Maximizing the Sum of Radii of Balls Inscribed in a Polyhedral Set Текст научной статьи по специальности «Математика»

КРАТКИЕ СООБЩЕНИЯ

Онлайн-доступ к журналу: http: / / mathizv.isu.ru

Серия «Математика»

2019. Т. 28. С. 138-145

УДК 518.517 MSG 90С26

DOI https://doi.org/10.26516/1997-7670.2019.28.138

Maximizing the Sum of Radii of Balls Inscribed in a Polyhedral Set

R. Enkhbat,

National University of Mongolia, Ulaanbaatar, Mongolia J. Davaadulam

National University of Mongolia, Ulaanbaatar, Mongolia

Abstract. The sphere packing problem is one of the most applicable areas in mathematics which finds numerous applications in science and technology [1-4; 8; 9; f 1-14]. We consider a maximization problem of a sum of radii of non-overlapping balls inscribed in a polyhedral set in Hilbert space. This problem is often formulated as the sphere packing problem. We extend the problem in Hilbert space as an optimal control problem with the terminal functional and constraints for the final moment. This problem belongs to a class of nonconvex optimal control problem and application of gradient methods does not always guarantee finding a global solution to the problem. We show that the problem in a finite dimensional case for three balls (spheres) is connected to well known Mal-fatti's problem [16]. Malfatti's generalized problem was examined in [6; 7] as the convex maximization problem employing the global optimality conditions of Strekalovsky [17].

Keywords: Hilbert space, maximization problem, optimality conditions, optimal control, sum of radii.

1. Statement of the problem and optimality conditions

Let X be a Hilbert space. We introduce the following sets. Denote by B(x°, r) a ball with a center x° € X and a radius r € IR :

B(x°, r) = {ж € X\ \\x - ж°|| < r}.

(1.1)

A polyhedral set D c X is given by

D = {xGX| (a\x) < bi, fl'Gl, bi G M, i = 1, m}, (1.2)

where ( , ) denotes the scalar product of two vectors in X, \ \ ■ \ | is the norm on X, and intD / 0. a% € X, i = 1, rri are linearly dependent.

Theorem 1. B(x°,r) C D if and only if

{a\x°) +r\\ai\\<bi, i = l~m. (1.3)

The proof can be done in a similar way as in [6], [7]. Denote by ul,u2,... ,uK centers of the balls inscribed in D defined by (1.2). Let r i, f2,... ,vk be their corresponding radii.

Now we consider a problem of maximizing sum of radii of K non-overlapping balls inscribed in D.

Then this problem is formulated as follows:

K

max/ = ^fj (1-4)

i= 1

(ai,uj) + ^11^11 < bi, i = 1 ,m; j = 1 ,K, (1.5)

Iul - u3\\2 > {n + r3)2, l,j = T^K; I / j, (1.6)

ri > 0,r2 > 0,...,rK > 0. (1.7)

The function / in (1.4) denotes a sum of radii of K balls which has to be maximized with respect to variables W 2 j * * * j W and n, r2,..., recondition (1.5) define that all balls are inscribed into a polyhedral set D. Conditions (1.6) describe non-overlapping balls.

In order to write optimality conditions for problem (1.4)-(1.7), we introduce the Lagrange function:

K m K

C(u,r) = + IV'^ + ^IMI ~bi\ +

i=i i=i j=l

K K

i=i j^i

K

Til — \ \u — u

I _„J||2

j=i

'3 3

Then the Karush-Kuhn-Tucker(KKT)conditions applied for the problem (1.4)-(1.7) are as follows:

f дС m K K

Q-. = -i + E +2 E +r3) - Ц = о ,j=XK,

j i=l 1=1 j^l ar 171 К К

= E +2 E E -= °> з=t7K,

i= 1 1=1 _

Лij{(a%,u3) + rj\\a%\ \ -bi) = 0, i = l,m, j = 1,K, Mi [in + r3)2-]\_ul - ||2] = 0, I, j = 1, К- l ф j . hri = 3 = K-

