'Fat' triangulations, or solving certain nonconvex matrix optimization ...

Mathematical Programming 31 (1985) 123-136 North-Holland

'FAT' TRIANGULATIONS, OR SOLVING CERTAIN NONCONVEX MATRIX OPTIMIZATION PROBLEMS M i c h a e l J. T O D D

School c~]" Operations Research, and Industrial Engineering, College of Engineering, Cornell University, Ithaca, N Y 14853, USA Received 24 June 1983 Revised manuscript received 1 I July [984

In an earlier paper, the author proposed the problems of determining 'optimal' linear transformations of the triangulations Jt and K~, in the sense of minimizing their average directional density for a given mesh size. These tasks were also formulated as optimization problems where the variable is a matrix. Here we solve these problems, and another one which is analogously related to finding an 'optimal' linear transformation of the new triangulation J'. We show that J~ and J' are themselves optimal, while the (~*,/3*)-transformation of K~ developed by van der Laan and Talman is optimal. The latter theorem extends partial results of van der Laan and Talman and Eaves. The optimality of these linear transformations is quite robust; we may change the objective function to maximizing the volume of each simplex, or the constraints to limiting the sum of squares of edge lengths of each simplex, or both, without changing the optimal solutions.

Key words: Triangulations, Nonconvex programming, Optimality conditions, Generalized singular value decomposition.

1. Introduction

This paper is concerned with the problem of finding 'optimal' linear transformations of certain regular triangulations of ~" according to various criteria. Such triangulations arise in a variety of applications. We are primarily interested in their use for solving systems of nonlinear equations via piecewise-linear homotopy methods (see, e.g., Allgower and Georg [1], Eaves [3], Todd [12]): they also arise in numerical integration (Whitney [16]) and in finite element methods (Zienkiewicz [18]). In most applications it is desirable to have triangulations with only 'fat' simplices to save in computation while avoiding inaccuracies or singularities. For example, the plane can be subdivided into equilateral triangles, which are, according to a n y r e a s o n a b l e c r i t e r i o n , t h e f a t t e s t 2 - s i m p l i c e s . F o r n / > 3, ~ n c a n n o t b e s u b d i v i d e d into equilateral n-simplices, and many criteria for fatness can be proposed. W e will c o n s i d e r t h r e e f a m i l i e s o f t r i a n g u l a t i o n s , n o n s i n g u l a r l i n e a r t r a n s f o r m a tions of three regular triangulations of ~", and four criteria. For any of the three triangulations, and any of the four criteria, the problem of choosing the best linear Research partially supported by a fellowship from the Alfred P. Sloan Foundation and by NSF Grant ENG82-I5361 123

M.J. Todd / "Fat" triangulations

124

transformation can be stated as a nonlinear programming problem, in which the variable is an n x n matrix. Moreover, all these problems are nonconvex. Nevertheless, we obtain globally optimal solutions with the robust property that, for each triangulation, a single transformation is simultaneously optimal for all four criteria. Two well-known triangulations of R" are K, and J~; see e.g. [10]. In KI, an initial vertex is chosen as any point in the integer lattice Z ' . Then each subsequent vertex is obtained from its predecessor by taking a unit step along a previously unused coordinate direction. The convex hull of the n + 1 vertices so formed is a particular simplex of K~, e.g. the convex hull of 0, e ~, e ~ + e : , . . . , e~+ . ' ' + e ' ' . K~ is the collection of all such simplices c7, starting at any initial vertex in Z" and choosing any permutation of the coordinate directions. For n = 2, the simplices (triangles) of K~ result from cutting unit squares with integral vertices in R e by diagonals running from southwest to northeast. The triangulation J~ is formed similarly, except that the initial vertex must lie in (2Z)" and each step is of the form • j. Thus, for instance the convex hull of 2e 3, 2e~+ e 2, e 3 + e -~, e3+ e 2 - e I is a simplex of J]. For n = 2, the simplices of J~ result from cutting the unit squares with integral vertices by diagonals running alternately from southwest to northeast and from northwest to southeast; each diagonal joins two vertices whose coordinate sums are even. The author has recently proposed a new triangulation J ' of R", for n~>3 [13]. Each simplex is the union of four simplices of Jj. A typical simplex of J ' (for n/> 4) takes the form of the convex hull of 0 ~ ' I) ~c

, cIq-c

2, . . . , cl-~ - . 9

.-}-c n 2 , c i+.

el+

.+e"

I+e'~.

