The Signomial Global Optimization Algorithm Some …users.abo.fi/alundell/files/EUROXXIV.pdf · The Signomial Global Optimization Algorithm – Some Recent Advances Andreas Lundell

The Signomial Global Optimization

Algorithm – Some Recent Advances

Andreas Lundell and Tapio Westerlund

Center of Excellence in Optimization and Systems EngineeringProcess Design & Systems EngineeringÅbo Akademi University, Finland

24th European Conference on Operations Research – EURO XXIV

July 11–14, Lisbon, Portugal

Introduction 2 | 18

The signomial global optimization algorithm

I Is a global optimization algorithm for solving MINLP problemscontaining signomial functions to global optimality.

I Convex underestimators for signomial functions throughsingle-variable transformations xi = T ji (X ji ) applied termwise.

. Transformations obtained by solving a MILP problem.

. Piecewise linear functions (PLFs) are used to approximateT ji (X ji ).

⇒ Gives a convex overestimation of the feasible region.

I The overestimated convex MINLP problem is solved using anyconvex solver.

I As the PLFs are updated, the approximations are improved.

Introduction 2 | 18









Introduction 2 | 18









Introduction 2 | 18









Introduction 2 | 18









Introduction 2 | 18









Introduction 2 | 18









Introduction 3 | 18

Flowchart of the SGO algorithm

Preprocessing step: Transformationsobtained by solving an MILP problem

The nonconvex terms are re-placed by convex underesti-

mators; the PLFs are initialized

The convexified and overestimated pro-blem is solved using a convex MINLP solver

Are thetermination

criteriafulfilled?

Updateapproximations

Optimal solutionno yes

Introduction 4 | 18

The considered class of MINLP problems

The mixed integer signomial programming (MISP) problem

formulation

minimize f(x) x= (x1,x2, . . . ,xI )

subject to Ax= a Bx ≤ bgn(x) ≤ 0 n = 1,2, . . . ,Jnqm(x)+ σm(x) ≤ 0 m = 1,2, . . . ,Jm

I The vector x can contain both continuous and integer-valuedvariables.

I The differentiable real functions f and g are (pseudo)convex,and the functions q and σ are convex and signomialrespectively.

Introduction 5 | 18

The generalized signomial functions q + σ

I A generalized signomial function is a sum of a convex functionq and a signomial function σ .

I A signomial function is a sum of signomial terms, where eachterm consists of products of power functions, i.e.,

σ(x) =J∑

j=1

cj

I∏i=1

xp jii , cj , pji ∈R.

The variables xi are positive.

Example

q(x1,x2)+ σ(x1,x2) = x21 + ex2︸︷︷︸q(x1,x2)

+2.3x0.351 −4x1x

0.52 + x1x2︸︷︷︸

σ(x1,x2)

.

Introduction 5 | 18




σ(x) =J∑

j=1

cj

I∏i=1



Example

q(x1,x2)+ σ(x1,x2) = x21 + ex2︸︷︷︸q(x1,x2)

+2.3x0.351 −4x1x

0.52 + x1x2︸︷︷︸

σ(x1,x2)

.

Introduction 5 | 18




σ(x) =J∑

j=1

cj

I∏i=1



Example

q(x1,x2)+ σ(x1,x2) = x21 + ex2︸︷︷︸q(x1,x2)

+2.3x0.351 −4x1x

0.52 + x1x2︸︷︷︸

σ(x1,x2)

.

The transformation approach 6 | 18

Convexification and relaxation of MISP problems

1. Convexification

I Every signomial term can be transformed to convex form byusing single variable transformations.

I In the process, additional variables and a number of nonlinearequality constraints defining the inverse transformations areobtained.

2. Underestimation, convexification and relaxation

I By using properly selected transformations, the convexifiedterms are underestimated when the inverse transformationsare approximated by piecewise linear functions.

I The MINLP problem is now convexified and relaxed.


Convexification and relaxation of MISP problems

1. Convexification

I Every signomial term can be transformed to convex form byusing single variable transformations.

I In the process, additional variables and a number of nonlinearequality constraints defining the inverse transformations areobtained.

2. Underestimation, convexification and relaxation

I By using properly selected transformations, the convexifiedterms are underestimated when the inverse transformationsare approximated by piecewise linear functions.

I The MINLP problem is now convexified and relaxed.


Transformation of the signomial constraints

qm(x)+ σm(x) ≤ 0

qm(x)+ σCm(x,X) ≤ 0

1.

qm(x)+ σCm(x,X̂) ≤ 0

2.

1. Convexification of σm by transformations xi = T ji (X ji ).Nonconvexities moved to Xji = T−1

ji (xi ).

