long and winding central paths - journée de rentrée du cmap

Introduction The tropical analogue of linear programming Tropicalizing the central path Tropical lower bound on the complexity of IPMs Conclusion

Long and winding central pathsJournée de rentrée du CMAP

Xavier Allamigeon1 Pascal Benchimol2 Stéphane Gaubert1Michael Joswig3

1INRIA Saclay – Ile-de-France and CMAP, Ecole Polytechnique, CNRS2EDF Lab

3Institut für Mathematik, Technische Universität Berlin

October 3rd, 2018

Long and winding central paths | Allamigeon, Benchimol, Gaubert, Joswig | 1/37

Motivation: the complexity of linear programming

DefinitionA linear program is an optimization problem of the form:

minimize c⊤xsubject to Ax ⩽ b, x ∈ Rn

where A ∈ Rm×n, b ∈ Rm, c ∈ Rn.

minimize x+ 3ysubject to x+ y ⩾ 3

23 ⩾ x+ 3y4x ⩾ 1+ y

11+ y ⩾ 2x2y ⩾ 2

Motivation: the complexity of linear programming (2)

Theorem (Khachiyan, 1980)Liner programming can be solved in polynomial time.

= execution time bounded by a polynomialP(m, n, L)

where:• m = nb of inequalities• n = dimension of the space• L = total size of the coefficients Aij, bi, cj in bits (sum of their log2).

= strongly polynomial complexity• polynomial time• number of arithmetic operations bounded by a polynomial in the dimension ofthe problem, i.e.∼ m× n

9th Smale’s ProblemIs there a strongly polynomial algorithm for linear programming?

9th Smale’s Problem for 21st CenturyIs there a strongly polynomial algorithm for linear programming?

Existing algorithms for LP:• simplex method (Dantzig, 1947)• ellipsoid method (Khachiyan, 1980)• interior-point method (Karmarkar, 1984)

In practiceThe simplex method and the interior-point method are very efficient:• scale to very large instances (105 variables)• usually perform at most 50-100 iterations

Purpose of this talk

What can we say about interior-point methods?

polynomial time

Log-barrier interior point methods for linear programming

Consider the following LP

minimize c⊤xsubject to Ax ⩽ b , x ∈ Rn (P)

Logarithmic barrier penalization

minimize c⊤x− µm∑i=1

log(bi − Aix)

subject to Aix < bi , i = 1, . . . ,m(Pµ)

where µ > 0.

For all µ > 0, Problem (Pµ) has a unique optimal solution xµ.Moreover, xµ converges to an optimal solution of (P) when µ → 0+.

DefinitionThe central path is the curve µ 7→ xµ.

Principle of log-barrier IPMFollow the central path with µ 0 up tothe solution of (P)

approximately

• stay in a certain “neighborhood” ofthe central path

• use Newton descent directions toiterate

• different choices of steps (short, long,predictor/corrector, etc)

log(bi − Aix)

where µ > 0.For all µ > 0, Problem (Pµ) has a unique optimal solution xµ.Moreover, xµ converges to an optimal solution of (P) when µ → 0+.

approximately

log(bi − Aix)

where µ > 0.For all µ > 0, Problem (Pµ) has a unique optimal solution xµ.Moreover, xµ converges to an optimal solution of (P) when µ → 0+.

approximately

log(bi − Aix)

where µ > 0.

approximately

log(bi − Aix)

where µ > 0.

approximately

log(bi − Aix)

where µ > 0.

approximately

log(bi − Aix)

where µ > 0.

Principle of log-barrier IPMFollow the central path with µ 0 up tothe solution of (P) approximately.

log(bi − Aix)

where µ > 0.

log(bi − Aix)

where µ > 0.

log(bi − Aix)

where µ > 0.

log(bi − Aix)

where µ > 0.

The curvature of the central path

Intimately related with the geometry of the central path!According to Bayer and Lagarias (1989), the central path is

[…] a fundamental mathematical object underlying Karmarkar’s algorithm and thatthe good convergence properties of Karmarkar’s algorithm arise from good geometricproperties […]

=⇒ motivated several works on the total curvature of the central path

Related workGiven a linear program defined by m inequalities in dimension n,• Dedieu and Shub (2005) conjectured that the total curvature is in O(n)

• Dedieu, Malajovich, and Shub (2005) showed that this is true “on average”, seealso (De Loera, Sturmfels, and Vinzant, 2012)

• Deza, Terlaky, and Zinchenko (2009) built a counter-example with total curvatureexponential in n, with m ∈ Ω(2n) constraints

Continuous analogue of Hirsch conjecture (Deza, Terlaky, and Zinchenko, 2009)The total curvature of the central path is bounded by O(m).