..

2+e'~

-+e" 2+e" i _ e ,"

Further details may be found in [13]. For any triangulation T of R", and any nonsingular n x n matrix A, A T denotes the triangulation {Act: ere T}. For example, if n = 2 and the columns of the 2 x 2 matrix A have equal Euclidean norm and make an angle of 2w/3, then AK~ subdivides R 2 into equilateral triangles. It is natural to ask which matrices A provide the 'best" linear transformations of Kt, JL and J'. To make this precise we must discuss various measures of triangulations. One measure that appears useful in evaluating the suitability of a triangulation for piecewise-linear homotopy algorithms is the average directional density, introduced in [10]. Roughly speaking, this is the rate per unit length at which a random straight line intersects simplices of a triangulation. Eaves and Yorke [5] have provided further insight into this measure by showing that it also measures the average surface density of a triangulation per unit volume. This interpretation suggests that it may be a useful measure in other contexts also. Thus we are interested in choosing a nonsingular matrix A to minimize the average direction density of AT, for Y = K~, J~, or J'. Of course, we must limit the mesh size of the resulting triangulation, i.e., the largest diameter of its simplices. It was shown in [11] that problems (PI) and (P2) below are formulations of this problem

M.J. Todd / 'Fat' triangulations

125

for T = K~ and T=J~ respectively. From the arguments in [1 l] and [13], it is clear that (P3) below is a corresponding formulation for T = J'. (P1)

minimize

f ~ ( A ) - - - E IIA lei]]+ Z IlA T(e,-ej)]l

A

i

i4. Moreover, equality is achieved when s is a (+ 1, - l)-vector - let T be the set of all such s's. Lemma

0)

3.3

E sc5

~-4'( n-1 D g ' ( / t ) = v [(n + l)/2J

) ( t r ( . ) + 8eT(')e) 1

where c~ is ~_ ' (fn is odd, 1 ifn is even. 2~

(ii)

,~rY~D g ' ( I ) = ~ n t r ( ' ) "

Proof. (i) Let p = [ i n + 1)/2J. For n odd we must add Dg~(A) for all (0, l)-vectors s with p ones; for n even those with p or p + 1 ones. Let the set o f (0, 1)-vectors with p ones be denoted SI, those with p + l S> Since, if s has ones in positions .j~. . . . . jp, A s = ej +. . .+ ej,,-pye, we obtain from (2.7)

E Dg,(e{)= sc&

(ej + . . . + % - p y e ) V ( . ) ( e i , + . . ' + e j , )

E J~ " " " "%it,

1

=-

0

{(e,,+.. "+%)T(')(ej+'''+%)

E

O jl,~...#j p

"

-- pyeT( . )( ej, +. 9 . + e6)}. In the first term, each e,f( 9)ei appears (p-l) "-~ times, and each e~( 9 )ei, i r Hence }~ Dg~(A) = -~ ~s,

tr(.)+ p

p --

eT(.)e-p7

(p-2) ,,-2 times.

ef( " )e}. p --

(3.l) Similarly, if n is even

Vs,_Dg~(A)=~l { ( n - 2 ) t r ( . ) + ( n - ~ ) Pe T ( . ) e - ( p + l ) y ( n - 1 )

P (3.2)

(,,-2) =(p_~) n-~ , .-,) For n odd, ,p-i =_~(p-i

thus (3.1) gives

,

{tr(.)+(1-2py)eT(.)e}.

v Dg~(A) = ~ s~.S

p-

But p = ( n + l ) / 2 , so 1 - 2 p y = 1 - ( n + l ) y = 6; thus (i) follows in this case. For n even, we add (3.1) and (3.2) to obtain 2 Dgs(5") = ~ ~s

p

')

tr(.)+

("i)

eT, ,e}

p --

eX(.)e--py

(;)

eT(')e

M.J. T o d d / "Fat"