2. Underestimation of σCm by approximating X ji = T ji

−1(xi )with PLFs X̂ ji . The integer-relaxed problem is now convexand overestimates the original problem.


Transformation of the signomial constraints

qm(x)+ σm(x) ≤ 0

qm(x)+ σCm(x,X) ≤ 0

1.

qm(x)+ σCm(x,X̂) ≤ 0

2.

1. Convexification of σm by transformations xi = T ji (X ji ).Nonconvexities moved to Xji = T−1

ji (xi ).

2. Underestimation of σCm by approximating X ji = T ji

−1(xi )with PLFs X̂ ji . The integer-relaxed problem is now convexand overestimates the original problem.


Convexifying nonconvex signomial terms

I Positive signomial terms can be convexified by. power transformations (PPT, NPT), xi = X ji

Q ji

. exponential transformation (ET), xi = eX ji

I Negative signomial terms can be convexified by. power transformations (PT), xi = X ji

Q ji

I A set of transformations for the nonconvex signomialterms is found by solving an MILP problem.

I The transformation sets are optimal w.r.t. certain strategyparameters. For example:. The number of transformations required is minimal.. The number of variables transformed is minimal.




Q ji



Q ji






Q ji



Q ji






Q ji



Q ji




Example: a nonconvex univariate signomial function

f(x) = −8x +0.05x3︸︷︷︸convex

+25x0.5︸︷︷︸nonconvex

, 1 ≤ x ≤ 7.

2 4 610

15

20

25

x

PPT Q = 2 ET NPT Q = −1

1 3 5 71

1.5

2

2.5

x

PPT Q = 2

1 3 5 70

0.51

1.5

x

ET

1 3 5 70

0.20.40.60.8

1

x

NPT Q = −1



f(x) = −8x +0.05x3︸︷︷︸convex


, 1 ≤ x ≤ 7.

2 4 610

15

20

25

x


1 3 5 71

1.5

2

2.5

x

PPT Q = 2

1 3 5 70

0.51

1.5

x

ET

1 3 5 70

0.20.40.60.8

1

x

NPT Q = −1



f(x) = −8x +0.05x3︸︷︷︸convex


, 1 ≤ x ≤ 7.

2 4 610

15

20

25

x


1 3 5 71

1.5

2

2.5

x

PPT Q = 2

1 3 5 70

0.51

1.5

x

ET

1 3 5 70

0.20.40.60.8

1

x

NPT Q = −1

Solving problems with negative domain using the SGO algorithm 10 | 18

Handling variables with nonpositive domains

I The inverse transformations X = T−1(x), x ∈ [x ,x], requirethat x is strictly positive.

I Can reformulate problems not defined in this way to aform solvable using the SGO algorithm:. If x < 0, it is possible to rewrite the variable according to

xp → (−1)p x̃p , where x̃ ∈ [−x ,−x].

. If 0 ∈ [x ,x], it is possible to do a translation

x̃ = x + τ, where τ > |x |,

i.e., x̃ ∈ [x + τ,x + τ].

I These reformulations are included in the preprocessingstep of the SGO algorithm.





xp → (−1)p x̃p , where x̃ ∈ [−x ,−x].


x̃ = x + τ, where τ > |x |,

i.e., x̃ ∈ [x + τ,x + τ].






xp → (−1)p x̃p , where x̃ ∈ [−x ,−x].


x̃ = x + τ, where τ > |x |,

i.e., x̃ ∈ [x + τ,x + τ].






xp → (−1)p x̃p , where x̃ ∈ [−x ,−x].


x̃ = x + τ, where τ > |x |,

i.e., x̃ ∈ [x + τ,x + τ].






xp → (−1)p x̃p , where x̃ ∈ [−x ,−x].


x̃ = x + τ, where τ > |x |,

i.e., x̃ ∈ [x + τ,x + τ].



Difficulties the translations lead to

I New nonconvex signomial terms appear, e.g.,

x21x2 − x1 − x2 ≤ 0, −3 ≤ x1 ≤ 1, 1 ≤ x2 ≤ 5,

x̃1 = x1 +4, 1 ≤ x̃1 ≤ 5

(x̃1 −4)2x2 − (x̃1 −4)− x2 = . . .=

x̃21x2 −8x̃1x2 − x̃1 +15x2 +4 ≤ 0

I Instead of one nonconvex term, we now have two thathave to be transformed.

