A barrier function method for minimax problems

Mathematical Programming 54 (1992) 155-176 North-Holland

155

A barrier function method for minimax problems E. Polak and J.E.

Higgins

Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkeley, CA 94720, USA

D.Q.

Mayne

Department of Electrical Engineering, Imperial College, London SW7 2BT, UK

Received 25 October 1988 Revised manuscript received 14 March 1990

This paper presents an algorithm based on barrier functions for solving semi-infinite minimax problems which arise in an engineering design setting. The algorithm bears a resemblance to some of the current interior penalty function methods used to solve constrained minimization problems. Global convergence is proven, and numerical results are reported which show that the algorithm is exceptionally robust, and that its performance is comparable, while its structure is simpler than that of current first-order minimax algorithms. Key' words: Barrier function methods, interior penalty methods, minimax algorithms, engineering design, nondifferentiable optimization.

I. Introduction F o l l o w i n g K a r m a r k a r ' s s p e c t a c u l a r success in utilizing a b a r r i e r f u n c t i o n t e c h n i q u e in his l i n e a r p r o g r a m m i n g a l g o r i t h m [ 17] there has b e e n a flutter o f activity r e e v a l u a t ing h o m o t o p y a n d b a r r i e r f u n c t i o n m e t h o d s for b o t h l i n e a r a n d n o n l i n e a r p r o g r a m m i n g (see, e.g., [8, 9, 14, 15, 30, 31, 35]). Barrier f u n c t i o n m e t h o d s have c o n s i d e r a b l e p o t e n t i a l for solving m i n i m a x p r o b l e m s arising in e n g i n e e r i n g design (see, e.g., [24] for a d i s c u s s i o n o f these p r o b l e m s ) . These p r o b l e m s are often semi-infinite; in a d d i t i o n , b e c a u s e o f their c o m p l e x i t y , c o m p u t a t i o n o f the g r a d i e n t s o f the c o m p o n e n t f u n c t i o n s (of the m a x f u n c t i o n ) is expensive, while c o m p u t a t i o n o f H e s s i a n s is often i m p r a c t i c a l . F r e q u e n t l y , only feasibility is r e q u i r e d , in w h i c h case the a d v a n t a g e s o f h i g h e r o r d e r m e t h o d s over first-order m e t h o d s are s u b s t a n t i a l l y r e d u c e d . In c o m p u t e r - a i d e d - d e s i g n a p p l i c a t i o n s , run times o f over 100 h o u r s are not i n f r e q u e n t ; h e n c e a l g o r i t h m s w h i c h m a y fail to converge, even to a l o c a l solution, are d e e m e d This research was supported by the National Science Foundation grant ECS-8517362, the Air Force Office Scientific Research grant 86-0116, the California State MICRO program, and the United Kingdom Science and Engineering Research Council.

E. Polak et al. / A barrier function method for minimax problems

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undesirable. Ondine applications are increasingly more common, for example, optimization for control of batch processes (see, e.g. [25]). In such applications, algorithms must be implemented using microprocessors or dedicated VLSI chips, and hence there is a premium on algorithms that are simple and that do not call massive subroutines. Very few algorithms successfully address these engineering problems. We present in this paper a barrier function method which constructs solutions to semi-infinite minimax problems under hypotheses which are much less restrictive than those required by nearly all existing algorithms. The new algorithm has a simple structure and it requires small memory (it does not utilize, for example, linear or quadratic programming subroutines). Furthermore, it has very strong theoretical and experimental convergence properties. Hence it meets the major criteria for engineering applications. The numerical performance of the new algorithm is, in most of the examples studied, superior or comparable to that of the only other first-order algorithm (Algorithm 5.2 in [24]) which can solve semi-infinite minimax problems of the same generality. Significantly, it is less affected by ill-conditioning than Algorithm 5.2 in [24]: it computes solutions for problems on which Algorithm 5.2 in [24] fails because of ill-conditioning. When applied to finite minimax problems, it is again distinguished by exceptional robustness: it does not fail on many problems which cause several excellent competing algorithms to fail. On more benign problems its performance is either comparable or not significantly inferior to that of other first-order minimax algorithms. We address the minimax problem: MMP:

min 0(x),

(1.1a)

XE~ n

where the function ~O: ~ ~ ~ is defined by

tO(x)&max(fl(x)'''" ' fro(x)' max, cEo.ll q~l(x' t)

. . . .