Example 1. Let X = Lp2[to,t\] be a space of functions

u(t) = (ui(t),...,up(t)),

("r \1/ 2

to < t < t\ with the norm ||«||l2 = / \u(t)\2dt and scalar product

ti

(u,v)l2 = J(u(t),v(t))dt, u,v el

to

Problem (1.4)-(1.7) has the form:

к

max / = ri,

i= 1

tl / tl \ V2

J/4- M2

(al(t),u](t))dt + rj / |a*(i)| di <h, i = l,m, j = 1 ,K,

j

to Vo

tl

I \ul(t)-u\t)\2dt > (n+rj)2, l,j = 1,K, l^j,

(1.8)

to

Г1 >0, r2 > 0,..., Гк > o, ueLP[i0,ii]

Introduce the new variables Xij(t), yij(t), Vi(t), Zi(t) and 7¿(t) as follows:

t

Xijit) = J{a\t),uj{t))dt, i = l~m, j = XK,

to

Xij(to) = 0, 7i(t) =(j \a\t)\2dt

\to t

Vij (t) = J \ul(t)-u3m2dt, = l^j,

to

yij(t0) = 0,_

vi = rh 1 = 1 ,K,

v'i(t) = 0, _

yi3{ti) > {Viit^+Vjin))2, l,j = l,K, I ¿j, t € [to,ti].

The problem (1.8) is formulated as the following optimal control problem with the terminal functional and constraints.

K

ma xj(u,v,y) = Y1 Vi(ti),

i= 1 _ _

x'ijit) = (ai{t),uj{t)),i = 1 ,m, j = 1 ,K, y'l3{t) = ||u\t)-u\t))\\2, l,j = 1 ,K, l^j,

v'S) = 0 ,i = 1 _

Xij(t0) =0, i = 1, m, j = 1 ,K, _

Xij{t{) + Vj{ti)ii < bj, i = 1, m, j = 1 ,K,

Vij(to) = 0, i = 1, m, j = 1, K, _

Vijih) > (Viitj + vjih))2, l,j = 1 ,K, Ijtj.

Existence of a solution of the above problem and numerical solutions will be discussed in a next paper.

Example 2. Let X = M2 and D be a triangle set. Assume that K = 3. In this case, we can reformulate problem (1.4)-(1.7) as the perimeter maximization problem:

3

max / = 2ttJ2 ri,

{a^u^ + rjWa'W < h, i,j =J3, (1.9)

IK -^||2 > (ri + rj)2, i,j = 1,3, ijij, n > 0, r2 > 0,r3 > 0.

/ ^

l,j = l,K, Ijtj,

This problem is maximization of a linear function over the nonconvex set and belongs to a class of nonconvex optimization problem.

2. Connection of the perimeter maximization problem to

Malfatti's problem

We also consider the following problem of maximizing total area of 3 balls inscribed in a triangle set. This problem in the literature [16], [18], [10] is called Malfatti's problem. In [5] it was shown that the global optimality conditions by Strekalovsky [17] can be applied to Malfatti's problem. Also, numerical methods and algorithms for solving Malfatti's problem have been developed in [6] and [7]. The Malfatti's problem first formulated in [5] as the convex maximization problem as:

3

maxS* = 7r E r2,

{a\v?) + r3\\al\\ < bi} 7= 173, (2.1)

(n + rj)2 - Wu1 - uj\\2 < 0, i,j = 1,3, i / j, n > 0, r2 > 0,r3 > 0.

Theorem 2. A solution of the perimeter maximization problem (1.9) is a stationary point of Malfatti's problem (2.1).

Proof We write down the lagrange functions for the problems (1.9) and (2.1), respectively

£Cu,r) = -27rEr,+ E £ Xij [{a\vP)+r3\\a%-bl] +

i=l i=lj=l

+ E E in, [{n + r3)2 ||2] - E ijfj]

t=lj^t 3=1

C(u, r) = -7Г E r2 + E E Ai,- [{a\ui) + г3\\аг\\ - Ъг] +

i=l i=lj=l 3 3 3

+ E E iHj [(ri + r3)2 -- ^'||2] - E ijry, i=lj^i j=1

Let a point z = (й1, и2, u3, f\, fs) be a solution to problem (1.9) with fi > 0, i = 1,2,3. Then there exist Lagrange multipliers (Лi3,jli3ll3) such that the optimality conditions (KKT) are satisfied as:

^^ = -2vr + E Ay | H | + E 2fiijin + fj) - k = 0, i = 1, 3,

1 3=1 j^i

i= i JV» _

Aijda1,^} +fj\\a%\\ - bi) = 0, i,j = 1,3,_

Uij [(h + fj)2^]!^ - uj\\2] = 0, i,j = 1,3; ijt j, , Ijrj = 0, j = 1,3.