I It is however often possible to use the sametransformations on the same variables in multiple terms.




x21x2 − x1 − x2 ≤ 0, −3 ≤ x1 ≤ 1, 1 ≤ x2 ≤ 5,

x̃1 = x1 +4, 1 ≤ x̃1 ≤ 5

(x̃1 −4)2x2 − (x̃1 −4)− x2 = . . .=

x̃21x2 −8x̃1x2 − x̃1 +15x2 +4 ≤ 0






x21x2 − x1 − x2 ≤ 0, −3 ≤ x1 ≤ 1, 1 ≤ x2 ≤ 5,

x̃1 = x1 +4, 1 ≤ x̃1 ≤ 5

(x̃1 −4)2x2 − (x̃1 −4)− x2 = . . .=

x̃21x2 −8x̃1x2 − x̃1 +15x2 +4 ≤ 0






x21x2 − x1 − x2 ≤ 0, −3 ≤ x1 ≤ 1, 1 ≤ x2 ≤ 5,

x̃1 = x1 +4, 1 ≤ x̃1 ≤ 5

(x̃1 −4)2x2 − (x̃1 −4)− x2 = . . .=

x̃21x2 −8x̃1x2 − x̃1 +15x2 +4 ≤ 0






x21x2 − x1 − x2 ≤ 0, −3 ≤ x1 ≤ 1, 1 ≤ x2 ≤ 5,

x̃1 = x1 +4, 1 ≤ x̃1 ≤ 5

(x̃1 −4)2x2 − (x̃1 −4)− x2 = . . .=

x̃21x2 −8x̃1x2 − x̃1 +15x2 +4 ≤ 0






x21x2 − x1 − x2 ≤ 0, −3 ≤ x1 ≤ 1, 1 ≤ x2 ≤ 5,

x̃1 = x1 +4, 1 ≤ x̃1 ≤ 5

(x̃1 −4)2x2 − (x̃1 −4)− x2 = . . .=

x̃21x2 −8x̃1x2 − x̃1 +15x2 +4 ≤ 0





I The new interval can become large, problematic forcertain transformations, e.g.,

−10 ≤ x1 ≤ 10 ⇒ 1 ≤ x̃1 ≤ 21.

If the transformation X = x4 is used, this leads toX ∈ [14,214] = [1,194481].

I Can use a scaling factor, to keep the transformationbounds “reasonable”, e.g.,

λ= 1/√x · x = 1/

√1 ·21 = 0.218

⇒ X ∈ [(0.218 ·1)4,(0.218 ·21)4] = [0.002,441].



I The new interval can become large, problematic forcertain transformations, e.g.,

−10 ≤ x1 ≤ 10 ⇒ 1 ≤ x̃1 ≤ 21.

If the transformation X = x4 is used, this leads toX ∈ [14,214] = [1,194481].

I Can use a scaling factor, to keep the transformationbounds “reasonable”, e.g.,

λ= 1/√x · x = 1/

√1 ·21 = 0.218

⇒ X ∈ [(0.218 ·1)4,(0.218 ·21)4] = [0.002,441].

Translations applied to an example 13 | 18

A bivariate MISP problem

minimize x21 + x2

2 −8x1 −2x2+0.1x1x22 +17,

subject to 0.1x21 +0.1x2−0.1x3

1x2 −0.05x32 +0.4x1x2 −0.05x2

2 ≤ 0.3

1 ≤ x1 ≤ 7, −2 ≤ x2 ≤ 2,

x1 ∈Z+, x2 ∈R.

1 2 3 4 5 6 7−2

−1

0

1

2

1 2 3 4 5 6 7

x1

x 2


Reformulating the nonconvex MISP problem

I Before reformulation:

minimize x21 + x2

2 −8x1 −2x2+0.1x1x22 +17,


1x2 −0.05x32 +0.4x1x2 −0.05x2

2 ≤ 0.3

1 ≤ x1 ≤ 7, −2 ≤ x2 ≤ 2,

x1 ∈Z+, x2 ∈R.

I After reformulation:

minimize µ,

subject to x21 + x2

2 +(0.1τ2 −8)x1 −2x2+0.1x1x22 −0.2τx1x̃2 −µ ≤ −17,

0.1τx31 +0.1x2

1 +0.15τ x̃22 −0.4τx1 +0.1τ x̃2 +0.1x2−0.1x3

1 x̃2

−0.05x̃32 +0.4x1x̃2 −0.05x̃2

2 −0.15τ2x̃2 ≤ 0.3−0.05(τ3 − τ2),

x2 = x̃2 − τ,1 ≤ x1 ≤ 7, −2 ≤ x2 ≤ 2, −2+ τ ≤ x̃2 ≤ 2+ τ,

x1 ∈Z+, x2 ∈R, x̃2 ∈R+, µ ∈R.

