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Approved for public releaseDistribution unlimited

Sponsored by

U. S. Army Research Office National Science Foundation

P. 0. Box 12211 Washing~ton, D. C. 20550

Research Triangle ParkNorth Carolina 27709

84 08 24 039

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UNIVERSITY OF WISCONSIN-MADISON

MATHEMATICS RESEARCH CENTER

QUASISTABLE PARAMETRIC OPTIMIZATION WITHOUT COMPACT LEVEL SETS

L. McLinden

Technical Summary Report #2708Jure 1984

ABSTRACT

The well-known perturbational duality theory for convex optimization is

refined to handle directly, in locally convex Hausdorff spaces, problems

involving noncoercive convex functionals together with unbounded densely

defined linear operators or, more generally, convex processes. The theory

presented includes conjugacy, recession, and C-subdifferential formulas for

the two fundamental pairs of dual operations and also includes systematic

treatment of C-solutions.

AMS (MOS) Subject Classifications: Primary: 49-00, 49A27, 49B27;Secondary: 65K10, 90C25, 90D05

Key Words: Convex optimization, noncoercive functionals, unbounded operators,convex processes, E-solutions, weak local compactness, recessiontheory, dual operations, c-subdifferentials, saddlepoint problems,perturbational duality, abstract Hamiltonian system, abstract

Euler-Lagrange system

Work Unit Number 5 (Optimization and Large Scale Systems)

t , r..

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041, andthe National Science Foundation under Grant No. MCS-8102684.

A-i

SIGNIFICANCE AND EXPLANATION

Dual variational principles, which arise classically in physics and

mechanics and typically involve partial differential operators, usually

correspond to dual pairs of constrained convex optimization problems in

various infinite-dimensional function space settings. Constrained

optimization denotes the search for an optimum, say a minimum, of some given

criterion functional over a set of candidate solutions which satisfy given

side conditions, or constraints. Convex problems form the simplest class of

nonlinear problems. Dual pairs of convex optimization problems are nonlinear

generalizations of the familiar, widely useful dual pairs of linear

programming problems. Dual pairs of constrained convex optimization problems

arise also in fields other than physics and mechanics, including mathematical

economics, operations research, management science, many subfields of

engineering, and mathematics itself.

The mathematical model for constrained convex optimization problems known

as perturbational duality theory, developed some 10 to 15 years ago, provides

a nearly complete mathematical treatment of those basic situations which

involve "nice" (i.e. coercive) convex functionals and "nice" (i.e. bounded)

linear operators. Much less developed is the treatment for problems involving

noncoercive functionals and unbounded operators only densely defined. Even

less understood are problems involving more complicated operators called

convex processes, which in general are multivalued and not even densely

defined, and which are important in problems from, e.g., mathematical

economics.

This paper presents a mathematical theory, valid in very general spaces,

for dual pairs of constrained convex optimization problems involving

noncoercive functionals and unbounded linear operators or, more generally,

convex processes. It constitutes a fundamental refinement of the well-known

perturbational duality model cited above. Systematic treatment of approximate

optimal solutions is also provided, rendering the model more amenable to

further, numerical analysis.

The responsibility for the wording and views expressed in this descriptivesummary lies with MRC, and not with the author of this report.

• - M

72 1i i

QUASISTABLE PARAMETRIC OPTIMIZATION WITHOUT COMPACT LEVEL SETS

L. McLinden

It is well known that a lower semicontinuous (l.s.c.) function

X + (-',r], # 0 c, achieves a finite infimum if the sets {xIO(x) ( a),

a e I, are compact. Convex conjugate duality establishes that, if 0 is convex,

a dual sufficient condition for such compactness is that the conjugate function 0

be continuous at 0. Much work in constrained optimization has dealt with

elaborating and implementing this result (see References). The present work is

rooted in the following generalization: a direct sufficient condition for such a

convex 0 to achieve a finite infimum is that (i) {xj$(x) 4 a}, aie R, are locally

compact and (11) 0 is constant along the whole line generated by any direction in

which 0 does not eventually increase. Here we present theorems embodying this

result in strengthened, augmented form tailored to basic structure occurring widely

in constrained optimization. They go well beyond the closest counterparts now

available (including, notably, (5], (10], (9], (1, Ch. 14]).

A word on notation. All spaces are locally convex Hausdorff topological vector

spaces over R; X, V are paired in duality, as are U, Y. For a multifunction

T from Z into W (i.e. T : Z + 2W), G(T) := {(z,w)lw e T(z)},

D(T) := {zIT(z) # 0}, R(T) :- (wlaz,w e T(z)}. For T and any 0 Z +

define TO on W by (TO)(w) : inf(O(z)Iz e T-1 (w)), with inf 0 0. For

l.s.c. proper convex, 00 +(z) sup{x-1(1(z + Xz) - 0(z))!O < X <} for any

fixed z e dom I := WW(z) < i}, and if Z is paired in duality with W,

0 (w) := sup{<z,w> - f(z)l. e ZI and G(O 0) :- {(z,w) (b*(w) 4 E + <z,w> - $(z))

for each fixed £ e (0,-). We abbreviate "weakly locally compact" by w.l.c.

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041, and theNational Science Foundation under Grant No. MCS-81026S4.

