A Polynomial Predictor-Corrector Trust-Region Algorithm for Linear Programming

Lan, Guanghui; Monteiro, Renato D. C.; Tsuchiya, Takashi

doi:10.1137/070693461

Cited by 10 publications

(26 citation statements)

References 30 publications

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“…Thus, Monteiro and Tsuchiya's question can be rephrased as to whether there exists an exact LP algorithm with running time poly(n, m, logχ * A ). Substantial progress on this question was made in the followup works [MT05,LMT09]. [MT05] showed that the number of iterations of the MTY predictor-corrector algorithm [MTY93] can get from µ 0 > 0 to η > 0 on the central path in O n 3.5 logχ * + min{n 2 log log(µ 0 /η), log(µ 0 /η)} iterations.…”

Section: The Question Of Scale Invariance Andχ *mentioning

confidence: 99%

“…Moreover, on the "straight" parts of the path, the rate of progress amplifies geometrically, thus attaining a log log convergence on these parts. Subsequently [LMT09] developed an affine invariant trust region step, which traverses the full path in O(n 3.5 log(χ * A + n)) iterations. However, each iteration is weakly polynomial in b and c. The question of developing an LP algorithm with complexity bound poly(n, m, logχ * A ) thus remained open.…”

Section: The Question Of Scale Invariance Andχ *mentioning

confidence: 99%

See 1 more Smart Citation

A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix

Dadush

Huiberts

Natura

et al. 2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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Following the breakthrough work of Tardos (Oper. Res. '86) in the bit-complexity model, Vavasis and Ye (Math. Prog. '96) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) max c ⊤ x, Ax = b, x ≥ 0, A ∈ R m×n , Vavasis and Ye developed a primal-dual interior point method using a 'layered least squares' (LLS) step, and showed that O(n 3.5 log(χA + n)) iterations suffice to solve (LP) exactly, whereχA is a condition measure controlling the size of solutions to linear systems related to A. Monteiro and Tsuchiya (SIAM J. Optim. '03), noting that the central path is invariant under rescalings of the columns of A and c, asked whether there exists an LP algorithm depending instead on the measureχ * A , defined as the minimumχAD value achievable by a column rescaling AD of A, and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an O(m 2 n 2 + n 3) time algorithm which works on the linear matroid of A to compute a nearly optimal diagonal rescaling D satisfyingχAD ≤ n(χ *) 3. This algorithm also allows us to approximate the value ofχA up to a factor n(χ *) 2. This result is in (surprising) contrast to that of Tunçel (Math. Prog. '99), who showed NPhardness for approximatingχA to within 2 poly(rank(A)). The key insight for our algorithm is to work with ratios gi/gj of circuits of A-i.e., minimal linear dependencies Ag = 0-which allow us to approximate the value ofχ * A by a maximum geometric mean cycle computation in what we call the 'circuit ratio digraph' of A. While this resolves Monteiro and Tsuchiya's question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved O(n 2.5 log n log(χ * A + n)) iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor n/ log n improvement on the iteration complexity bound of the original Vavasis-Ye algorithm. * Supported by the ERC Starting Grants ScaleOpt and QIP.

show abstract

Section: The Question Of Scale Invariance Andχ *mentioning

confidence: 99%

Section: The Question Of Scale Invariance Andχ *mentioning

confidence: 99%

A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix

Dadush

Huiberts

Natura

et al. 2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

View full text Add to dashboard Cite

show abstract

“…Thus, Monteiro and Tsuchiya's question can be rephrased as to whether there exists an exact LP algorithm with running time poly(n, m, log χ * A ). Substantial progress on this question was made in the followup works [MT05,LMT09]. [MT05] showed that the number of iterations of the MTY predictor-corrector algorithm [MTY93] can get from µ 0 > 0 to η > 0 on the central path in O n 3.5 log χ * + min{n 2 log log(µ 0 /η), log(µ 0 /η)} iterations.…”

Section: The Question Of Scale Invariance and χ *mentioning

confidence: 99%

“…Moreover, on the "straight" parts of the path, the rate of progress amplifies geometrically, thus attaining a log log convergence on these parts. Subsequently [LMT09] developed an affine invariant trust region step, which traverses the full path in O(n 3.5 log( χ * A + n)) iterations. However, each iteration is weakly polynomial in b and c. The question of developing an LP algorithm with complexity bound poly(n, m, log χ * A ) thus remained open.…”