'

4)t(x,t)),

,c[o,llmax

(1.1b)

and the component functions f i :N,, ~ R and 4)j :R n x [0, 1] -~ ~ satisfy certain continuity hypotheses. Engineering design problems often have at least one max-function ( 4 / ( . , . ) ) , arising from, for example, constraints on time or frequency responses. In addition, in many cases the presence of measurement and modeling inaccuracies often means that it is not worthwhile obtaining accurate solutions. The essential features of the barrier function algorithm can be explained by considering the case of MMP where O(" ) is defined in terms of a single function, i.e., O(x)& max qS(x, t).

(1.2a)

tc[O,l]

The barrier function employed in our algorithm is defined by

p(x, ~) ~ JEo,ll ( (c~ -

1

&(x, t)) dt,

(1.2b)

E. Polak et al. / A barrierfunction methodfor minimax problems

157

where a > ~(x). For such a, the function p ( - , a ) is continuously differentiable on the set C ( ~ ) a {x ~ 0~" I ~(x ) < 4 .

(1.3)

It is shown later that p ( . , a ) is a barrier function for the set C(cQ, so that, if {cei} is a monotone decreasing sequence which converges to minx~e- g,(x), then the sets argminx,_,,p(x, ai) must converge to argminx~0~, O(x). The conceptual algorithm which, at iteration i, sets ai = 4'(xi) and selects Xi+l as any element in argminx~- p(x, o~) generates such a sequence. Our algorithm is a practical version of the conceptual algorithm, in which, inter alia, exact minimization of p(x, ~ ) is replaced by approximate minimization. Under mild continuity assumptions, all accumulation points generated by the algorithm satisfy first order optimality conditions. The early literature on semi-infinite optimization was devoted to linear problems. A conceptual algorithm for solving nonlinear semi-infinite optimation problems is presented in [23]. The first implementable algorithm for solving nonlinear, nonconvex semi-infinite programs, such as m i n { f ( x ) lO(x) ~ ~ , ( x ) - e } ,

(2.5)

Lemma 2.1. Suppose that Assumption 1 holds, and that Co is a bounded subset of C. Then there exists a constant L > O, such that for all (x, a ) c Co,

p(x,

L ~)~Z 1 log( 1-~ (~ --~(x)))"

(2.6)

Proof. Since Co is bounded, so is the projection H of Co onto ~". Hence by assumption there exists a Lipschitz constant, L ~ a - f J ( x ) , for all ( x , a ) ~ C o . Let k 6 J be such that Ako(X) is nonempty, let t~ c Ako(x) be given and let t c [0, 1]. Then we have that ~k(x, t)/> ~k(X, t~) -- Lit - txl = tb(x) - Lit - t×[.

(2.7)

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E. Polak et al. / A barrier function method for minimax problems

Consequently, we have

p(x,~)>~[

1

(2.8)

~ ~o,,~ ( ~ - 6 ~ ( x , t)) dt

o,~l (o, - 4 , ( x ) + L [ t -

i log(( -

1 (

I>-L log

(2.9)

t~l) at

+

+ L(I-

(2.10)

L )

1~

(~ -7(x)}

(2.11)

'

where (2.11) is obtained by minimizing (2.10) with respect to tx.

[]

Consequently, if ~ c R is such that C ( a ) # 0 and {x~}~:~oc C ( a ) is a bounded sequence such that 4,(x~)-+ c~ as i-+oo, then p(xi, c~)-+oo as i-+oo, i.e., p ( . , a ) is indeed a barrier function for C ( a ) . We now establish our conjecture, made in Section 1, that the following conceptual algorithm constructs a minimizing sequence.

Algorithm 1 (Minimizes q,(.)). Data: xo c ~". Step O. Set i = 0. Step 1. Set c~i & q,(x~). Step 2. Compute x~+~e arg minx~c(~,) p(x, c~). Step 3. Replace i by i + 1 and go to Step 1. Let G ~ arg minx~u,, g,(x). We see that Algorithm 1 defines the iteration function A : N" ~ 2 u", defined on the complement of G as follows: A(x)=arg

min

x'EC(4,(x))

p(x',O(x))

ifx~G.

(2.12)

To complete the definition of A(. ), we set

a(x) = d

if x ~ d.

(2.13)

Now suppose that the set C(O(xo)) is bounded, and consider the sequence {xi}~'-o constructed by Algorithm 1. Then, there is an infinite subset K c N and vectors x*, x**, such that xi K_~X* and xi+l ~ x** as i-->co. Since the sequence {O(x~)}~o is monotone decreasing, and since #/(- ) is continuous, we must have that g,(xi) -+ q,(x*), as i-+ co, and hence that q,(x*) = O(x**). For the sake of contradiction, suppose that x * ~ G . Then the set C(O(x*)) is nonempty, and hence for any x'eA(x*), ~(x') < ~0(x*) must hold. Let x'~ A(x*), then it follows that x'c C(~O(xi)) for all i. By construction, p(x~+l, qJ(xi)) p(x', O(x*)). However, L e m m a 2.1 implies that p(x~+~, q,(x~))-+oo, which yields a contradiction. Hence we must have that x* < G.