ObstacleThere is no explicit relation between the curvature of the central path and thecomplexity of interior point methods.

Intimately related with the geometry of the central path!=⇒ motivated several works on the total curvature of the central path

Intimately related with the geometry of the central path!=⇒ motivated several works on the total curvature of the central pathRelated workGiven a linear program defined by m inequalities in dimension n,• Dedieu and Shub (2005) conjectured that the total curvature is in O(n)• Dedieu, Malajovich, and Shub (2005) showed that this is true “on average”, seealso (De Loera, Sturmfels, and Vinzant, 2012)

Related workGiven a linear program defined by m inequalities in dimension n,• Dedieu and Shub (2005) conjectured that the total curvature is in O(n)• Dedieu, Malajovich, and Shub (2005) showed that this is true “on average”, seealso (De Loera, Sturmfels, and Vinzant, 2012)

This talk

Main resultLog-barrier interior point methods are not strongly polynomial.

Consider the following parametric family of LPs:minimize x1subject to x1 ⩽ t2

x2 ⩽ tx2j+1 ⩽ t x2j−1 , x2j+1 ⩽ t x2jx2j+2 ⩽ t1−1/2j (x2j−1 + x2j)x2r−1 ⩾ 0 , x2r ⩾ 0

1 ⩽ j < rLWr(t)

CorollaryThe number of iterations of any log-barrier interior point algorithm is exponential in r on thelinear program LWr(t), provided that t > 1 is sufficiently large.

RemarkThe total curvature of the central path of LWr(t) is also exponential.

This talk

1 ⩽ j < rLWr(t)

This talk

1 ⩽ j < rLWr(t)

This talk

1 ⩽ j < rLWr(t)

This talk

1 ⩽ j < rLWr(t)

Outline of the talk

Key ingredientWe set up a tropical lower bound on the iteration complexity of log-barrier IPM.

1 The tropical analogue of linear programming

2 Tropicalizing the central path

3 Tropical lower bound on the complexity of IPMs

4 Conclusion

Outline of the talk

4 Conclusion

Tropical algebra and tropical polyhedra

Tropical algebra refers to the semiring T := R ∪ −∞ where:• the addition x⊕ y is max(x, y)• the multiplication x⊙ y is x+ y

Tropical operations extend to matrices and vectors:

A⊕ B = (Aij ⊕ Bij)ij A⊙ B =(⊕

kAik ⊙ Bkj

Tropical algebra and tropical polyhedra

Tropical algebra refers to the semiring T := R ∪ −∞ where:• the addition x⊕ y is max(x, y)• the multiplication x⊙ y is x+ y

Tropical operations extend to matrices and vectors:

A⊕ B = (Aij ⊕ Bij)ij A⊙ B =(⊕

kAik ⊙ Bkj

Tropical algebra and tropical polyhedra (2)

A tropical polyhedron is the set of solutions x ∈ Tn of a system of the form:

A+ ⊙ x⊕ b+ ⩾ A− ⊙ x⊕ b−

with A+,A− ∈ Tm×n and b+, b− ∈ Tm.

A+ ⊙ x⊕ b+ ⩾ A− ⊙ x⊕ b−

( 0 1−∞ 0−∞ 0−∞ −∞4 −∞

)⊙(x1x2

)⊕(−∞

−∞48

−∞

⩾( −∞ −∞

−10 −∞−3 −∞0 2

−∞ 0

)⊙(x1x2

)⊕( 3

1−∞−∞5

A+ ⊙ x⊕ b+ ⩾ A− ⊙ x⊕ b−

max(x1, 1+ x2) ⩾ 3x2 ⩾ max(−10+ x1, 1)

max(x2, 4) ⩾ −3+ x18 ⩾ max(x1, 2+ x2)

4+ x1 ⩾ max(x2, 5)

A+ ⊙ x⊕ b+ ⩾ A− ⊙ x⊕ b−

max(x1, 1+ x2) ⩾ 3x2 ⩾ max(−10+ x1, 1)

max(x2, 4) ⩾ −3+ x18 ⩾ max(x1, 2+ x2)