Section: The Question Of Scale Invariance and χ *mentioning

confidence: 99%

A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix

Dadush¹,

Huiberts²,

Natura³

et al. 2019

Preprint

View full text Add to dashboard Cite

Following the breakthrough work of Tardos (Oper. Res. '86) in the bit-complexity model, Vavasis and Ye (Math. Prog. '96) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) max c ⊤ x, Ax = b, x ≥ 0, A ∈ R m×n , Vavasis and Ye developed a primal-dual interior point method using a 'layered least squares' (LLS) step, and showed that O(n 3.5 log( χA + n)) iterations suffice to solve (LP) exactly, where χA is a condition measure controlling the size of solutions to linear systems related to A.Monteiro and Tsuchiya (SIAM J. Optim. '03), noting that the central path is invariant under rescalings of the columns of A and c, asked whether there exists an LP algorithm depending instead on the measure χ * A , defined as the minimum χAD value achievable by a column rescaling AD of A, and gave strong evidence that this should be the case. We resolve this open question affirmatively.Our first main contribution is an O(m 2 n 2 + n 3 ) time algorithm which works on the linear matroid of A to compute a nearly optimal diagonal rescaling D satisfying χAD ≤ n( χ * ) 3 . This algorithm also allows us to approximate the value of χA up to a factor n( χ * ) 2 . This result is in (surprising) contrast to that of Tunçel (Math. Prog. '99), who showed NPhardness for approximating χA to within 2 poly(rank(A)) . The key insight for our algorithm is to work with ratios gi/gj of circuits of A-i.e., minimal linear dependencies Ag = 0-which allow us to approximate the value of χ *A by a maximum geometric mean cycle computation in what we call the 'circuit ratio digraph' of A.While this resolves Monteiro and Tsuchiya's question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved O(n 2.5 log n log( χ * A + n)) iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor n/ log n improvement on the iteration complexity bound of the original Vavasis-Ye algorithm.

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“…Lan, Monteiro and Tsuchiya [22] gave a scaling invariant trust region IPM taking O(n 3.5 log( χ * A + n)) iterations. Here, χ *…”

Section: Introductionmentioning

confidence: 99%

Interior point methods are not worse than Simplex

Allamigeon¹,

Dadush²,

Loho³

et al. 2022

Preprint

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Whereas interior point methods provide polynomial-time linear programming algorithms, the running time bounds depend on bit-complexity or condition measures that can be unbounded in the problem dimension. This is in contrast with the simplex method that always admits an exponential bound. We introduce a new polynomial-time path-following interior point method where the number of iterations also admits a combinatorial upper bound O(2 n n 1.5 log n) for an n-variable linear program in standard form. This complements previous work by Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018) that exhibited a family of instances where any path-following method must take exponentially many iterations.The number of iterations of our algorithm is at most O(n 1.5 log n) times the number of segments of any piecewise linear curve in the wide neighborhood of the central path. In particular, it matches the number of iterations of any path following interior point method up to this polynomial factor. The overall exponential upper bound derives from studying the 'max central path', a piecewise-linear curve with the number of pieces bounded by the total length of 2n shadow vertex simplex paths.From the existence of a line segment in the wide neighborhood we derive strong implications on the structure of the corresponding segment of the central path. Our algorithm is able to detect this structure from the local geometry at the current iterate, and constructs a step direction that descends along this segment. The bound O(n 1.5 log n) that applies for arbitrarily long line segments is derived from a combinatorial progress measure.Our algorithm falls into the family of layered least squares interior point methods introduced by Vavasis and Ye (Math. Prog. 1996). In contrast to previous layered least squares methods that partition the kernel of the constraint matrix into coordinate subspaces, our method creates layers based on a general subspace providing more flexibility. Our result also implies the same bound on the number of iterations of the trust region interior point method by Lan, Monteiro, and Tsuchiya (SIOPT 2009).

show abstract

A Polynomial Predictor-Corrector Trust-Region Algorithm for Linear Programming

Cited by 10 publications

References 30 publications

A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix

A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix

A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix

Interior point methods are not worse than Simplex

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