Predictor-corrector interior-point algorithm for sufficient linear complementarity problems based on a new type of algebraic equivalent transformation technique

We propose a new predictor-corrector (PC) interior-point algorithm (IPA) for solving linear complementarity problem (LCP) with P_* (κ)-matrices. The introduced IPA uses a new type of algebraic equivalent transformation (AET) on the centering equations of the system defining the central path. The new technique was introduced by Darvay et al. [21] for linear optimization. The search direction discussed in this paper can be derived from positive-asymptotic kernel function using the function φ(t)=t^2 in the new type of AET. We prove that the IPA has O(1+4κ)√n log⁡〖(3nμ^0)/ε〗 iteration complexity, where κ is an upper bound of the handicap of the input matrix. To the best of our knowledge, this is the first PC IPA for P_* (κ)-LCPs which is based on this search direction

Unified Analysis of Kernel-Based Interior-Point Methods for \u3cem\u3eP\u3c/em\u3e *(κ)-LCP

We present an interior-point method for the P∗(κ)-linear complementarity problem (LCP) that is based on barrier functions which are defined by a large class of univariate functions called eligible kernel functions. This class is fairly general and includes the classical logarithmic function and the self-regular functions, as well as many non-self-regular functions as special cases. We provide a unified analysis of the method and give a general scheme on how to calculate the iteration bounds for the entire class. We also calculate the iteration bounds of both long-step and short-step versions of the method for several specific eligible kernel functions. For some of them we match the best known iteration bounds for the long-step method, while for the short-step method the iteration bounds are of the same order of magnitude. As far as we know, this is the first paper that provides a unified approach and comprehensive treatment of interior-point methods for P∗(κ)-LCPs based on the entire class of eligible kernel functions

Predictor-corrector interior-point algorithm based on a new search direction working in a wide neighbourhood of the central path

We introduce a new predictor-corrector interior-point algorithm for solving P_*(κ)-linear complementarity problems which works in a wide neighbourhood of the central path. We use the technique of algebraic equivalent transformation of the centering equations of the central path system. In this technique, we apply the function φ(t)=√t in order to obtain the new search directions. We define the new wide neighbourhood D_φ. In this way, we obtain the first interior-point algorithm, where not only the central path system is transformed, but the definition of the neighbourhood is also modified taking into consideration the algebraic equivalent transformation technique. This gives a new direction in the research of interior-point methods. We prove that the IPA has O((1+κ)n log⁡((〖〖(x〗^0)〗^T s^0)/ϵ) ) iteration complexity. Furtermore, we show the efficiency of the proposed predictor-corrector interior-point method by providing numerical results. Up to our best knowledge, this is the first predictor-corrector interior-point algorithm which works in the D_φ neighbourhood using φ(t)=√t

New predictor-corrector interior-point algorithm with AET function having inflection point

In this paper we introduce a new predictor-corrector interior-point algorithm for solving P_* (κ)-linear complementarity problems. For the determination of search directions we use the algebraically equivalent transformation (AET) technique. In this method we apply the function φ(t)=t^2-t+√t which has inflection point. It is interesting that the kernel corresponding to this AET function is neither self-regular, nor eligible. We present the complexity analysis of the proposed interior-point algorithm and we show that it's iteration bound matches the best known iteration bound for this type of PC IPAs given in the literature. It should be mentioned that usually the iteration bound is given for a fixed update and proximity parameter. In this paper we provide a set of parameters for which the PC IPA is well defined. Moreover, we also show the efficiency of the algorithm by providing numerical results

A new Ai-Zhang type interior point algorithm for sufficient linear complementarity problems

In this paper, we propose a new long-step interior point method for solving sufficient linear complementarity problems. The new algorithm combines two important approaches from the literature: the main ideas of the long-step interior point algorithm introduced by Ai and Zhang, and the algebraic equivalent transformation technique proposed by Darvay. Similarly to the method of Ai and Zhang, our algorithm also works in a wide neighbourhood of the central path and has the best known iteration complexity of short-step variants. We implemented the new method in Matlab and tested its efficiency on both sufficient and non-sufficient problem instances. In addition to presenting our numerical results, we also make some interesting observations regarding the analysis of Ai-Zhang type methods

Introducing Interior-Point Methods for Introductory Operations Research Courses and/or Linear Programming Courses

In recent years the introduction and development of Interior-Point Methods has had a profound impact on optimization theory as well as practice, influencing the field of Operations Research and related areas. Development of these methods has quickly led to the design of new and efficient optimization codes particularly for Linear Programming. Consequently, there has been an increasing need to introduce theory and methods of this new area in optimization into the appropriate undergraduate and first year graduate courses such as introductory Operations Research and/or Linear Programming courses, Industrial Engineering courses and Math Modeling courses. The objective of this paper is to discuss the ways of simplifying the introduction of Interior-Point Methods for students who have various backgrounds or who are not necessarily mathematics majors

Asymptotic behavior of underlying NT paths in interior point methods for monotone semidefinite linear complementarity problems

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