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

New predictor-corrector interior-point algorithm for symmetric cone horizontal linear complementarity problems

In this paper we propose a new predictor-corrector interior-point algorithm for solving P_* (κ) horizontal linear complementarity problems defined on a Cartesian product of symmetric cones, which is not based on a usual barrier function. We generalize the predictor-corrector algorithm introduced in [13] to P_* (κ)-linear horizontal complementarity problems on a Cartesian product of symmetric cones. We apply the algebraic equivalent transformation technique proposed by Darvay [9] and we use the function φ(t)=t-√t in order to determine the new search directions. In each iteration the proposed algorithm performs one predictor and one corrector step. We prove that the predictor-corrector interior-point algorithm has the same complexity bound as the best known interior-point algorithms for solving these types of problems. Furthermore, we provide a condition related to the proximity and update parameters for which the introduced predictor-corrector algorithm is well defined

New Predictor-Corrector Algorithm for Symmetric Cone Horizontal Linear Complementarity Problems

We propose a new predictor-corrector interior-point algorithm for solving Cartesian symmetric cone horizontal linear complementarity problems, which is not based on a usual barrier function. We generalize the predictor-corrector algorithm introduced in Darvay et al. (SIAM J Optim 30:2628-2658, 2020) to horizontal linear complementarity problems on a Cartesian product of symmetric cones. We apply the algebraically equivalent transformation technique proposed by Darvay (Adv Model Optim 5:51-92, 2003), and we use the difference of the identity and the square root function to determine the new search directions. In each iteration, the proposed algorithm performs one predictor and one corrector step. We prove that the predictor-corrector interior-point algorithm has the same complexity bound as the best known interior-point methods for solving these types of problems. Furthermore, we provide a condition related to the proximity and update parameters for which the introduced predictor-corrector algorithm is well defined

Unified approach of primal-dual interior-point algorithms for a new class of AET functions

We propose new short-step interior-point algorithms (IPAs) for solving P_* (κ)-linear complementarity problems (LCPs). In order to define the search directions we use the algebraic equivalent transformation technique (AET) of the system which characterizes the central path. A novelty of the paper is that we introduce a new class of AET functions. We present the complexity analysis of the IPAs that use this general class of functions in the AET technique. Furthermore, we also deal with a special case, namely φ(t)=t^2-t+√t. This function differs from the ones used in the literature in the sense that it has inflection point. It does not belong to the class of concave functions determined by Haddou et al. Furthermore, the kernel function corresponding to this AET function is neither eligible nor self-regular kernel function. We prove that the IPAs using any member φ of this new class of AET functions have polynomial iteration complexity in the size of the problem, bit length of the integral data and in the parameter κ. Beside this, we also provide numerical results that show the efficiency of the introduced methods

Unified Approach of Interior-Point Algorithms for P_*(\kappa )-LCPs Using a New Class of Algebraically Equivalent Transformations

We propose new short-step interior-point algorithms (IPAs) for solving P_*(\kappa ) P ∗ ( κ ) -linear complementarity problems (LCPs). In order to define the search directions, we use the algebraic equivalent transformation (AET) technique of the system describing the central path. A novelty of the paper is that we introduce a whole, new class of AET functions for which a unified complexity analysis of the IPAs is presented. This class of functions differs from the ones used in the literature for determining search directions, like the class of concave functions determined by Haddou, Migot and Omer, self-regular functions, eligible kernel and self-concordant functions. We prove that the IPAs using any member \varphi φ of the new class of AET functions have polynomial iteration complexity in the size of the problem, in starting point’s duality gap, in the accuracy parameter and in the parameter \kappa κ

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

Interior-point algorithms for symmetric cone horizontal linear complementarity problems based on a new class of algebraically equivalent transformations

We introduce interior-point algorithms (IPAs) for solving P_* (κ)-horizontal linear complementarity problems over Cartesian product of symmetric cones. We generalize the primal-dual IPAs proposed recently by Illés et al. [21] to P_* (κ)-horizontal linear complementarity problems over Cartesian product of symmetric cones. In the algebraic equivalent transformation (AET) technique we use a modification of the class of AET functions proposed by Illés et al. [21]. In the literature, there are only few classes of functions for determination of search directions. The class of AET functions used in this paper differs from the other classes appeared in the literature. We prove that the proposed IPAs have the same complexity bound as the best known interior-point methods for solving these types of problems