Geometric Programming with Applications in Power Control · 2020. 12. 28. · i are the...

Geometric Programming with Applicationsin Power Control

Prof. Daniel P. Palomar

ELEC5470/IEDA6100A - Convex Optimization

The Hong Kong University of Science and Technology (HKUST)

Fall 2020-21

Outline of Lecture

• GP

• Generalized GP

• Examples

• More transformations for GP

• Power control via GP

• Complementary GP and Signomial Programming

• Summary

Daniel P. Palomar 1

GP

• A geometric program (GP) is a type of mathematical optimizationproblem with objective and constraint functions with a special form.

• Importance of GP:

– new numerical methods can solve large-scale GPs extremely effi-ciently and reliably

– a number of practical problems have recently been found tobe equivalent to (or well approximated) by GPs (e.g., circuit

design, geometric placement problems, power control in wireless

communications, duality in information theory).

• GP modeling is not a brainless employment of a software package; itinvolves some knowledge, as well as creativity, to be done effectively.

Daniel P. Palomar 2

Monomials

• A monomial is a real-valued function f of x of the form1

f (x) = cxa11 xa22 · · ·xann

where c > 0 and ai ∈ R.

• Calculus for monomials:

– if f and g are both monomials then so are fg and f/g.– a monomial raised to any power is also a monomial:

fγ (x) = (cxa11 xa22 · · ·xann )

γ= cγxγa11 x

γa22 · · ·xγann .

1The term monomial used in the context of GP differs from the standard definition in algebra,

where a monomial is similarly defined but with nonnegative integer exponents ai and c = 1.

Daniel P. Palomar 3

Posynomials

• A posynomial is a sum of monomials:

f (x) =

K∑k=1

ckxa1k1 x

a2k2 · · ·xankn .

• Calculus for posynomials:

– posynomials are closed under addition, multiplication, and positivescaling

– if f is a posynomial and g is a monomial, then f/g is a posynomial– if γ is a nonnegative integer and f is a posynomial, then fγ always

makes sense and is a posynomial.

Daniel P. Palomar 4

Examples

• Monomials (assume x, y, and z positive variables):

2x, 0.23, 2z√x/y, 3x2y−0.12z.

• Posynomials:

0.23 + x/y, 2 (1 + xy)3, 2x+ 3y + 2z.

• What about these?:

−1.1, 2 (1 + xy)3.1 , 2x+ 3y − 2z, x2 + tanx.

Daniel P. Palomar 5

GP in Standard Form

• A GP in standard form is a problem of the form

minimizex

f0 (x)

subject to fi (x) ≤ 1 i = 1, · · · ,mgi (x) = 1 i = 1, · · · , p.

where fi are posynomials, gi are monomials, and xi are the opti-

mization variables (with the implicit assumption that the variables

are positive, i.e., xi > 0).

• Observe that posynomials and monomials in the definition of aGP seem to play the role of convex functions and affine functions,

respectively, in the definition of a convex problem.

Daniel P. Palomar 6

• Note that a GP in standard form is nonconvex!

– Ex.:√x or x3.

• A simple example of a GP in standard form is

minimize x−1y−1/2z−1 + 2.3xz + 4xyz

subject to (1/3)x−2y−2 + (4/3) y1/2z−1 ≤ 1x+ 2y + 3z ≤ 1(1/2)xy = 1

Daniel P. Palomar 7

A Simple Extension of GP

• If f is a posynomial and g is a monomial, then the constraintf (x) ≤ g (x) can be handled by expressing it as f (x) /g (x) ≤ 1which is a valid posynomial inequality constraint.

• Example: maximize x/ysubject to 2 ≤ x ≤ 3

x2 + 3y/z ≤ √yx/y = z2

can be rewritten as the GP

minimize x−1y

subject to 2x−1 ≤ 1, (1/3)x ≤ 1x2y−1/2 + 3y1/2z−1 ≤ 1xy−1z−2 = 1.

Daniel P. Palomar 8

Simple Application

• Optimize the shape of a box with height h, width w, and depth d:

– objective: maximize the volume– limit on the total wall area of Awall– limit on the floor area of Afloor– lower and upper bounds on the aspect ratios h/w and w/d.

• Problem formulation:

maximize hwd

subject to 2 (hw + hd) ≤ Awall, wd ≤ Afloorα ≤ h/w ≤ β, γ ≤ w/d ≤ δ

which can be rewritten as a GP in standard form.

Daniel P. Palomar 9

How GPs Are Solved

• Standard interior-point algorithms can solve a GP with 1,000 vari-ables and 10,000 constraints in under a minute, on a small desktop

computer.