E. Polak et al. / A barrier function method for minimax problems

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To convert Algorithm 1 into an implementable form, we introduce two modifications. First, we relax Step 2, which requires that x~+lc arg minx~c(~0 p(x, cq), by accepting any x~+lc C(a~) which satisfies IIVxp(&+l, ai)[[ ~< K, for some fixed K > 0. Clearly, such an xi+l can be computed in a finite number of operations by means of any number of descent algorithms. Next, because x ~ C(c~) and since an initial point in C(a~) is needed for the computation of x~+l by means of a descent method, we replace the construction in A1 Step 1 by setting ai =~(0(x/ 1) -k- ~b(xi)), and using either xi or xi 1 (whichever has the smaller qJ(. ) value) as an initial point for the routine that computes &+l. It is still possible that 0(x~ 1) = ~b(x~), in which case we increase a~ by a suitably small amount. The resulting implementable algorithm is as follows: Algorithm 2 (Minimizes 0 ( . ) ) . Data: x ~, XoC~ ~, K ~ O , {T~k}kC°=0 such that T / k > 0 , and ~oo k=o Step O. Set i = 0. Step 1. Set

~l(qJ(xi 1) + ~b(x~)) ai =[½(0(xi 1)+ ~b(x,))+ r/i

if q/(Xg_l) ~ tk(xi), if 0 ( x , - 0 = tp(xi),

{x~ if 0(xi-1) I> 0(xi), Yg~ xg_, if ~l(X i 1) ( ~ / ( x i ) "

~ k ~ OO.

(2.14)

(2.15)

Step 2. Using y~ as an initial point, use any method to generate a x~+~c C ( a i ) satisfying IlV~p(x~+,, a~)ll 0. In iteration i, if O(x~-l) ~ O(xi), then set 7h = 0 and u~+~= ~,f,otherwise set Th = vi and u~+l = ~,~/1.1. Clearly, the resulting sequence {~/k}k~0 has a convergent sum. (vi) Rather than using y~ as an initial point in Step 2, a variant of a h o m o t o p y method can be used to generate an alternative starting point. Ideally, we would like to compute a new starting point x ' c C ( a i ) such that IlVxP( x', ~)ll ~< K.

(2.18)

In this case, Step 2 would require zero iterations to satisfy (2.16). Since at the previous iteration, we have computed a pair (xi, ~ - 1 ) satisfying

nVxp(xi,

1)11 K.

(2.19)

a sufficient condition which guarantees satisfaction of (2.18) is that x' satisfies V x p ( x ' , c~i)= Vxp(x~, c~ 1).

(2.20)

Using Taylor's Theorem (under suitable differentiability hypotheses and ignoring higher order terms) we expand the function Vxp( ", • ) about the point (x~, c~_1) to get

VxP( xt, O£i)~Vxp(Xi, 0£i 1)~-Pxx(Xi, Oli--1)(Xt--Xi)~-Pax(Xi, O[i 1)(OLi--O(i 1)" (2.21) This suggests that x ' & x~-(px~(xi, c~ 1 ) ) a P ~ ( x ~ , c ~ i)(c~-c~_1)

(2.22)

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E. Polak et al. / A barrier function method for minimax problems

might be a reasonable starting point for Step 2 (assuming that x ' c C(~i)). A straightforward computation shows that (under smoothness assumptions) the Hessian (with respect to x) of the barrier function is given by

px~(x,

t)Vx0k(x, t) T _~"x(x,t)_ ] o~)a~!It o.,j FL2 Vx0k(x, (-~a~--~,t~ ~(a-Ok(x,t))2Jdt,

(2.23)

Since we wish to avoid computing Hessian information (and making an additional smoothness hypothesis), we approximate the Hessian p~x(xi, ~_~) in expression (2.22) by the positive definite matrix /4(x,

T a)~ Z f[[Vxcbk(x,t)VxOk(x,t) 2 ~ k~,

o.1]

( a - 0 k ( x , t)) 3

~rl] ( a - 0 k ( x , t)) 2

dt,

(2.24)

where cr > 0 is some fixed constant. This yields the following formula X' ~ X i -- ( I~I ( x i , Oli--1) )--l poex(X,, Ogi--' )( 0l i -- a~i_l).

