Complexity of large-update interior point algorithm for P*(K) linear complementarity problems
Computers & Mathematics with Applications
A new large-update interior point algorithm for P*(κ) linear complementarity problems
Journal of Computational and Applied Mathematics
A polynomial-time algorithm for linear optimization based on a new class of kernel functions
Journal of Computational and Applied Mathematics
ISNN 2009 Proceedings of the 6th International Symposium on Neural Networks: Advances in Neural Networks - Part III
Polynomial interior-point algorithms for P*(K) horizontal linear complementarity problem
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
Interior point algorithm for P* nonlinear complementarity problems
Journal of Computational and Applied Mathematics
SIAM Journal on Optimization
Journal of Computational and Applied Mathematics
New parameterized kernel functions for linear optimization
Journal of Global Optimization
Journal of Computational and Applied Mathematics
Interior-point algorithms for $$P_{*}(\kappa )$$-LCP based on a new class of kernel functions
Journal of Global Optimization
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Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) for linear optimization were introduced. Each such barrier function is determined by its (univariate) self-regular kernel function. We introduce a new class of kernel functions. The class is defined by some simple conditions on the kernel function and its derivatives. These properties enable us to derive many new and tight estimates that greatly simplify the analysis of IPMs based on these kernel functions. In both the algorithm and its analysis we use a single neighborhood of the central path; the neighborhood naturally depends on the kernel function. An important conclusion is that inverse functions of suitable restrictions of the kernel function and its first derivative more or less determine the behavior of the corresponding IPMs. Based on the new estimates we present a simple and unified computational scheme for the complexity analysis of kernel function in the new class. We apply this scheme to seven specific kernel functions. Some of these functions are self-regular, and others are not. One of the functions differs from the others, and from all self-regular functions, in the sense that its growth term is linear. Iteration bounds for both large- and small-update methods are derived. It is shown that small-update methods based on the new kernel functions all have the same complexity as the classical primal-dual IPM, namely, $O(\sqrt{n}\log\frac{n}{\e})$. For large-update methods the best obtained bound is $O(\sqrt{n}(\log n)\log\frac{n}{\e})$, which until now has been the best known bound for such methods.