• For sparse problems, far larger problems can be readily solved, e.g.,a problem with 10,000 variables and 1,000,000 constraints can be

solved in minutes.

• It is also possible to optimize a GP solver for a particular applica-tion, exploiting special structure to gain in efficiency or in size of

manageable problems.

Daniel P. Palomar 10

• Interior-point methods for GPs are also very robust; they require noalgorithm parameter tuning and no starting point or initial guess of

the optimal solution.

• They always find the (global) optimal solution and, when the problemis infeasible, they provide a certificate of infeasibility.

• The main trick to solving a GP efficiently is to convert it to a convexoptimization problem.


GP in Convex Form

• Start with the GP in standard form

minimizex

f0 (x)

subject to fi (x) ≤ 1 i = 1, · · · ,mgi (x) = 1 i = 1, · · · , p.

• Then, use a logarithmic change of variables and a logarithmictransformation of the objective and constraint functions:

– in place of the original variables xi we use their logarithms x̃i =log xi (so xi = e

x̃i)

– instead of minimizing the objective f0, we minimize its logarithmlog f0


– we replace the inequality constraints fi ≤ 1 with log fi ≤ 0 andthe equality constraints gi = 1 with log gi = 0

to obtain the convex problem

minimizex̃

log f0(ex̃)

subject to log fi(ex̃)≤ 0 i = 1, · · · ,m

log gi(ex̃)= 0 i = 1, · · · , p.

Proof. (of convexity)

The monomial g (x) = cxa11 xa22 · · ·xann under the transformation becomes

log g(ex̃)

= log c+ a1 log x1 + a2 log x2 + · · ·+ an log xn= c̃+ a1x̃1 + a2x̃2 + · · ·+ anx̃n


which is an affine function of the variables x̃i.

The posynomial f (x) =∑k ckx

a1k1 x

a2k2 · · ·x

ankn under the transformation be-

comes

log f(ex̃)

= log∑k

ckea1kx̃1ea2kx̃2 · · · eankx̃n

= log∑k

ec̃k+a1kx̃1+a2kx̃2+···+ankx̃n

which is the convex log-sum-exp function.

Therefore, the monomial equality constraints become affine constrains and the

posynomial inequality constraints become convex constraints. 2


Generalized GP: Fractional Powers of Posynomials

• We know that posynomials are preserved under positive integerpowers, e.g., the constraint

f1 (x)2+ f2 (x)

3 ≤ 1

is a standard posynomial inequality.

• However,f1 (x)

2.2+ f2 (x)

3.1 ≤ 1is not a valid posynomial constraint.

• Nevertheless, we can handle it using a trick:


– introduce new variables t1 and t2, along with the inequalityconstraints

f1 (x) ≤ t1, f2 (x) ≤ t2which are both valid posynomial inequality constraints, and

– replace the original nonposynomial inequality with

t2.21 + t3.12 ≤ 1

which is a valid posynomial inequality.

• This relies on the fact that the posynomial t2.21 +t3.12 is an increasingfunction of t1 and t2.

• This method can be used to handle any number of positive fractionalpowers.


• Example: the problem

minimize√1 + x2 + (1 + y/z)

3.1

subject to 1/x+ z/y ≤ 1(x/y + y/z)

2.2+ x+ y ≤ 1

is not a GP because neither the objective nor the second constraint

function are posynomials, but can be rewritten as the GP

minimize t0.51 + t3.12

subject to 1 + x2 ≤ t11 + y/z ≤ t21/x+ z/y ≤ 1t2.23 + x+ y ≤ 1x/y + y/z ≤ t3.


Generalized GP: Composite Functions of Posynomials

• Same idea also applies to other composite functions of posynomials.

• If f0 is a posynomial of k variables, with all its exponents positive(or zero), and f1, · · · , fk are posynomials, then the inequality

f0 (f1 (x) , · · · , fk (x)) ≤ 1

can be handled by replacing it with

f0 (t1, · · · , tk) ≤ 1

andf1 (x) ≤ t1, · · · , fk (x) ≤ tk.

• This shows that products, as well as sums, of fractional positivepowers of posynomials can be handled.


Generalized GP: Maximum of Posynomials

• Suppose f1, f2, and f3 are posynomials. The inequality constraint

max {f1 (x) , f2 (x)}+ f3 (x) ≤ 1

is certainly not a posynomial inequality.

• We can introduce a new variable t and two new inequalities toobtain:

t+ f3 (x) ≤ 1and

f1 (x) ≤ t, f2 (x) ≤ twhich are valid posynomial inequality constraints.