I Minimal number of transformations required to transform the problem is 4:x1 = X2/9

1,1 , x1 = X−11,2, x̃2 = X−1

2,1 and x̃2 = X1/32,2 .




minimize x21 + x2

2 −8x1 −2x2+0.1x1x22 +17,


1x2 −0.05x32 +0.4x1x2 −0.05x2

2 ≤ 0.3

1 ≤ x1 ≤ 7, −2 ≤ x2 ≤ 2,

x1 ∈Z+, x2 ∈R.


minimize µ,

subject to x21 + x2

2 +(0.1τ2 −8)x1 −2x2+0.1x1x22 −0.2τx1x̃2 −µ ≤ −17,

0.1τx31 +0.1x2

1 +0.15τ x̃22 −0.4τx1 +0.1τ x̃2 +0.1x2−0.1x3

1 x̃2

−0.05x̃32 +0.4x1x̃2 −0.05x̃2

2 −0.15τ2x̃2 ≤ 0.3−0.05(τ3 − τ2),

x2 = x̃2 − τ,1 ≤ x1 ≤ 7, −2 ≤ x2 ≤ 2, −2+ τ ≤ x̃2 ≤ 2+ τ,

x1 ∈Z+, x2 ∈R, x̃2 ∈R+, µ ∈R.


1,1 , x1 = X−11,2, x̃2 = X−1

2,1 and x̃2 = X1/32,2 .




minimize x21 + x2

2 −8x1 −2x2+0.1x1x22 +17,


1x2 −0.05x32 +0.4x1x2 −0.05x2

2 ≤ 0.3

1 ≤ x1 ≤ 7, −2 ≤ x2 ≤ 2,

x1 ∈Z+, x2 ∈R.


minimize µ,

subject to x21 + x2

2 +(0.1τ2 −8)x1 −2x2+0.1x1x22 −0.2τx1x̃2 −µ ≤ −17,

0.1τx31 +0.1x2

1 +0.15τ x̃22 −0.4τx1 +0.1τ x̃2 +0.1x2−0.1x3

1 x̃2

−0.05x̃32 +0.4x1x̃2 −0.05x̃2

2 −0.15τ2x̃2 ≤ 0.3−0.05(τ3 − τ2),

x2 = x̃2 − τ,1 ≤ x1 ≤ 7, −2 ≤ x2 ≤ 2, −2+ τ ≤ x̃2 ≤ 2+ τ,

x1 ∈Z+, x2 ∈R, x̃2 ∈R+, µ ∈R.


1,1 , x1 = X−11,2, x̃2 = X−1

2,1 and x̃2 = X1/32,2 .


The impact of the translation constant τ on the LB

0 0.5 1 1.5 2 2.5−20−18−16−14−12−10−8−6

Translation constant, τ

Low

erbo

und

1st

iter

atio

n

I In this case the smaller the value of τ , the tighterunderestimator.

I Problem-specific, not a general result.



0 0.5 1 1.5 2 2.5−20−18−16−14−12−10−8−6


Low

erbo

und

1st

iter

atio

n





0 0.5 1 1.5 2 2.5−20−18−16−14−12−10−8−6


Low

erbo

und

1st

iter

atio

n




The impact of τ on the iterations

1 2 3 4 5 6 7

−13

−11

−9

−7

−5

−3

−1

10.29

Iteration

Obj

ecti

vefu

ncti

onva

lue

τ = 1.5, iterations = 6τ = 1.0, iterations = 6τ = 0.5, iterations = 6τ = 0.3, iterations = 6τ = 0.1, iterations = 7

Conclusions 17 | 18

Some final comments

I The SGO algorithm is an global optimization algorithm forMINLP problems containing signomial functions.

. Convex underestimators through single-variabletransformations

. The set of transformations is obtained by solving a MILPproblem

I Problems with variable with negative bounds can be handledthrough translations.

. The parameter τ impacts the time and iterations requiredto solve the reformulated problem

. Open question how to choose τ in an optimal way

Conclusions 17 | 18

Some final comments

I The SGO algorithm is an global optimization algorithm forMINLP problems containing signomial functions.

. Convex underestimators through single-variabletransformations

. The set of transformations is obtained by solving a MILPproblem

I Problems with variable with negative bounds can be handledthrough translations.

. The parameter τ impacts the time and iterations requiredto solve the reformulated problem

. Open question how to choose τ in an optimal way

Thank you for your attention!

Any questions?

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The Signomial Global Optimization Algorithm Some …users.abo.fi/alundell/files/EUROXXIV.pdf · The Signomial Global Optimization Algorithm – Some Recent Advances Andreas Lundell