Abnormal extremal problems and optimality conditions

Nonlinear

Analysis,

Theory, Methods

r

& Applicotiom-. Vol. 30, No. 4, pp. 2461-2467.1997 Proc. 2nd World Congress of Nonlinear Anolysrs Q 1997 Ekvier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.M) + 0.00

PII:SO362-546X(97)00008-4 ABNORMAL

EXTREMAL

PROBLEMS

AND OPTIMALITY

CONDITIONS

A. V. ARUTYUNOV Department

of Differentional

equations and Functional analysis, Peoples Friendship Mikluho-Maklaia, 6, Moscow 117198

This work was supported

by Russian Fund of Fundamental

Key words: abnormal point, Lagrange multipliers, form, 2-normal mapping. In this article

the following

minimization

second-order

problem

University of Russia,

Researches, project N 96-01-00800 necessary conditions, index of quadratic

is considered:

f(z) + min, 2 E X { F(z) E c

(1)

Here X is a vector space, C is a closed convex cone with a finite number arithmetic space Y = Rk, and f :X+R1,F:X+Rk

of faces in the k-dimensional

are smooth mappings. This problem includes the problems of mathematical programming with finite number of equality and inequality constraints. The principal assential a priori assumption is finite dimensionality of the space Y that is the image of the map F. Let x0 be a solution of problem (1). It is well known (see, for example, [l]), that if one wants to obtain the necessary extremum conditions for finite-dimensional as well as for infinite-dimensional problems, the most unpleasant difficulties may appear when the point x0 is abnormal’. This means that 3yo E Nc(F(z,)) n (-Nc(F(x,))) : yo # 0, F’(zo)*yo = 0. (2) Here No(y) is the normal cone to the set C at the point y and No(y) n (-N,(y)) is the largest subspace that is contained in the cone NC(~). The fact is that the abnormal points are singular for the manifold {Z E X : F(z) E C } and the Lagrange multipliers rule holds trivially at an abnormal point, that is, the Lagrange multipliers are X = (Aa, y), where X0 = 0, y = ys, and vector y0 satisfies conditions (2). So the Lagrange multipliers rule holds at an abnormal point for any arbitrary minimized functional f and does not give any information beside a trivial consequence of the abnormality condition. Moreover, the classical second order necessary condition [3] that the second derivative of the Lagrange function is nonnegative on the cone

{ h E X : (f’(d, ‘If C = 0, then it is said that Lyusternic condition ‘Here Tc(y) = N:(y) is the tangent cone to the

h) 2 0, F’(Q)~ E Tc(F(xo)) 1” is violated set C at the

2461

at the point point y.

zo

Second World Congress of Nonlinear Analysts

2462

generally

does not hold at an abnormal

point.

The following

example

illustrates

this assertion:

F(z) = (F,(z),&(z)) = 0, F,(z) = 2122, F2(2) = x; - z;; 20 = 0. Let us cite one of the specialists in the second-order conditions about the normality condition: “It is natural to ask what means we have at our disposal if the normality hypothesis is not satisfied. The generalization of the theory seems difficult or even impossible” [2]. In this paper we have obtained second-order necessary conditions that are informative for abnormal points as well. If the point under examination for extremum is normal (i.e. vector y,,, satisfying (2) does not exist), then these conditions turn out to be the well-known ones. Denote by angular brackets ( , ) t h e scalar product and the action of a linear functional; ]z] denotes the magnitude of a vector z; conv denotes the convex hull of a set; NC(Y) = {F E Y : ((, z - y) I ovz E C} is the normal cone to a convex set C at a point y E C, and To(y) = N:(y) is the tangent cone to C at the point y, where ’ is the cone polar and by * we denote the conjugate operator and algebraically dual space. If A is a quadratic form or a quadratic mapping, then let us set A[z]’ = (AZ, x). Let y be a quadratic form and 2 a subspace of X. Then ind zq is the index of the form q at 2 (i.e. the maximum of dimensions of subspa.ces N C 2 such that form q is negative definite on them). Let us introduce the so-called finite topology in the vector space X. In this topology we consider as open sets only those ones whose intersection with any finite-dimensional subspace M is open in the linear topology of M (which is the unique Hausdorff one). Let us denote the finite topology by r. It is the strongest topology among those which turn X to topological vector space. On the other hand a minimum which is local with respect to the finite topology is the weakest one among all studied kinds of minimum. Let us fix the point x0 E X. Functions f and F = (F,, . . , Fk) are assumed twice continuously differentiable in a neighborhood of the point 2 ,, with respect to the finite topology T. This means that for an arbitrary finite-dimensional linear subspace M the restrictions of f and F at M are twice continuously differentiable in a neighborhood of 50 (which depends on M). So there exist a linear functional a E X‘, a linear operator A : X + Y, a quadratic form mapping Q : X -+ Y, as well as mappings + RI, and a quadratic 4 : X o,,:X+R’,o:X-tRksuchthat f(x)

= f(xo)

+ (a,z - zo) + 1/2q(s

- zo) + ao(z - zo).