Generalized Posynomials

• A function f of positive variables x1, · · · , xn is a generalized posyn-omial if it can be formed from posynomials using the operations of

addition, multiplication, positive (fractional) powers, and maximum.

• Examples of generalized posynomials:

max{1 + x1, 2x1 + x

0.22 x

−3.93

}(1 + max {x1, x2})

(max

{1 + x1, 2x1 + x

0.22 x

−3.93

}+(0.1x1x

−0.53 + x

1.72 x

0.73

)1.5 )1.7• What about the following inequality?:

x+ y + z −min{√

xy, (1 + xy)−0.3

}≤ 0


GGP Example: Floor Planning• The problem is to configure and place some rectangles in such a

way they do not overlap:

– objective: minimize the area of the bounding box– fixed areas for the four rectangles– upper and lower bounds on the aspect ratio– placement of rectangles as indicated in the figure


• We can identify the following terms:

– width of the bounding box: W = max {wA + wB, wC + wD}– height of the bounding box: H = max {hA, hB}+max {hC, hD}– area of the bounding box: WH– areas of rectangles: hiwi– aspect rations of rectangles: hi/wi


minimizemax {wA + wB, wC + wD}

× (max {hA, hB}+max {hC, hD})subject to hAwA = a, hBwB = b, hCwC = c, hDwD = d

1/αmax ≤ hA/wA ≤ αmax, · · ·

which is a GGP.


• Numerical results: Optimal trade-off curves of minimum boundingbox area versus maximum aspect ratio αmax:


GGP Example: Digital Circuit Gate Sizing

• Consider a digital circuit consisting of a set of gates (that performlogic functions) interconnected by wires.

• Each gate is to be sized; in the simplest case, each gate has avariable xi ≥ 1 associated with it, which gives a scale factor for thesize of the transistors it is made from.

• The scale factors of the gates, which are the optimization variables,affect the total circuit area, the power consumed by the circuit, and

the speed of the circuit.


• We can identify the following terms:

– total circuit area:

A =

n∑i=1

aixi

where ai is the area of gate i with unit scaling

– total power consumed by the circuit:

P =

n∑i=1

fieixi

where fi is the frequency of transition of the gate and ei is the

energy lost when the gate transitions

– each gate has an input capacitance:

Ci = αi + βixi


and driving resistance:

Ri = γi/xi

– delay of a gate is the product of its driving resistance and the sumof the input capacitances:

Di =

{Ri∑j Cj for i not an output gate

RiCouti for i an output gate

which is a posynomial function of the scale factors

– speed of the circuit is its maximum or worst-case delay D whichis a generalized posynomial.



minimize D

subject to P ≤ Pmax, A ≤ Amaxxi ≥ 1, i = 1, · · · , n

which is a GGP.


• The worst-case delay for the circuit in the figure is

D = max

{D1 +D4 +D6, D1 +D4 +D7, D2 +D4 +D6,

D2 +D4 +D7, D2 +D5 +D7, D3 +D5 +D7

}Daniel P. Palomar 28

• In larger circuits, the number of paths can be very large and it isnot practical to list all the possible paths like before.

• A simple recursion for D can be used to avoid enumerating all pahts:define Ti as the maximum delay over all paths that end with gate i

and compute it via recursion.

• For the previous example, the recursion isTi = Di, i = 1, 2, 3

T4 = max {T1, T2}+D4T5 = max {T2, T3}+D5T6 = T4 +D6

T7 = max {T3, T4, T5}+D7D = max {T6, T7} .


• Numerical results: Optimal trade-off curves of minimum delay Dminversus maximum power Pmax, for three values of maximum area

Amax:


More Transformations in GP: Function Composition

• We have already seen composition with the positive power functionor the maximum function in generalized GP.

• It is possible to handle composition with many other functions.

• Composition with function 1/ (1− z): the constraint

1

1− q (x)+ f (x) ≤ 1,

where f and q generalized posynomials (with the implicit constraint

q (x) < 1), can be expressed as the two generalized posynomial

constraints

t+ f (x) ≤ 1, q (x) + 1/t ≤ 1.


• A more general variation: the constraint

p (x)

r (x)− q (x)+ f (x) ≤ 1,

where r is a monomial, p, q, and f are generalized posynomials

(with the implicit constraint q (x) < r (x)), can be expressed as

t+ f (x) ≤ 1, q (x) + p (x) /t ≤ r (x) .

• Another example is the exponential function; one good approxima-tion is

ef(x) ≈ (1 + f (x) /a)a

where a is large. If f is a generalized posynomial, then the right-hand

side is a generalized posynomial.


More Transformations in GP: Additive log Terms

• Suppose we have a constraint of the form:

f (x) + log q (x) ≤ 1

where f and q are generalized posynomials. (Note that the left-hand

side can be negative.)

