A little theorem of the bigℳ in interior point algorithms

When we apply interior point algorithms to various problems including linear programs, convex quadratic programs, convex programs and complementarity problems, we often embed an original problem to be solved in an artificial problem having a known interior feasible solution from which we start the algorithm. The artificial problem involves a constantℳ (or constants) which we need to choose large enough to ensure the equivalence between the artificial problem and the original problem. Theoretically, we can always assign a positive number of the order O(2L) toℳ in linear cases, whereL denotes the input size of the problem. Practically, however, such a large number is impossible to implement on computers. If we choose too largeℳ, we may have numerical instability and/or computational inefficiency, while the artificial problem withℳ not large enough will never lead to any solution of the original problem. To solve this difficulty, this paper presents “a little theorem of the bigℳ”, which will enable us to find whetherℳ is not large enough, and to updateℳ during the iterations of the algorithm even if we start with a smallerℳ. Applications of the theorem are given to a polynomial-time potential reduction algorithm for positive semi-definite linear complementarity problems, and to an artificial self-dual linear program which has a close relation with the primal—dual interior point algorithm using Lustig's limiting feasible direction vector.