(2.25)

Before using the x' estimated by this calculation, we must, of course, verify that similar type of initialization may be obtained by replacing (2.20) by

x'~ C(a~). A

V~p(x', a~)~ hV~p(x~,

a~ 0,

(2.26)

where h c [0, 1), and repeating the above expansions. We have found empirically that using h = 0 does not yield as good results as using a value of h close to one. Another possible approach would be to use extrapolation based on the past points {(xi-k, ai ,-k)}k~,~, for some m > 0. In [16], an analysis of extrapolation methods for a specific barrier function arising in constrained optimization is presented. However, in [16], the desired final value of the extrapolation parameter (i.e. a) is known, whereas for problem MMP, the final value is unknown. Before concluding this section, we note that we have tacitly assumed that 4:(" ), p ( . , . ) and V~p(.,-) can be evaluated exactly. Consequently, Algorithm 2 should be viewed as a conceptual algorithm. An implementable algorithm may be developed in a manner similar to that presented in [18], by adopting a suitable discretization scheme for the interval [0, 1].

3. Proof of convergence The main theorem of this section shows that any accumulation point ~, of a sequence produced by Algorithm 2 satisfies 0 c 00(~) (where 04:(~) denotes the Clarke generalized gradient [3] of 0(" ) at ~). Our proof requires the following definition of the set valued function G4: : R" -->2~"+1:

k~J \

tc[0,1]

vx&k(x, t)

"

(3.1)

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E. Polak et aL / A barrierfunction methodfor minimax problems

It is straightforward to show that Gq~(.) is an a u g m e n t e d convergent direction finding (a.c.d.f.) m a p for 6 ( . ) (see [24, Definition 5.1]). In particular, we use two properties of ( 3 6 ( . ) : (i) ( 3 6 ( . ) is u p p e r semi-continuous (see [2]), and (ii) 0 6 (36(~) if and only if 0 ~ Oq,(~). The p r o o f of convergence d e p e n d s on the following two technical lemmas, which generalize the fact that a decreasing sequence either converges or diverges p r o p e r l y to -oo. L e m m a 3.1. Suppose that the sequences o f real numbers {/3i}~o and {~}~-o satisfy the following conditions: (i) rh/> Of o r all i c N, (ii) Y ~o rh < co, and (iii)/3~+1 ~/3. Thus, let e > 0 be arbitrary, and suppose that there is no io such that/3~ > fi - e for all i > i0. Clearly, there exists an co il such that Yk=~ *lk~il . It follows from our hypothesis that there exists an i2 >i i~, such that/3i2 ~< fi - e. It follows from (3.2) that for i > i2, i 1 /3i--/3i2 A

v---" ~ k~i 2

(/3k+1--/3k)~

co ~, '?~k~/e.

(3.3)

k=i 2

1

Hence /3~ ~< 13-~e for all i sufficiently large, which contradicts the definition of ft. It follows that lim~oo/3~ = ft. [] Lemma 3.2. Suppose that the sequences o f real numbers (%}i~_j and {~7i}i°~o satisfy the following conditions: (i) ~7~~> 0, f o r all i c N, (ii) ~o~ i=o ~i < oc, and (iii) %+1 f i - e for all i ~> io. Clearly, there exists an il such oo that ~k:~ ~k i 1 . By assumption, there exists an i ~> il, such that T~~< f i - e. Hence, by definition of/3i, we must have that 7~-1 =/3~. H e n c e we obtain that ')/i+1 ~ 2( 'Yi -~- ")/i 1) -[- T]i ~ 2(/3 -- E -[- ~ -~-IE) -]-IE ~ ~ --~6 E.

(3.5)

E. Polak et al. / A barrier function method for minimax problems

165

Since Y~~ i + 1, /3J ~ O, and that (x, ee) c C. Then for each t ~ A ~ ( x ) , with t c [ 0 , 1] and k~J,

~-O(x) (c~

1

4,k~x, t))2 ~ (a - O(x)) ~ 5

(3.8)

Proof. Since t~ A~(x), we have that &k(x, t) < O(x) - e. Hence,

-,~(x, t)> ~-0(x)+~

>

~,

from which the desired inequality follows.

(3.9) []

Lemma 3.4. Suppose that Assumption 1 holds, and that Co is a bounded subset of C.