• We can use the approximation (valid for large a)

log u ≈ a(u1/a − 1

)to obtain

f (x) + a(q (x)

1/a − 1)≤ 1

and then the generalized posynomial inequality

f (x) + aq (x)1/a ≤ 1 + a.


Power Control via GP

• Consider a wireless network with n logical transmitter/receiver pairs.

• The power received from transmitter j at receiver i is given by Gijpjwhere Gij includes the path gain and fading.

• The SINR for the receiver on the ith link is

sinri =piGii∑

j 6=i pjGij + σ2i

.

• Constraints:– we require the SINR for all links to be larger than a threshold:

sinri ≥ sinrmini– we impose limits on the transmitted powers: pmini ≤ pi ≤ pmaxi .


• Problem formulation (to minimize the total power):

minimizep

p1 + · · ·+ pnsubject to piGii∑

j 6=i pjGij+σ2i≥ sinrmini i = 1, · · · , n

pmini ≤ pi ≤ pmaxi .

• This problem is not a GP, but can be rewritten as a GP taking theinverse of the SINR constraints:∑

j 6=i pjGij + σ2i

piGii≤ 1/sinrmini

which are valid posynomial constraints.


• However, we are “killing flies with canons” here as this problem isin fact an LP: linear objective, linear power bounds, and the SINR

constraints can be written as:∑j 6=i

pjGij + σ2i ≤ piGii

(1/sinrmini

).

• We can add constraints to control the near-far problem:pi1Gi1 = pi2Gi2

which are also linear.

• Alternatively, we could maximize the SINR of a particular user i∗ asa GP:

minimizep

∑j 6=i∗ pjGi∗j+σ

2i∗

pi∗Gi∗i∗

subject to (other constraints).


• Another interesting formulation related to fairness is to maximizethe minimum SINR:

maximizep

minipiGii∑

j 6=i pjGij+σ2i

subject to (other constraints)

or, in epigraph form,

maximizet,p

t

subject to t ≤ piGii∑j 6=i pjGij+σ

2i

∀i(other constraints).

• It can be rewritten as a GP:minimize

t,pt−1

subject to∑j 6=i tpjGij+tσ

2i

piGii≤ 1 ∀i

(other constraints).


• Yet another important parameter is the rate:

Ri = log (1 + sinri) .

• Maximizing the rate of the whole system R =∑iRi =∑

i log (1 + sinri) is equivalent to

maximizep

∏i (1 + sinri)


• The difficulty with the maximization of the rate is that 1 + sinri isa quotient of posynomials ...

• An engineering solution is to make the low-SNR approximation:

log (1 + sinri) ≈ log (sinri) .


• The problem then becomes

minimizep

∏i (1/sinri)

subject to (other constraints)

which a GP because we know that the inverse of the SINR is a

posynomial and so it the profuct of inverses of SINRs.

• We can also include individual rate constraints of the form Ri ≥Rmini ; these constraints are easy as they can be rewritten as

1/sinri ≤1

eRmini − 1

which is a valid posynomial inequality constraint.


• Yet another interesting type of constraints are outage probabilityconstraints (modeling the channel gain as GijFij, where Fij is a

random variable with exponential distribution that models Rayleigh

fading, and ignoring the noise):

Pout,i = Pr[sinri ≤ sinrthi

]= Pr

piGii ≤ sinrthi ∑j 6=i

pjGij

= 1−

∏j 6=i

1

1 +sinrthi pjGijpiGii

and then we can write constraints like Pout,i ≤ Pmaxout,i as∏j 6=i

(1 +

sinrthi pjGijpiGii

)≤ 1

1− Pmaxout,i

which is a valid posynomial inequality constraint.


Complementary GP and Signomial Programming

• So far, all the considered constraints and objectives can be formu-lated as a GP except for the maximization of the total rate that

requires the approximation log (1 + sinri) ≈ log (sinri).

• However, for medium-to-low SINR, the approximation does not holdand we have to deal with a ratio of posynomials:

minimizep

∏i

11+sinri

=∏i

∑j 6=i pjGij+σ

2i∑

j pjGij+σ2i


• Minimizing or upper bounding a ratio of posynomials is a trulynonconvex problem that is intrinsically intractable.


• This leads to– complementary GP : allows upper bound constraints on the ratio

between posynomials.

– signomial programming : involving signomials, which are likeposynomials but possibly with negative coefficients.

• A signomial can be expressed as the difference of two posynomials,f (x) − g (x) ≤ 1, which can always be rewritten as a ratio of twoposynomials:

f (x)

1 + g (x)≤ 1.