4+ x1 ⩾ max(x2, 5)

A+ ⊙ x⊕ b+ ⩾ A− ⊙ x⊕ b−

max(x1, 1+ x2) ⩾ 3x2 ⩾ max(−10+ x1, 1)

max(x2, 4) ⩾ −3+ x18 ⩾ max(x1, 2+ x2)

4+ x1 ⩾ max(x2, 5)

A+ ⊙ x⊕ b+ ⩾ A− ⊙ x⊕ b−

max(x1, 1+ x2) ⩾ 3x2 ⩾ max(−10+ x1, 1)

max(x2, 4) ⩾ −3+ x18 ⩾ max(x1, 2+ x2)

4+ x1 ⩾ max(x2, 5)

A+ ⊙ x⊕ b+ ⩾ A− ⊙ x⊕ b−

max(x1, 1+ x2) ⩾ 3x2 ⩾ max(−10+ x1, 1)

max(x2, 4) ⩾ −3+ x18 ⩾ max(x1, 2+ x2)

4+ x1 ⩾ max(x2, 5)

A+ ⊙ x⊕ b+ ⩾ A− ⊙ x⊕ b−

max(x1, 1+ x2) ⩾ 3x2 ⩾ max(−10+ x1, 1)

max(x2, 4) ⩾ −3+ x18 ⩾ max(x1, 2+ x2)

4+ x1 ⩾ max(x2, 5)

Tropical polyhedra vs convex polyhedra

Alternative definitionTropical polyhedra= limits of deformations of classical polyhedra through the map

logt : x 7→ log xlog t

Maslov dequantization

max(logt x, logt y) ⩽ logt(x+ y) ⩽ max(logt x, logt y) + logt 2

logt(x · y) = logt x+ logt y

Our goal: tropicalizing the central pathStudy the central path of a parametric family of LPs:

minimize c(t)⊤xsubject to A(t)x ⩽ b(t) , x ⩾ 0

and its deformation by the map logt(·), when t goes to+∞.

x1 + tx2 ⩾ t3

x2 ⩾ t−10x1 + tx2 + t4 ⩾ t−3x1

t8 ⩾ x1 + t2x2t4x1 ⩾ x2 + t5

logt x1

logt x2

x1 + tx2 ⩾ t3

x2 ⩾ t−10x1 + tx2 + t4 ⩾ t−3x1

t8 ⩾ x1 + t2x2t4x1 ⩾ x2 + t5

logt(·)t = 5

logt x1

logt x2

x1 + tx2 ⩾ t3

x2 ⩾ t−10x1 + tx2 + t4 ⩾ t−3x1

t8 ⩾ x1 + t2x2t4x1 ⩾ x2 + t5

logt(·)t = 10

logt x1

logt x2

x1 + tx2 ⩾ t3

x2 ⩾ t−10x1 + tx2 + t4 ⩾ t−3x1

t8 ⩾ x1 + t2x2t4x1 ⩾ x2 + t5

logt(·)t = 100

logt x1

logt x2

x1 + tx2 ⩾ t3

x2 ⩾ t−10x1 + tx2 + t4 ⩾ t−3x1

t8 ⩾ x1 + t2x2t4x1 ⩾ x2 + t5

logt(·)t = 1000

logt x1

logt x2

x1 + tx2 ⩾ t3

x2 ⩾ t−10x1 + tx2 + t4 ⩾ t−3x1

t8 ⩾ x1 + t2x2t4x1 ⩾ x2 + t5

logt(·)t → +∞

logt x1

logt x2

A possible setting for tropicalizationThe entries of A, b and c are taken from a field K of “Puiseux series”.

DefinitionAn absolutely convergent generalized real Puiseux series is a series of the form:

x = c1tλ1 + c2tλ2 + . . . ,

where the ci are nonzero reals, and:• λ1 > λ2 > . . . is a strictly decreasing sequence of real numbers that is eitherfinite or unbounded;

• the series is absolutely convergent for t large enough.

The field K is totally ordered:

x > 0 if the coefficient of the leading term in x is positive.

RemarkThe order over K is compatible with the asymptotic ordering:

x ⩽ y iff x(t) ⩽ y(t) for all t ≫ 1 .

x = c1tλ1 + c2tλ2 + . . . ,

A possible setting for tropicalization (2)

The field K is totally ordered:x > 0 if the coefficient of the leading term in x is positive.