F(z) = F(q) + A(” - z,,) + 1/‘2Q(z - x0) + a(ci-- q,) V’z E X and 1142 - ZOIIM i o I --) 20, z E &f 2 lb - ~0112 ’ holds true for an arbitrary finite-dimensional subspace 44, where ]] ]],+r is a norm in M and the same is true for oo. The mappings a and q are denoted by f’(zo) (or $(zO)) and f”(zo) (or %(x0)) and are called the first and the second derivatives of f. We use analogous notations for the mapping F and other functions. with respect to the Consider problem (1). Let us assume that zo is a point of local minimum finite topology T and f, F are twice continuously differentiable functions in a finite neighborhood of z. with respect to the finite topology.

Second World Congress of Nonlinear Analysts Consider

the Lagrange

2463

function

c(z, A) = x”j(z)

+ (y, F(z)); A = (A”, Y), X0 E R’> Y E Y.

According to the Lagrange multipliers exist Lagrange multipliers X for the point

rule (it gives the necessary first-order of local minimum z0 such that

conditions)

there

g(zo,x) = 0,x02 0,YENc(F(zo))3 x # 0. The set of all Lagrange cone

A = h(x,). Let us consider

lI C X (that depends

multipliers

corresponding

the set of those Lagrange on X) such that

multipliers

codim 11 5 k; II C Ker F’(so); We shall denote this set of Lagrange multipliers cone. Observe that (4) is equivalent to ind zg(z,,, Consider

to the point

x0 forms a nonempty

X E A for which

%(x0.

X)[z]*

by A, = h,(so).

X) 5 codim ( Im F’(z,)),

(nonclosed)

there exists a. subspace

2 0 Vz E Il.

(4)

It is clear that A, c A and A, is a

2 = Ker F’(z,).

the cone X: = /c(sO):

K(zo)= {h Ex : (f’(20), h) I 0, F’(zo)hE Tc(F(~o))). Let us formulate

the main result.

THEOREM 1. Let z. be a local minimum the set A,(zO) is not empty. Moreover,

for problem

(1) with respect to the finite topology

T. Then

(5)

The proof of Theorem 1 is based on the method of perturbations. At the begining we prove that the set A,(z,) is not empty with the help of the penalty method (see [6], [7]). After that we prove (6) using the result obtained above and considering a family of perturbed extremal problems which includes (1) (see [9]). Now let us consider problem (1) ,assuming that the point z. is not simply abnormal but there is a “complete degeneracy” at this point, i.e. f’(Q)

= 0; F’(z,)

= 0.

(6)

In spite of the fact that “complete degeneracy” seems unnatural there are problems that can be reduced to case (6), for example, the problem of studying the fixed sign property of a quadratic form over the intersection of a finite number of quadrics [4]. It turns out possible to complete the necessary conditions of Theorem 1 in the case of a “complete degeneracy”.

2464

Second

THEOREM multipliers

Congress of Nonlinear Analysts

World

2. Assume that dimX < 00, C = 0 and (6) holds true. X = (X0, y) E A, and a vector 1 E X such that

$x0,

an example

there exist Lagrange

X)l = 0; y = $(xo)[q2,

g(20)[612 2 0; ~(zo, A)[$ + 21~(zo)[l,a]l~ REMARK. There exist Theorem 2 is important.

Then

showing

2 0 vx E x;

that the assumption

(8)

x0 + III # 0.

