Unified Approach of Interior-Point Algorithms for $$P_*(\kappa )$$-LCPs Using a New Class of Algebraically Equivalent Transformations
提出一类新的代数等价变换函数，用于定义内点算法的搜索方向，并证明使用该类函数中任意成员的短步内点算法在求解P_*(κ)-线性互补问题时具有多项式迭代复杂度。
Abstract We propose new short-step interior-point algorithms (IPAs) for solving $$P_*(\kappa )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>κ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -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 $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> 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 $$\kappa $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> .