A New Complexity Result on Solving the Markov Decision Problem
Mathematics of Operations Research
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In this paper we present a new iteration-complexity bound for the Mizuno--Todd--Ye predictor-corrector (MTY P-C) primal-dual interior-point algorithm for linear programming. The analysis of the paper is based on the important notion of crossover events introduced by Vavasis and Ye. For a standard form linear program min{cTx:Ax = b, x \ge 0} with decision variable $x\in \Re^n$, e show that the MTY P-C algorithm, started from a well-centered interior-feasible solution with duality gap $n \mu_0$, finds an interior-feasible solution with duality gap less than $n\eta$ in ${\cal O}(T(\mu_0/\eta)+n^{3.5}\log({{\bar\chi^*_A}}))$ iterations, where $T(t)\equiv\min\{n^2\log(\log t) ,\log t\}$ for all $t0$ and ${{\bar\chi^*_A}}$ is a scaling invariant condition number associated with the matrix $A$. More specifically, ${{\bar\chi^*_A}}$ is the infimum of all the conditions numbers ${\bar\chi}_{AD}$, where D varies over the set of positive diagonal matrices. Under the setting of the Turing machine model, our analysis yields an ${\cal O}(n^{3.5}L_A+ \min\{n^2\log L,L\})$ iteration-complexity bound for the MTY P-C algorithm to find a primal-dual optimal solution, where LA and L are the input sizes of the matrix A and the data (A,b,c), respectively. This contrasts well with the classical iteration-complexity bound for the MTY P-C algorithm, which depends linearly on L instead of log L.