of finite-dimensionality

of X in

The second-order necessary condition that the set Aa is not empty was first obtained for a time- optimal problem in [5]. This condition was further generalized for a vast class of extremal problems and optimal control problems [6, 7,8].The second order necessary conditions which are also informative in the abnormal case were obtained by A.A.Agrachev [9] and A.A.Milyutin [lo] but all these results are covered by the assertion of Theorem 1. First and second order necessary conditions for the abnormal extremal problems were otained by E.R.Avakov in [ll]. The most important characteristic of the extremum second-order conditions is the value of the gap between the necessary and the sufficient conditions. It seems natural to consider this gap to be minimal with respect to the second-order necessary conditions if the necessary conditions turn into sufficient ones after arbitrarily small in C2 perturbations of the minimized function and the map representing constraints in such a way that they do not alter the values off, F and the values of their first derivatives at the point under examnination. Our aim is to find out when this gap is minimal with respect to the second-order necessary conditions obtained in Theorem 1. It turns out that everything depends on whether the cone conv A,(zo) is pointed 3 or not. The fact is that the maximum on set {X E A, : 1x1 = l} in (5) coincides with the maximum on its convex hull. Therefore if the cone conv Aa is not pointed (hence, it contains X = (X’,y) such that (-X) E conv A,) then it is obvious that the condition (5) holds automatically. In this case condition (5) holds for any minimized function f and therefore it does not provide us with any useful information for the minimization problem (1). So when the cone conv A, is not pointed, one can not hope, generally speaking, that the point x0 is a local minimum for the perturbed problem after small in C2 perturbations of F and f in such a way that these perturbations do not alter the values of f(xo), F(xo), f’(xo), F’(xo). Th e f o 11owing example illustrates what was said above: X = R”, Y = R’, C = 0, f(z) F(x)

= -1~1~ + min;

= x1x2 = 0, x = (x1,. . .,x,).

The cone A, = A,(O) is not empty at the point 2 o = 0 and convA, is not pointed. Hence conditions (5) is valid at the origin. Nevertheless it follows from Theorem 2 that x0 is not a local minimum for any arbitrarily small in C2 perturbations of f and F such that F(0) = F’(0) = 0. The situation is quite different when the cone convA1, is pointed. DEFINITION. Let us denote for 2 E X by Fz( x ) a cone which consists of y E Y, y # 0 such that y E in X such that codimll 5 NC(F(x)); (F’(x))‘y = 0 and th ereexists a closed subspace 17 c K~TF’(x) 3The too.

convex

cone

is called

pointed,

if it does

not

contain

nontrivial

subspaces.

The

empty

set is called

pointed

cone,

Second World Congress of Nonlinear

k; gF(z),y)[h12

Analysts

2465

L OVh E 17.

The mapping F is called 2-normal with respect to the cone C at the point 2 if the cone convFz(z) is pointed (we do not exclude the case &(z) = 0 since the empty cone is pointed by definition.) The mapping F is called 2-normal if it is 2-normal with respect to an arbitrary cone C at any point z E X. This definition is a geometric one and is not useful to verify P-normality. Therefore we shall give below the criterion of 2-normality and study the properties of 2-normal map. Here we shall only note that if f is an arbitrary smooth function then the map F is 2-normal with respect to C at the point 5 if and only if the cone R,(z) is pointed. Let X be a Banach space and there exists a continuous quadratic form y : y(z) > 0 t/z # 0. The mappings f and F are assumed to be twice continuously differentiable in the neighbourhood of x0 and to satisfy the following:

IIf(x) - (f(Zo) + (f’(20), 1:- 4 + ffWb

- ~ol”)ll= O(Y(Z- 20)),

IIF

- zol”)II = O(Y(z - 20)).

- (F(Q)

+ F’(z,)(z

- 20) + ;f”(+

THEOREM 3. Assume that the mapping and the following second-order necessary

Then

F is 2-normal with conditions hold :

there exists a vector jj E Y such that in the perturbed

respect

to the cone C at the point

~0

problem

fE(x)= f(z)+&~(a: - 20)+ min F,(z) = F(z) t q(~ - zo)Y E (2’ z0 is a point main properties

of strict

local minimum

of 2-normal

mappings.

for any E > 0. To conclude Consider

d = dim( Im F’(Q))*

a point

this paper

let us investigate

the

20 E X and put

n Nc(F(zo))

rl (-Nc(F(zo)).

LEMMA. The mapping F is 2-normal with respect to C at the point z0 if and only if for any number s : 1 5 s 5 d there are no linearly independent vectors yi E Y, i = 1, s such that yi E Ker (F’(Q)*)

n Nc(F(z,))

ind z&(Y,, F(~o)) 5 4

n (-N,(F(q,));

i = -1,s; ind s(-

e

&(y;,

i = G, F(q)))

(9) 5 d.

i=l

Here 2 = Ker F’(q). According vice versa.

to the Lemma,

if the point z is normal

then the mapping

F is 2-normal

at it, but not

2466

Second

World

Congress

of Nonlinear Analysts

Let us analyze the sufficient conditions of 2-normality of the mapping with respect to the cone C. The first one is that the interior of the cone C is not empty (i.e. there are only inequality constraints in the minimization problem (1)). Th en, it is clear that F is 2-normal at every point. The following sufficient conditions of ‘L-normality are already not obvious. Let us formulate them. The symmetric bilinear mapping g(~s) is defined over the subspace Ker F’(s,). It generates a symmetric bilinear form Q = ys(zO) for every y E Y by the formula

Q(z~,Q) = ~((1/.F(z,)))[zl,z,]Vz,,2,

E Ker F’(G).