Theorem (van den Dries and Speissegger, 1998)K is a real closed field.

By Tarki’s principleAll the “good” properties known on polyhedra and linear programming remain validover the field K.

=⇒ a LP over K encodes a parametric family of LPs over R:

minimize c⊤xsubject to Ax ⩽ b , x ∈ Kn

minimize c(t)⊤xsubject to A(t)x ⩽ b(t) , x ∈ Rn

⩾0Long and winding central paths | Allamigeon, Benchimol, Gaubert, Joswig | 14/37

A possible setting for tropicalization (3)K is a nonarchimedean field:The valuation of x ∈ K is defined as the leading exponent in the series, orequivalently:

val(x) := limt→+∞

logt |x(t)| .

Example

x = c1tλ1 + c2tλ2 + . . . where λ1 > λ2 > . . . val(x) = λ1

The valuation maps the “classical” laws to the tropical ones:

∀x, y ∈ K⩾0,

val(x+ y) = max(val(x), val(y))val(x · y) = val(x) + val(y)

max(logt x(t), logt y(t)) ⩽ logt(x(t) + y(t)) ⩽ max(logt x(t), logt y(t)) + logt 2

logt(x(t) · y(t)) = logt x(t) + logt y(t)

logt |x(t)| .

Example

∀x, y ∈ K⩾0,

logt |x(t)| .

Example

∀x, y ∈ K⩾0,

logt |x(t)| .

Example

∀x, y ∈ K⩾0,

logt(x(t) · y(t)) = logt x(t) + logt y(t)Long and winding central paths | Allamigeon, Benchimol, Gaubert, Joswig | 15/37

Proposition (Develin and Yu, 2007)LetP ⊂ Kn

⩾0 be a convex polyhedron. Then val(P) is a tropical polyhedron.

Every polyhedronP := x ∈ Kn⩾0 : Ax ⩽ b gives rise to a parametric family of real

convex polyhedra:

P(t) :=x ∈ Rn

⩾0 : A(t)x ⩽ b(t)

Theorem

val(P) = limt→+∞

logt(P(t)) .

Moreover, when the entries of A and b are monomials, we have

d(logt P(t), val(P)) ⩽ logt((n+ 1)2(n!)4

)provided that t is large enough.

convex polyhedra:

P(t) :=x ∈ Rn

⩾0 : A(t)x ⩽ b(t)

Theorem

logt(P(t)) .

convex polyhedra:

P(t) :=x ∈ Rn

⩾0 : A(t)x ⩽ b(t)

Theorem

logt(P(t)) .

convex polyhedra:

P(t) :=x ∈ Rn

⩾0 : A(t)x ⩽ b(t)

Theorem

logt(P(t)) .

Outline of the talk

4 Conclusion

The central path over Puiseux series

Given A ∈ Km×n, b ∈ Km and c ∈ Kn, consider the following LP:minimize c⊤xsubject to Ax ⩽ b , x ⩾ 0

, w ⩾ 0

LP(A, b, c)

and its dual:maximize −b⊤ysubject to −A⊤y+ s = c , s ⩾ 0 , y ⩾ 0

DualLP(A, b, c)

Proposition-DefinitionUnder mild conditions, for all µ ∈ K>0, the system

Ax+w = b−A⊤y+ s = c

∀j, xjsj = µ

∀i, wiyi = µ

has a unique solution C(µ) = (xµ,wµ, sµ, yµ)

which is called the primal-dual centralpath at µ.

“µ-perturbed” optimality conditions

Given A ∈ Km×n, b ∈ Km and c ∈ Kn, consider the following LP:minimize c⊤xsubject to Ax+w = b , x ⩾ 0 , w ⩾ 0

LP(A, b, c)

DualLP(A, b, c)

∀j, xjsj = µ

∀i, wiyi = µ

Given A ∈ Rm×n, b ∈ Rm and c ∈ Rn, consider the following LP:minimize c⊤xsubject to Ax+w = b , x ⩾ 0 , w ⩾ 0

LP(A, b, c)

DualLP(A, b, c)

Proposition-DefinitionUnder mild conditions, for all µ ∈ R>0, the system

∀j, xjsj = µ

∀i, wiyi = µ

LP(A, b, c)

DualLP(A, b, c)

∀j, xjsj = µ

∀i, wiyi = µ

LP(A, b, c)

DualLP(A, b, c)

∀j, xjsj = µ

∀i, wiyi = µ

LP(A, b, c)

DualLP(A, b, c)

∀j, xjsj = µ

∀i, wiyi = µ

has a unique solution C(µ) = (xµ,wµ, sµ, yµ) which is called the primal-dual centralpath at µ.