132

triangulations

Now, since (%-') = (;_l) and p(;) = ,,(~ l), we deduce E D g , ( A ) = O1 ( n - 1- 1) { t r ( . ) + ( l - v - n v ) e r ( . ) e } .

sr

p

But since 1 - y - n y = 1-(n+ (ii) We have from (2.7)

1)3,= 6, (i) is established also for n even.

iT(

E Dg,(l)= E t~T

t;-T

") t

,-- . .On

But since, for i # j , e~(.)e i and -e~X(.)ej occur the same n u m b e r of times, and e,r( 9)e~ occurs in each term, we deduce 2 n

Y Dg,(l) - - ~ = t r ( . ) . e, T

",/1l

Note that A P = P.4 for all P e l/ and that I P D is orthogonal for all P e II, D e ,=1. Thus hp.l(A) is the same for all P e l l , and hp.D(I) and h'p.z)(l) the same for all P e I1, D r A. Thus, because of the way we have chosen the right hand side constants, all constraints are active in ( R I ) - ( S 3 ) . Lemma 3.4 (i)

~

Dhe.l(,4)=(n+2)" (tr(.)+aer(.)e).

p.n

(ii)

12 2"(n+2)?

Y_ 2 Dhp,.(I) P e r t Dr

(iii)

2 " ( n - 1 ) ? ( 1 7 + 2 ) ( n 2 + n + 121)

~-v Dh'E,D(I)=

~

6

P~I1 Dr=_A

Proof. (i) From (2.8), using fi,= l - y e e

2

tr(. ).

6

r, we have

D h n .; ( A ) ( E ) = E 2 siiP r TE P % - y

Pr

tr(.).

P=FI i-~-i

v

x;/ sTpTeeTEPsq.

(3.3)

P~I1 i ~ j

Consider the first term. It is straightforward to check that

E PTEP = (11 - 2 ) [ ( ( n t r ( E ) -

eTEe)I + (eTEe--tr(E))eeT).

(3.4)

P~ll

Now note that

y, s~lsij= V ( j - i + l ) = n ( n + l ) ( n + 2 ) / 6 i~---.j

(3.5)

i~j

and T

9

.

Y (s!ie)-= Y (j-t+l) i~ j

i~ J

2

=

n(n+l)2(n+2)/12.

(3.6)

M.J.

133

Todd / 'Fat" triangulations

Thus the first term simplifies to

~ s~pTEPsi i P~tt i~ O, where H stands for the appropriate choice of K, J, or J'. This will imply as desired that JH(A2)>~ f n ( A l ) and f(A2)>~f(Ai) as long as ./H(Y.'X) and f ( X X ) are p s e u d o c o n v e x in c r = ( c r , , . . . , o ' ~ ) where X = d i a g ( c r ) ['or o r > 0 . Note that ( X X ) - T w - X '(X-Tw) and that f ( Z X ) = - d e t 2;]det X]. Hence the following result will complete the proof.

Lemma 4.1. The function &(o-) = IIZ -~ v{[ is convex on {o- 6 ~": cr > 0} jbr any v e R" and the function X(cr) = - d e t Z is pseudoconvex on {or c R": o" > 0}, where Z = diag(cr). = (~, v~/cr~) 1/: we obtain

Proof. F r o m (b(r qjr

-

-

- 4(o-)

o-~

135

M.J. Todd / "Fat" triangulations

and hence

a~aiO(~

vi2

v~ vj2

1

8~ (~(o~))3 a-~ @ I)i

Di

: (o-~(4, (o-))3/z) (3 (q5(~

~j

- ~, ~2~) (o-~(4~(-o-))3/J)"

Hence it is sufficient to show that 3 0 : I - u u v is positive semi-definite, where (b = qS(o-) and u = (ui)= (vi/o-i). But as a symmetric rank-one modification of the (scaled) identity, this matrix is positive semi-definite if[ its determinant is nonnegative. Now det(34,2I - uu r) = (3(b2)"det(I

-

uuT/3d) 2)

=

(302)"(1 - uru/3c52)

= 2~z(3~2) " I ~ o . Next we consider X. Suppose X ( ~ ' ) < x ( c r ) , where ~ ' > 0 and ~r>O. Then the product of the terms (o-'J~r~) is greater than one. Thus their arithmetic mean is also greater than one, so that ~i'~ (o-'Jo-,)> n. Hence - t r ( Z - ~ ( I ; ' - Z ) ) < O , and X is pseudoconvex. From the discussion preceding the lemma, we have now established Theorem 4.2. A, I and I are the globally, optimal solutions to (P1)-(SI), (P2)-($2)

and (P3)-($3) respectively'.