This bilinear form generates a linear operator Q : Ker F’(z,) + X’ by the formula (Qz,[) Q(z,[)Vz E Ker F’(z,), [ E X. Denote the bilinear form ys(~,) and the operator generated the same symbol.Thus we have Ker (YE(G))

= {cc E Ker F’(G)

: (y, ~(~)[z,f)

= OVE E Ker F/(x,)}

THEOREM 4. Let for every positive integer s 5 d and any linearly satisfying (9), (10) the following condition hold codim (t-l:=, Ker (yig(xO)))

= by

independent

vectors yi, i = 1, s

> d(s + 1).

(Here codim denotes codimension with respect to the subspace is a-normal with respect to the cone C at the point 2s.

(11)

Ker F’(Q).)

Then

the mapping

F

Let X = R”. Let us consider the linear topological space C:(R”, Rk) that consists of three times continuously differentiable mappings F : R” + Rk and provided with the Whitney topology [12, chapter 2, $11. THEOREM

5. Let n > 2(k - 2); (n - I)(?2 - 2) > 2(k - 1).

Then the set of 2-normal mappings is massive (i.e. it contains of everywhere dense open subsets) and hence it is everywhere

the intersection dense in C:(R*,

of a countable Rk).

family

Theorem 5 shows the main difference between the 2-normality condition and the normality condition (Lyusternik condition) as well as the 2-regularity condition [ll], [13]. Unlike the degeneracy conditions mentioned above the 2-normality condition is generic in the space of functions. REFERENCES

1. BLISS G.A., Lectures on the Calculus of Variations,

Univ. Chicago Press, Chicago (1946).

2. HENRION R., La Theorie de la Segonde Variation et ses Applications Roy. Be/g. Cl. Sci. Mem. Coil. 8 (2), 41 No ‘7 (1975). 3. IOFFE A.D., TIKHOMIROV

V.M., Theory of Extremal

en Comande Optimale,

Problems, North - Holland, Amsterdam

in Acad. (1979).

Second World 4. DINES

L.C.,

On

the

Mapping

of n Quadratic

A.V., TYNYANSKII Soviet Math.Dokl.,

6. ARUTYUNOV Conditions,

A.V., Perturbations Sov.Math., V. 56, A.V., Math.

8. ARUTYUNOV Math. Rvkl.,

A.V., On Necessary Optimality No 1, 174-177 (1985). More

10. LEVITIN E.S., MILYUTIN in Problems with Constraints,

Condition

13. AVAKOV neighborhood 455-466

W. HIRSCH, E.R., (1992).

Differential

AGRACHEV of a singular

Analysts Amer.

2467

Math.

for

Condit,ional

33,

with

SIX,

Problem

in a Problem

Extremum,

Minimum

in Optimal

and

Necessary

Optimality

with

Constraints

A.A., ARUTYUNOV A.V., point, and zeros of a quadratic

New

York

The level mapping,

of Equal-

Phase

Constraints,

lisp.

Math.

Naak,

N.P., Higher Order Conditions No 6, 85-148 (1978) (Russian).

Springer-Verlag,

(1942).

with

for smooth abnormal extremal problems 45, No 6, 3-11 (1989) (Russian). topology,

V. 48, 467-471

for a Local

Constraints

in Abnormal (Russian).

Conditions

A.A., OSMOLOVSKII lisp. Math. Nauk,

11. AVAKOV E.R., Necessary conditions and inequality types, Malh.Zamelkz, MORRIS

Bull.

Necessary Extremum Conditions Nauk, 45, No 5, 181-182 (1990)

9. AGRACHEV A.A., One 1533154 (1989) (Russian).

12.

Forms,

of Extremal Problems No 6, 1342-1400 (1991).

7. ARUTYUNOV ity Type, usp.

31.

of Nonlinear

(londitions N.T., On N ecessary 29, No 2, 176-179 (1984).

5. ARUTYUNOV Control Theory,

J.of

Congress

with

Heidelberg

of Local

ronstrants

Berlin

Sot:ret

44,

No 5,

Minimum

of equaht,y

(1976).

set of a smooth mapping Math. USSR Sbornik, 73,

in a No 2.