LP(A, b, c)

DualLP(A, b, c)

∀j, xjsj = µ

∀i, wiyi = µ

Given A ∈ Km×n, b ∈ Km and c ∈ Kn, consider the following LP:minimize c⊤xsubject to Ax+w = b , x ⩾ 0 , w ⩾ 0

LP(A, b, c)

DualLP(A, b, c)

∀j, xjsj = µ

∀i, wiyi = µ

The tropical central path

Natural question: what is the image under the valuation of the central path?

Relies on the notion of barycenter of a (compact)tropical polyhedron P= greatest point of the set P w.r.t. thecoordinate-wise order⩽

LetP be the feasible set of LP(A, b, c):

P := (x,w) ∈ Kn+m⩾0 : Ax+w = b .

Assume, for simplicity, b, c ⩾ 0.

TheoremThe image under val of the point (xµ,wµ) of the primal central path is given by thebarycenter of the tropical polyhedron:

val(P) ∩ (x,w) ∈ Tn+m : val(c)⊤ ⊙ x ⩽ val(µ) .

P := (x,w) ∈ Kn+m⩾0 : Ax+w = b .

The tropical central path (2)

LetP be the feasible set of LP(A, b, c).Assume, for simplicity, b, c ⩾ 0.

The image under val of the point (sµ, yµ) of the dual central path is given by the barycenter ofthe tropical polyhedron:

val(Q) ∩ (s, y) ∈ Tn+m : val(b)⊤ ⊙ y ⩽ val(µ) .

=⇒ yields a tropical central path λ 7→ C trop(λ):C trop(λ) := val(C(µ)) for any µ such that val(µ) = λ .

RemarkThe tropical central path does not depend on the representation ofP andQ.

LetP be the feasible set of LP(A, b, c) andQ be the feasible set of DualLP(A, b, c).Assume, for simplicity, b, c ⩾ 0.

Example

minimize x1 + t3x2

x1 + x2 ⩽ 2tx1 ⩽ 1+ t2x2tx2 ⩽ 1+ t3x1x1 ⩽ t2x2

x1, x2 ⩾ 0

max(x1, 3+ x2) ⩽ λ

val(P) :

max(x1, x2) ⩽ 01+ x1 ⩽ max(0, 2+ x2)1+ x2 ⩽ max(0, 3+ x1)

x1 ⩽ 2+ x2

λ = 4λ = 3λ = 2λ = 1λ = 0λ = −1

Exampleminimize x1 + t3x2

x1 + x2 ⩽ 2tx1 ⩽ 1+ t2x2tx2 ⩽ 1+ t3x1x1 ⩽ t2x2

x1, x2 ⩾ 0

max(x1, 3+ x2)

⩽ λ

val(P) :

x1 ⩽ 2+ x2

λ = 4λ = 3λ = 2λ = 1λ = 0λ = −1

x1 + x2 ⩽ 2tx1 ⩽ 1+ t2x2tx2 ⩽ 1+ t3x1x1 ⩽ t2x2

x1, x2 ⩾ 0

max(x1, 3+ x2) ⩽ λ

val(P) :

x1 ⩽ 2+ x2

λ = 4λ = 3λ = 2λ = 1λ = 0λ = −1

x1 + x2 ⩽ 2tx1 ⩽ 1+ t2x2tx2 ⩽ 1+ t3x1x1 ⩽ t2x2

x1, x2 ⩾ 0

max(x1, 3+ x2) ⩽ λ

val(P) :

x1 ⩽ 2+ x2

λ = 4

λ = 3λ = 2λ = 1λ = 0λ = −1

x1 + x2 ⩽ 2tx1 ⩽ 1+ t2x2tx2 ⩽ 1+ t3x1x1 ⩽ t2x2

x1, x2 ⩾ 0

max(x1, 3+ x2) ⩽ λ

val(P) :

x1 ⩽ 2+ x2

λ = 4

λ = 3

λ = 2λ = 1λ = 0λ = −1

x1 + x2 ⩽ 2tx1 ⩽ 1+ t2x2tx2 ⩽ 1+ t3x1x1 ⩽ t2x2

x1, x2 ⩾ 0

max(x1, 3+ x2) ⩽ λ

val(P) :