• Similarly, a ratio between posynomials f (x) /g (x) ≤ 1 can alwaysbe rewritten as a signomial:

1 + f (x)− g (x) ≤ 1.


• One way to deal with these problems is with a successive con-vex approximation, known as condensation method in operations

research:

– single condensation method: approximate the denominator of aratio between two posynomials as a monomial (it becomes then a

posynomial)

– double condensation method: approximate both numerator anddenominator (the constraint becomes linear!).

• Successive approximation:1. Start with k = 0.

2. Approximate difficult function f (x) with simple function f̃ (x)

(typically convex) around some nominal point x(k).

3. Solve the approximate (and simple) problem with solution x(k+1).

4. Go to step 2 to refine the approximation around the new point.


Monomial Approximation of a Posynomial

• Recall the arithmetic/geometric mean inequality:∑i

αivi ≥∏i

vαii

where v > 0, α ≥ 0, and 1Tα = 1.

• Define ui = αivi and write∑i

ui ≥∏i

(uiαi

)αi.


• Now, let {ui (x)} be the monomial terms in a posynomial f (x) =∑i ui (x) to finally get the desired result:

f (x) =∑i

ui (x) ≥∏i

(ui (x)

αi

)αiwith equality for x such that the terms ui (x) /αi are equal. One way

to force equality at one particular nominal point x0 is by choosing

αi =ui (x0)

f (x0).


• With the previous result, we can then substitute a lower boundinequality on a posynomial

f (x) =∑i

ui (x) ≥ t

by a tightened lower inequality on a monomial

∏i

(ui (x)

αi

)αi≥ t

which we can rewrite as a valid monomial inequality

∏i

(ui (x)

αi

)−αi≤ t−1.


• As an example, we can now deal with an upper bound inequalityconstraint of a ratio between posynomials

f (x)

g (x)≤ 1

by first rewritting it as

f (x)

t≤ 1, t ≤ g (x)

and then using the monomial approximation for the constraint

t ≤ g (x).


Approximation by Monomials and Posynomials

• We have seen how to approximate a posynomial by a monomial. Wewill address now a more general question.

• Can we approximate an arbitrary function f (x) with a monomial orposynomial?

• Let F (y) = log f (ey). Then,

– f can be approximated by a monomial if and only if F can beapproximated by an affine function

– f can be approximated by a posynomial if and only if F can beapproximated by a convex function.


Local Monomial Approximation

• Let’s find a local monomial approximation of a general function faround the point x.

• In other words, we want to find an affine approximation of F (y) =log f (ey) around y = log x.

• This is given by the first-order Taylor approximation of F :

F (z) ≈ F (y) +n∑i=1

∂F

∂yi(zi − yi)

which is valid for z ≈ y.


• Using

∂F

∂yi=

1

f (ey)

∂f (ey)

∂yi=

1

f (ey)

∂f

∂xieyi =

xif (ey)

∂f

∂xi

we can write the Taylor approximation (after exponentiating) as

eF (z) ≈ eF (y)n∏i=1

e∂F∂yi

(zi−yi)

or

f (ez) ≈ f (ey)n∏i=1

exi

f(ey)∂f∂xi

(zi−yi).


Defining wi = ezi we finally have

f (w) ≈ f (x)n∏i=1

(wixi

)aiwhere ai =

xif(ey)

∂f∂xi

.


Summary

• We have introduced GP as a class of nonconvex problems that canbe transformed into convex form ones and, hence, optimally solved.

• We have seen several applications: floor planning, gate sizing incircuit design, and power control.

• We have also learned variations of GP termed Generalized GP aswell as Complementary GP and Signomial Programming.

• Some applications cannot be written as GP but still can be wellapproximated by GP and then successive approximation methods

can be used.


References

• Stephen Boyd, Seung-Jean Kim, Lieven Vandenberghe, and Arash Hassibi, “Atutorial on geometric programming,” Optim. Eng., Springer, vol. 8, April 2007.

• Mung Chiang, “Geometric programming for communication systems,” Founda-tions and Trends in Communications and Information Theory , Now Publishers,vol. 2, no. 1-2, Aug. 2006.

• Mung Chiang, Chee Wei Tan, Daniel P. Palomar, Daniel O’Neill, and DavidJulian, “Power Control by Geometric Programming,” IEEE Trans. on WirelessCommunications, vol. 6, no. 7, pp. 2640-2651, July 2007.

• S. Kandukuri and Stephen Boyd, “Optimal power control in interference-limitedfading wireless channels with outage-probability specifications,” IEEE Trans. onWireless Communications, vol. 1, no. 1, pp. 46–55, 2002.


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