5. Concluding remarks

We have shown that particular linear transformations of the triangulations Kb J~ and J' simultaneously minimize average directional (or surface) density and maximize the volume of an individual simplex, subject to a restriction on either the mesh size or the sum of squares of edge lengths of each simplex. Note that since the surface density is the average of the total surface area divided by the volume of each simplex for these triangulations, we can alternatively minimize the average of the sum of the inverses of the altitudes of each simplex. While this provides a different interpretation, it does not contribute further to the robustness of our optimal linear transformations; however, we hope that this has been amply demonstrated by their simultaneous optimality according to four apparently distinct criteria.

References [I] E. Allgower and K. Georg, "Simplicial and continuation methods for approximating fixed points", S I A M Review 22 (I980) 28-85.

136

M.J. Todd / "Fat' triangulations

[2] J.E. Dennis, Jr., and R.B. Schnabel, "'Least change secant updates for quasi-Newton methods", SIAM Review 21 (1979) 443-459. [3] B.C. Eaves, "'A short course in solving equations with PL homotopies", in: R.W. Cottle and C.E. Lemke, eds., Nonlinear Programming, ?roceedings of the Ninth SIAM-AMS .~vmposium in Applied Mathematics (SIAM, Philadelphia, PA, 1976) pp. 73-143. [4] B.C. Eaves, "Permutation congruent transformations of the Freudenthal triangulation with minimal surface density", Mathematical Programming 29 (1984) 77-99. [5] B.C. Eaves and J. Yorke, "Equivalence of surface density and average directional density", Mathematics c~f Operations Research 9 (1984) 363-375. [6] G.H. Golub and C.F. Van Loan, Matrix computations (The Johns Hopkins Press, Baltimore, 1983). [7] G. van der Laan and A.J.J. Talman, "'An improvement of fixed point algorithms by using a good triangulation", Mathematical Programming 18 (1980) 274-285. [8] J.M. Ortega and W.C. Rheinboldt, lterative solutions of" nonlinear equations of se~eral variables (Academic Press, New York, 1970). [9] A.J.J. Talman (with the collaboration of G. van der Laan I, "'Variable dimension fixed point algorithms and triangulations", Mathematical Centre Tracts No. 128, Amsterdam, 1980. [10] M.J. Todd, "On triangulations for computing fixed points", Mathematical Programming I0 (1976) 322-346. [11] M.J. Todd, "'Improving the convergence of fixed-point algorithms", Mathematical Programming Study 7 (1978) 151-169. [12] M.J. Todd, "An introduction to piecewise-linear homotopy algorithms for solving systems of equations," in: P.R. Turner. ed., Topics in Numerical Analysis, Lecture Notes in Mathematics No. 965 (Springer-Verlag, Berlin-Heidelberg-New York, 1982), pp. 149-202. [13] M.J. Todd, "J': A new triangulation of ~"", to appear in S4AM Journal on Algebraic and Discrete Methods 5 (1984) 244-254. [14] M.J. Todd, "Erratum in 'Improving the convergence of fixed-point algorithms'", manuscript, February 1984. [15] C.F. Van Loan, "Generalizing the singular value decomposition", SIAM Journal on Numerical Ana(vsis 13 (1976) 76-83. [16] H. Whitney, Geometric integration theory (Princeton University Press, Princeton, NJ, 1957). [17] A.H. Wright, "'The octahedral algorithm, a new simplicial fixed point algorithm", Mathematical Programming 21 (1981) 47-69. [ 18] O.C. Zienkiewicz, The finite element method in engineering science (McGraw-Hill, London, 1971).