x1 ⩽ 2+ x2

λ = 4λ = 3

λ = 2

λ = 1λ = 0λ = −1

x1 + x2 ⩽ 2tx1 ⩽ 1+ t2x2tx2 ⩽ 1+ t3x1x1 ⩽ t2x2

x1, x2 ⩾ 0

max(x1, 3+ x2) ⩽ λ

val(P) :

x1 ⩽ 2+ x2

λ = 4λ = 3λ = 2

λ = 1

λ = 0λ = −1

x1 + x2 ⩽ 2tx1 ⩽ 1+ t2x2tx2 ⩽ 1+ t3x1x1 ⩽ t2x2

x1, x2 ⩾ 0

max(x1, 3+ x2) ⩽ λ

val(P) :

x1 ⩽ 2+ x2

λ = 4λ = 3λ = 2λ = 1

λ = 0

λ = −1

x1 + x2 ⩽ 2tx1 ⩽ 1+ t2x2tx2 ⩽ 1+ t3x1x1 ⩽ t2x2

x1, x2 ⩾ 0

max(x1, 3+ x2) ⩽ λ

val(P) :

x1 ⩽ 2+ x2

λ = 4λ = 3λ = 2λ = 1λ = 0

λ = −1Long and winding central paths | Allamigeon, Benchimol, Gaubert, Joswig | 21/37

x1 + x2 ⩽ 2tx1 ⩽ 1+ t2x2tx2 ⩽ 1+ t3x1x1 ⩽ t2x2

x1, x2 ⩾ 0

max(x1, 3+ x2) ⩽ λ

val(P) :

x1 ⩽ 2+ x2

λ = 4λ = 3λ = 2λ = 1λ = 0λ = −1

Pathological example of tropical central path

minimize x1subject to x1 ⩽ t2

1 ⩽ j < r

Tropical central pathThe x-component of C trop(λ) is given by

xλ1 = min(λ, 2)

xλ2 = 1

xλ2j+1 = 1+ min(xλ

2j−1, xλ2j)

xλ2j+2 = (1− 1/2j) + max(xλ

2j−1, xλ2j)

1 ⩽ j < r

0 1 20

1 ⩽ j < r

Tropical central pathThe x-component of C trop(λ) is the greatest point of

x1 ⩽ λ

x1 ⩽ 2 , x2 ⩽ 1x2j+1 ⩽ 1+ x2j−1 , x2j+1 ⩽ 1+ x2jx2j+2 ⩽ (1− 1/2j) + max(x2j−1, x2j)

1 ⩽ j < r ,

0 1 20

1 ⩽ j < r

xλ1 = min(λ, 2)

xλ2 = 1

2j−1, xλ2j)

xλ2j+2 = (1− 1/2j) + max(xλ

2j−1, xλ2j)

1 ⩽ j < r

0 1 20

1 ⩽ j < r

xλ1 = min(λ, 2)

xλ2 = 1

2j−1, xλ2j)

xλ2j+2 = (1− 1/2j) + max(xλ

2j−1, xλ2j)

1 ⩽ j < r

0 1 20

1 ⩽ j < r

xλ1 = min(λ, 2)

xλ2 = 1

2j−1, xλ2j)

xλ2j+2 = (1− 1/2j) + max(xλ

2j−1, xλ2j)

1 ⩽ j < r

0 1 20

1 ⩽ j < r

xλ1 = min(λ, 2)

xλ2 = 1

2j−1, xλ2j)

xλ2j+2 = (1− 1/2j) + max(xλ

2j−1, xλ2j)

1 ⩽ j < r

0 1 20

1 ⩽ j < r

xλ1 = min(λ, 2)

xλ2 = 1

2j−1, xλ2j)

xλ2j+2 = (1− 1/2j) + max(xλ

2j−1, xλ2j)

1 ⩽ j < r

0 1 20

Pathological example of tropical central pathProjection onto the (x2r−1, x2r)-plane: staircase with 2r−1 steps

x2r−1

r− 1r− 1(r− 1)

