithm based on the Laplacian consensus to solve such a problem in a distributed manner by introducing the local estimates of the dual variable. We theoretically show the geometric convergence of the primal-dual gradient optimization algorithm by the contraction theory in the context of discrete-time updating dynamics. The exact convergence rate is obtained, leading the stopping number of iterations to be bounded. The recursive feasibility of the proposed DMPC strategy and the stability of the closed-loop system can be established pursuant to the inexact solution. Numerical simulation demonstrates the performance of the proposed strategy.

In this paper we present the theory and practical aspects of implementing the path following interior point methods for linear optimization, based on kernel functions. We will investigate the influence of the choice of the kernel function on the computational behavior of the generic primal-dual algorithm for Linear Optimization. We find that the finite kernel function gives the best results for more than 50 % of the tested problems compared to the standard log-barrier method.

Primal–dual interior-point algorithms for second-order cone optimization based on kernel functions

The parameters selection and optimization of kernel function is the core of support vector machine (SVM), which is closely related to the distribution of datasets. It can be obtained a series of different group index, either mapped by different kernel functions on the same dataset or did by same kernel function on the different subsets. We analyze the impact of the choice of kernel functions and parameters on the performance of SVM, and propose a GA-optimized weighted mixed kernel function of SVM based on information entropy (GA-IE-RBF-SVM). The algorithm uses the information entropy to improve the contribution of the features that are conducive to classification firstly to mitigate falling into a local optimum, then learn from the idea of multi-core learning to enhance the adaptability of the algorithm. The optimal genetic algorithm (GA) is used to select the type of mixed kernel function, kernel function parameters and error penalty factor. The experimental results show that compared with other similar algorithms, this algorithm has a higher classification accuracy rate and faster convergence speed.

Complexity analysis and numerical implementation of a short-step primal-dual algorithm for linear complementarity problems