Unified Approach of Interior-Point Algorithms for P∗(κ)-LCPs Using a New Class of Algebraically Equivalent Transformations

• Published in: Journal of optimization theory and applications Vol. 202; no. 1; pp. 27 - 49
• Main Authors: Illés, Tibor, Rigó, Petra Renáta, Török, Roland
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2024; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Complexity; Engineering; Equivalence; Linear programming; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Parameters; Polynomials; Theory of Computation; Transformations (mathematics); Interior-point algorithm; Algebraic equivalent transformation technique; Linear complementarity problems; New class of AET functions
• ISSN: 0022-3239; 1573-2878; 1573-2878
• DOI: 10.1007/s10957-023-02232-1

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 (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 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 κ .