Then there exists a constant A > O, such that jor all ( x, a ) c Co,

(o, -

O(x))

Y f[

k~J

0,1]

(,~ - ~ ( 1 x,

t)) 2dt~>A>0"

(3.10)

Proof. Since Co is bounded, so is the projection H of Co onto R". Hence by assumption there exists a Lipschitz constant, L < e c , such that each &k(x,.) is uniformly Lipschitz in t on [0, 1], for all x c H. Without loss of generality, we may assume (since C o is bounded) that L>~ c e - O ( x ) , for all (x, c~)6 Co. Let k ~ l be such that A~(x) is nonempty, let t~ c A~(x) he given and let t c [0, 1]. Then we have that &k(x, t) >~& k(x, tx) - Lit - tx] = O(x) - Lit - t~l.

(3.11)

Now suppose that O<e ~ e / L , where m ( . ) denotes the Lebesgue measure on N. Hence we conclude that

~"

k~l

fr o,1]

e~--O(X) " >- fA c~-O(X) dt>-e c~-O(x) (c~-&k(x,t)) 2dt~ )(x)-(c~ &k(x,t))2 ~L(c~y~-~+e)

Setting e = c~- 0(x), and A =¼L, we obtain the desired result.

2.

(3.12t

[Y

The essence of the proof of Theorem 3.5 is to show that, if xi ~ )~, there exist elements ~:i c GO(x~) such that ~ & 0. Upper semi-continuity of G0(" ) allows us to conclude that 0 ~ G0(~), which is equivalent to 0 ~ 00()~).

E. Polak et al. / A barrier function method for minimax problems

166

oo Theorem 3.5. Suppose that Assumption 1 holds. I f {X i}i=-i is any seque~ce produced by Algorithm 2, when applied to problem M M P , then any accumulation point ~, of { xi } i~- l , satisfies 0 c a~(~).

Proof. Suppose that x~ s> ~, as i-> ~ for some infinite subset S c N. By construction x,+~ e C(a,) for all i e N, and hence it follows that I~(Xi+I) < O/i ~ l(I/t (X,-1) -~- ~ ( X i ) ) ~- T~i.

(3.13)

Therefore the sequence {qJ(x,)}~ ~ satisfies the conditions of L e m m a 3.2. Since ~ ( . ) is continuous, we must have that 0(x,) s_~ tp(~), and hence, because of L e m m a 3.2, the whole sequence {q~(x,)},~-i converges to tp(~). As a consequence, the sequence {ai};~0 also converges to ~0(~). By construction, we have for all i • N, i > 0, IlVxp(x,, oh-011 ~< g .

(3.14)

Since (~, 1 - q J ( x , ) ) ~ 0 as i ~ o o , it follows that lim (ai_l - O(x,))V~p(x,, cq ,) = 0. ,-~cO

(3.15)

For each k e J , define pk:[0, 1 ] ~ N by

pki(t) A= cei_,--tP(Xi) (a,_x - Ok(x,, t) ) 2"

(3.16)

Since {(~,-1, x,)}i~s is contained in some b o u n d e d subset of C, we conclude from L e m m a 3.4 that there exists a A > 0 such that for all i • S,

~',

2

pk(t) dt ~ A.

k~l

(3.17)

0,1]

It follows from (3.15) and (3.17) that -

1 2

pk(t)Vxc~k(x, t) dt--~ O.

1)i kEI

(3.18)

0,1]

Furthermore, since {x,},~s is b o u n d e d , there exists some constant B such that 6(xi) - 4~k(x~, t) 0,

2 [

pk(t)(qs(X,)-- chk(xi, t)) dt

(3.19)

k~l J[0,1]

I

A~(xi)

+fA~(x;)c P)(t)(O(Xi)-c~k(xi't))dt) ~;~,j ~- 1, 2, are defined by

ft(x) ~ (X 1 --~/(xl)2-~ (X2) 2 COS(~/(X1)2-~ (X2)2))2"~- 0.005((XI)2"~ (X2)2),

(4.11)

/ 2 ( x ) & (x 2 _ x/(x ~)2 + (x2)2 sin(~/(xl)2 + (x2)2))2 + 0.005((x t)2 + (x2)2).

(4.12)

The solution of this problem is ~ = (0, 0) T, and we used the initial points x-1 = x0 = (1.41831, - 4 . 7 9 4 62) T.

E. Polak et aL / A barrier function method Jbr minimax problems

171

~J-~({1,3,5,7,9})n[-5,5]x[-5,5]

Fig. 3. Contours of ~h(. ).

Table 2 gives the number of iterations (I), the time in seconds (T), and the number of function evaluations (NF) and gradient evaluations (NG) required to satisfy I[xi-.~ll2