On the other hand, it can be checked that the point z satisfies the KKT conditions for problem (2.1) with the Lagrange multipliers Xij =

fi

_fl>jj j _ h_ ■ ■_ï—o

Hij — —, tj — —, l,J — 1,0. Ti Vi

Indeed, we have

^^- = -2vr + E Aiillail + + ~k = 0,i= 173,

OTi

1=1 3^1

¿=1 jy» _

\ij{(a\u>) +fj\\a%\\ - bi) = 0, i,j = 1,3,_

. p>ij [{n + fj)2 - W^-u^l2] = 0, i,j = 1,3; ijij.

which means that the point z is a stationary point of Malfatti's problem.

Inversely, we can show that if z is a solution to Malfatti's problem then z is a stationary point of problem (1.9). □

3. Conclusion

In this paper, we formulate the problem of maximizing a sum of radii of non-overlapping balls inscribed in a polyhedral set in Hilbert space as an optimal control problem with the terminal functional and constraints for the final moment. The problem belongs to a class of global optimization. We show that the problem in a finite dimensional case is connected to Malfatti's problem via its optimality conditions.

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Rentsen Enkhbat, Doctor of Sciences (Physics and Mathematics), Professor, Institute of Mathematics, National University of Mongolia, 4, Baga toiruu, Sukhbaatar district, Ulaanbaatar, Mongolia, tel.: 976-99278403, (e-mail: renkhbat46@yahoo.com)

Jamsranjav Davaadulam, Ph. D. (Mathematics), Professor, Institute of Mathematics, National University of Mongolia, 4, Baga toiruu,

Sukhbaatar district, Ulaanbaatar, Mongolia,tel.: 976-99141976, (e-mail: jdavaadulam@yahoo.com)

Received 04.02.19

Максимизация суммы радиусов шаров вписанных в многогранник

Р. Энхбат

Национальный университет Монголии, Улан-Батор, Монголия

Ж. Даваадулам

Национальный университет Монголии, Улан-Батор, Монголия

Аннотация. Задача упаковки шаров имеет множество приложений в различных областях науки и техники. Мы рассматриваем задачу максимизации суммы радиусов непересекающихся шаров, вписанных в многогранное множество в гильбертовом пространстве. Такая задача часто формулируется как задача упаковки. Рассматривая задачу в гильбертовом пространстве, мы формулируем ее как задачу оптимального управления с терминальным функционалом и терминальными ограничениями на конечный момент времени. Эта задача принадлежит к классу невыпуклых задач оптимального управления, поэтому применение градиентного метода не всегда гарантирует нахождения глобального решения для данной задачи. В работе показано, что задача для трех кругов в конечномерном пространстве является хорошо известной задачей Мальфатти [16]. Дополнительно доказано, что максимизация суммы радиусов кругов, вписанных в треугольник, эквивалентна задаче Мальфатти. Обобщенная задача Мальфатти рассматривалась как задача выпуклой максимизации в работах [6; 7] с применением условия глобальной оптимальности А. С. Стрекаловского [17].

Ключевые слова: гильбертово пространство, выпуклая максимизация, условия оптимальности, оптимальное управление, радиус шаров.

Рэнцэн Энхбат, доктор физико-математических наук, профессор, Национальный университет Монголии, Монголия, г. Улан-Батор, ул. Бага Тойру, 4, тел.: 976-99278403, (e-mail: renkhbat46@yahoo.com)

Жамсранжав Даваадулам, Ph. D., профессор, Национальный университет Монголии, Монголия, г. Улан-Батор, ул. Бага Тойру, 4, тел.: 976-99278403 (e-mail: jdavaadulam@yahoo.com)

Поступила в редакцию 04-02.19