+ 22r−1

(r− 1)+ 4

2r−1

(r− 1)+ 6

2r−1

(r− 1)+ 1

2r−1

(r− 1)+ 3

2r−1

(r− 1)+ 5

2r−1

(r− 1)+ 7

2r−1

λ = 0λ = 1

2r−2

λ = 22r−2

λ = 32r−2

λ = 42r−2

λ = 52r−2

λ = 62r−2

Outline of the talk

4 Conclusion

Classical central path vs tropical central

So far, we looked at the central path of the linear program LP(A, b, c) over Puiseuxseries:

µ ∈ K>0 7→ C(µ)

Associated family of (real) central pathsCentral path of

LP(A(t), b(t), c(t)) ≡ minc(t)⊤x : A(t)x+ w = b(t) , x,w ⩾ 0

denoted by

µ ∈ R>0 7→ Ct(µ)

QuestionConvergence of the image under logt of:• the family of central paths Ct

• the neighborhood of these central paths

to the tropical central path?

µ ∈ K>0 7→ C(µ)

denoted by

µ ∈ R>0 7→ Ct(µ)

to the tropical central path?

µ ∈ K>0 7→ C(µ)

denoted by

µ ∈ R>0 7→ Ct(µ)

to the tropical central path?Long and winding central paths | Allamigeon, Benchimol, Gaubert, Joswig | 25/37

µ ∈ K>0 7→ C(µ)

denoted by

µ ∈ R>0 7→ Ct(µ)

QuestionConvergence of the image under logt of:• the family of central paths Ct• the neighborhood of these central paths

to the tropical central path?Long and winding central paths | Allamigeon, Benchimol, Gaubert, Joswig | 25/37

Neighborhoods of the classical central

Several kinds of neighborhoods used by interior point methods, which result indifferent complexity bounds.

We consider the widest kind of neighborhoods studied in the literature:

N−∞t (µ) :=

(x,w, s, y) ∈ P(t)×Q(t) : µ(x,w, s, y) = µ

and ( xswy ) ⩾ (1− θ)µe

where θ ∈ ]0, 1[ parametrizes the size of the neighborhood, and:

• µ(x,w, s, y) := 1m+n

(x⊤s+ w⊤y

)• e is the all-1 vector.

To summarizeN−∞

t (µ) is a neighborhood of the point Ct(µ) of the central path of

LP(A(t), b(t), c(t)) ≡ minc(t)⊤x | A(t)x+ w = b(t) , x,w ⩾ 0 .

and θ is a fixed parameter in ]0, 1[ related to the size of the neighborhood.

N−∞t (µ) :=

(x,w, s, y) ∈ P(t)×Q(t) : µ(x,w, s, y) = µ

where θ ∈ ]0, 1[ parametrizes the size of the neighborhood, and:

• µ(x,w, s, y) := 1m+n

(x⊤s+ w⊤y

To summarizeN−∞

N−∞t (µ) :=

(x,w, s, y) ∈ P(t)×Q(t) : µ(x,w, s, y) = µ

where θ ∈ ]0, 1[ parametrizes the size of the neighborhood, and:• µ(x,w, s, y) := 1

m+n(x⊤s+ w⊤y

To summarizeN−∞

N−∞t (µ) :=

(x,w, s, y) ∈ P(t)×Q(t) : µ(x,w, s, y) = µ

where θ ∈ ]0, 1[ parametrizes the size of the neighborhood, and:• µ(x,w, s, y) := 1

m+n(x⊤s+ w⊤y

To summarizeN−∞

Uniform convergence to the tropical central

TheoremThe image under logt of the neighborhoods of the central paths

(Ct(·)

)t≫1 uniformly

converges to the tropical central path:

∀µ ∈ R>0 , d∞(logt N−∞t (µ) , C trop(logt µ)) ⩽ logt

(m+ n1− θ

)+ δ(t) .

RemarkThe quantity δ(t) is built from the Hausdorff distance between the sets logt P(t) andvalP and their dual counterparts.

Uniform convergence to the tropical central

TheoremThe image under logt of the neighborhoods of the central paths

(Ct(·)

)t≫1 uniformly

converges to the tropical central path:

∀µ ∈ R>0 , d∞(logt N−∞t (µ) , C trop(logt µ)) ⩽ logt

(m+ n1− θ

)+ δ(t) .

RemarkThe quantity δ(t) is built from the Hausdorff distance between the sets logt P(t) andvalP and their dual counterparts.

