Introduction Central path behavior Central path total curvature

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions Central path curvature and iteration-complexity for redundant Klee-Minty cubes A. Deza T. Terlaky Y. Zinchenko May 2006 Yuriy. Zinchenko, Mc. Master. University Yuriy Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions Introduction linear programming and central path curvature and path-following “average” total curvature of the central path Central path behavior redundant Klee-Minty cubes redundancy effects bending the central path Central path total curvature for redundant cubes getting the bound likely algorithmic implications Conclusions Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions linear programming and central path curvature and path-following “average” total curvature of the central path Linear programming and central path Feasible interior-point path-following methods • start from the analytic center • follow the central path to an optimal solution. are polynomial time algorithms for linear optimization number of iterations N: number of inequalities L: input-data bit-length analytic center optimal solution central path 0 : central path parameter Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions linear programming and central path curvature and path-following “average” total curvature of the central path Linear programming, curvature of the central path – related work • Nesterov and Todd – Riemannian geometry of the central path • Todd and Ye – order-N 1/3 lower bound iteration-complexity for a certain wide-neighborhood interior-point method • Megiddo and Shub – order-n central path total curvature lower bound • Dedieu, Malajovich, Shub – central path total “average” curvature • Sonnevend et al – a curvature measure for the central path • Vavasis and Ye – cross-over events • Monteiro and Tsuchiya – relationship between the two above • many others … Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions linear programming and central path curvature and path-following “average” total curvature of the central path Curvature and path-following Definition: Let : [a, b] n be C 2, parameterized by its arc length t (note (t) 0, t [a, b], in fact, ║ (t)║ = 1) The curvature at t is y = sin (x) and the total curvature Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions linear programming and central path curvature and path-following “average” total curvature of the central path Curvature and path-following K Predictor-corrector type path-following in the Euclidean neighborhood make a tangent predictor step form an orthogonal corrector step iterate Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions linear programming and central path curvature and path-following “average” total curvature of the central path “Average” total curvature of the central path Theorem (Dedieu, Malajovich, Shub): If A is injective, then the average total curvature of the central path is ≤ 2 n (3) (1) (2) c Average over bounded polytopes Conjecture (Dedieu, Shub): The worst-case total curvature is O(n) Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions redundant Klee-Minty cubes redundancy effects bending the central path Redundant Klee-Minty cubes min subject to for repeated h 1 times repeated h 2 times repeated hn times n variables 2 n + h 1 + h 2 +…+ hn constrains We denote its feasible region C Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions redundant Klee-Minty cubes redundancy effects bending the central path Redundant Klee-Minty cubes : small positive factor by which the Klee-Minty cube is squashed : defines the polyhedral neighborhoods h = (h 1, . . . , hn ) : number of redundant constraints d : distance of the redundant constraints h 1 d d Yuriy Zinchenko, Mc. Master University h 2 Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions redundant Klee-Minty cubes redundancy effects bending the central path Redundancy effects Animation 1: effect of d Animation 2: effect of h = (h 1, . . . , hn ) Animation 3: effect of Animation 4: effect of Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions redundant Klee-Minty cubes redundancy effects bending the central path Redundancy effects Animation 1: effect of d Animation 2: effect of h = (h 1, . . . , hn ) Animation 3: effect of Animation 4: effect of Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions redundant Klee-Minty cubes redundancy effects bending the central path Redundancy effects Animation 1: effect of d Animation 2: effect of h = (h 1, . . . , hn ) Animation 3: effect of Animation 4: effect of Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions redundant Klee-Minty cubes redundancy effects bending the central path Redundancy effects Animation 1: effect of d Animation 2: effect of h = (h 1, . . . , hn ) Animation 3: effect of Animation 4: effect of Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions redundant Klee-Minty cubes redundancy effects bending the central path Redundancy effects Animation 1: effect of d Animation 2: effect of h = (h 1, . . . , hn ) Animation 3: effect of Animation 4: effect of Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions redundant Klee-Minty cubes redundancy effects bending the central path Bending the central path Q: Can the central path be bent along the edge-path followed by the simplex method on the Klee-Minty cube? (can the central path visit an arbitrary small neighborhood of all 2 n vertices? ) Starting point Yes! - if we an of carefully exponential redundant 1 add number constrains 0. 5 Optimal point 0 0 In particular, suffices N = O(n 322 n) 1 0. 5 1 Yuriy Zinchenko, Mc. Master University 0 Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions redundant Klee-Minty cubes redundancy effects bending the central path Bending the. For central Theorem: , > 0 path such that + < ½, if satisfies and where 0 A= 0 0 0 0 then the central path visits the - neighborhoods of all the vertices of C. Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions getting the bound likely algorithmic implications Bounding the total central path curvature from below 3 Recall: (a) and (b) 2 1 for our construction, adjacent to each vertex have a “polyhedral elbow” Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions getting the bound likely algorithmic implications Bounding the total central path curvature from below 2 2 D: 2 b (0, 0) a 1 (t 2) By Mean-Value Theorem, • 1(t 1) = a implies t 1 > a Thus, t 2 (< t 1) such that | 2(t 2)| < b/a • similarly, t 3 such that | 1(t 3)| < b/a Yuriy Zinchenko, Mc. Master University 1 (t 3) K > /2 – b/a Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions getting the bound likely algorithmic implications Bounding the total central path curvature from below 3 3 D: K > arccos(1 -1/(n-1))– b/a (t 1) 2 1 (t 2) (t 3) Theorem: The total curvature of the central path may be exponential in n (e. g. , 1. 5 n ) Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions getting the bound likely algorithmic implications Likely algorithmic implications theoretical iteration-complexity upper bound: likely redundant Klee-Minty iteration-complexity lower bound: the gap might be essentially closed Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes

Introduction Central path behavior Central path total curvature for redundant cubes Conclusions A counterexample to O(n) conjecture on the worst-case total curvature of the central path Likely to be the worst-case known iteration-complexity instance for linear programming, matching theoretical bound Yuriy Zinchenko, Mc. Master University Central path curvature and iteration-complexity for Klee-Minty cubes