Uniform convergence to the tropical central (2)

For all radius ε > 0, if t is sufficiently large, we have

logt( )

Uniform convergence to the tropical central (2)

For all radius ε > 0, if t is sufficiently large, we have

logt( )

Pathological tropical central pathIn the (x2r−1, x2r)-plane,

x2r−1

For t sufficiently large:• the image under logt of the pointsarising in the iterations of the IPM

• as well as the image under logt of thesegments between successive points

• as well as the sequence of tropicalsegments between successive points

are contained in the neighborhood of thetropical central path.

x2r−1

Classical and tropical segmentsDefinitionGiven x, y ∈ Tn, the tropical segment between x and y is defined as:

tsegm(x, y) :=(

λ ⊙ x)⊕(µ ⊙ y

)| λ ⊕ µ = 0

PropositionGiven x, y ∈ Kn

[x(t), y(t)

], tsegm

((logt x(t), logt y(t)

( 1log t

tsegm(x, y) :=(

λ ⊙ x)⊕(µ ⊙ y

)| λ ⊕ µ = 0

[x(t), y(t)

], tsegm

( 1log t

tsegm(x, y) :=(

λ ⊙ x)⊕(µ ⊙ y

)| λ ⊕ µ = 0

[x(t), y(t)

], tsegm

( 1log t

Pathological tropical central path (2)In the (x2r−1, x2r)-plane,

x2r−1

Pathological tropical central path (2)In the (x2r−1, x2r)-plane,

x2r−1

Pathological tropical central path (3)

Let us set the radius of the neighborhood to ε < 12r+1 .

x2r−1

There is no tropical segment contained inthe neighborhood between any two pointsin distinct red parts.

There is no tropical segment contained inthe neighborhood between any two pointsin distinct green parts.

=⇒ in the best case, points have to al-ternate between successive red andgreen parts.

x2r−1

Tropical lower bound on the number of iterations (2)

LemmaThe tropical central path is a union of tropical segments.

γ([λ, λ]) := smallest number of tropical segments needed to describe C trop([λ, λ]).

PropositionLet tsegm(z0, z1) ∪ · · · ∪ tsegm(zp−1, zp) be asequence of tropical segments contained in theneighborhood∪

λ⩽λ⩽λ

B∞(C trop(λ); ϵ)

of C trop([λ, λ]), and such that

z0 ∈ B∞(C trop(λ)) , zp ∈ B∞(C trop(λ)) .

If ϵ ≪ 1, we have p ⩾ γ([λ, λ]).

C trop(λ)

λ⩽λ⩽λ

C trop(λ)

λ⩽λ⩽λ

C trop(λ)

Application to the pathological tropical central path

1 ⩽ j < r

LWr(t) x2r−1

CorollaryLet 0 < θ < 1, and suppose that

(10r− 1)!)8

1− θ

Then, any log-barrier interior point method which describes a trajectory contained in theneighborhoodN−∞

θ,t of the primal-dual central path of LWr(t), needs to perform at least 2r−1

iterations to reduce the duality measure from t2 to 1.

1 ⩽ j < r

LWr(t) x2r−1

TheoremLet 0 < θ < 1, and suppose that

(10r− 1)!)8

1− θ

Then, every polygonal curve [z0, z1] ∪ [z1, z2] ∪ · · · ∪ [zp−1, zp] contained in theneighborhoodN−∞

θ,t of the primal-dual central path of LWr(t), with µ(z0) ⩽ 1 andµ(zp) ⩾ t2, contains at least 2r−1 segments.

(10r− 1)!)8

1− θ

iterations to reduce the duality measure from t2 to 1.

1 ⩽ j < r

LWr(t) x2r−1

(10r− 1)!)8

1− θ

iterations to reduce the duality measure from t2 to 1.Long and winding central paths | Allamigeon, Benchimol, Gaubert, Joswig | 34/37

Outline of the talk

4 Conclusion

Conclusion

Contribution

• we have used tropical geometry to analyze the complexity of log-barrier interiorpoints

• evaluate the complexity of algorithms over numerical instances with differentorders of magnitude

PerspectivesCan we generalize this approach?• other barrier functions• other optimization problems

Can we fix interior point methods?

Conclusion

Contribution

• we have used tropical geometry to analyze the complexity of log-barrier interiorpoints

• evaluate the complexity of algorithms over numerical instances with differentorders of magnitude

PerspectivesCan we generalize this approach?• other barrier functions• other optimization problems

Can we fix interior point methods?

Thank you!

Log-barrier interior point methods are not strongly polynomial, arXiv:1708.01544

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