I e

THEOREM I. Let h be l.s.c. proper convex on X with k h*. Let A be a

densely defined linear operator from X into U with closed graph and adjoint

B. Define kB on Y by (kB)(y) := k(By) if y e D(B) and := if y 0 D(B).

Assume (i) D(A) n dom h ia nonempty and w.l.c. and (ii) x e A-1 (0), hO(x) 4 0

==> hO+ (-x) ( 0. Then (1) (kB)*(u) = (Ah)(u) > -- and (2) (Ah)O+(u) =

(A(hO+))(u) > -0, with both infima on the right attained whenever not

vacuously. For any E 6 [0,-), y e D (Ah)(u) <==> u e a (kB)(y) <==>

y e D(B), u e Aa Ck(By).

Notice (1) ==> p R(B) n dom k. Further, by (1), (2) is equivalent to:

(2') sup <u,y> = inf sup <x,v> > -,

yeB-dom k xeA 1 (u) vedom k

with the infimum attained whenever u e R(A). If (ii) holds in the form

"x e A-1(0), h0+(x) ( 0 ==> x = 0," the infimum problems on the right of (1), (2),

(2') have weakly compact s-approximate solution sets for all C e (0,-).

"Quasistable" in the title denotes the fact that, whenever (Ah)(u) < - in (1), one

can prove the u.s.c. proper concave q(v) := inf{h(x) - <x,v>Ix e A- 1(u)} on V is

Mackey quasicontinuous [5) and that the "inf" defining q is attained for all v

(e.g., v = 0) in the Mackey relative interior of fvlq(v) > -). Counterparts of

these remarks apply below.

THEOREM II. For j = 1,...,m let fj be l.s.c. proper convex on X withtJ

gj := fj. Assume (1) dom f2,...,dom fm are w.l.c. and (ii) x, + ... + x= 0,

f10+(x1 ) + ... + fm0 + (xm) 4 => f10+(-Xl) + ... + fm+ (-xm) < 0. Then (1)

(g, + ... + gm )*(x) = (f, 0 ... 0 fm)(x) > -- , and (2) (fl 0 ... 0 fm)0 +(x) =

(f10+ E3 ... 0 fm0+)(x) > --, with both infima on the right attained. [0 denotes

inf-convolution.] For any £ e [0,c), v e a(f 1 ... 0 fm)(x) <==> x e a5 (gl +

+ gm)(V) <==> x e u 0 g1 (v) + ... + ac gm(v)lj 0, C1 + ... + sm = ) .

h*THEOREM III. Let h be l.s.c. proper convex on X with k := . Let A be

a closed convex process C8, §39] from X into U (i.e. G(A) is a nonempty closed

j-2-

* *

convex cone) with adjoint A from Y into V given by v e A (y) <==> <u,y>

<x,v> for all u e A(x). Assume (i) 0 ' D(A) n dom h and dom h is w.l.c. and

(ii) x e A-1(0), h0+(x) 4 0 ==> hO+(-x) 4 0. Then, for B := A*-', (1) (Bk)*(u)

= (Ah)(u) > -- and (2) (Ah)0+(u) = (A(h0+))(u) > .1o, with both infima on the

right attained whenever not vacuously. Further, y e D (Ah)(u) <==>0

u e 3 (Bk)(y) <=> [ ax e A-1 (u) av e B-1(y) such that <u,y> ; <xv>,0v e a h(x)]. For any c,6 e [0,-) and (v,y) e G(B) with k(v) 4 6 + (Bk)(y) <

y e 3>C (Ah)(u) u e a (Pk)(y) ==> [u e R(A) and 3x e A-1(u) aa > 0 a8 ) 0

such that a + 8 e + 6, <u,y> - <x,v> -8, x e a k(v)] -> u e a +6(Bk)(y) <>

y e 3C+6 (Ah)(u).

THEOREM IV. Let F be l.s.c. proper convex on X x U. Let A be a linear

transformation from X into U with either A continuous or A densely defined

with closed graph. In the latter case, assume [F(a,u) => F(o,u) somewhere

continuous], or dually, [F (v,-) i f == F (v,-) somewhere continuous]. Assume

(i) {x e D(A)Iau, F(x,u) < -} is nonempty and w.l.c. and (ii) x e D(A),

F0 +(x,Ax) 4 0 -> F0 +(-x,-Ax) 4 0. Then

-, < min F(x,u + Ax) = sup* {-F (A y,-y) - <uy>} =: p(u),

xeD(A) yeD(A

-m < min sup <(x,u),(v,-y)> = sup <u,-y> = p0+(u)xeD(A) (y,v)eA yeA

where A {(y,v)ly 8 D(A*), F*(A*y,-y) < -1, A 0 (yI(y,O) e Al. For any0A

£ 8 (0,-), -y e 8cp(u) <=-> y e Arg C-max{-F (A y,-y) - <u,y>ly e D(A <==>

fy e D(A*) and Zx e D(A) such that (A*y,-y) e CF(x,u + Ax)], and any such x

belongs to Arg £-min{F(x,u + Ax)jx e D(A)}. If also (iii) A is w.l.c. then, foro

each u satisfying [y e D(A }, F*0 +(A*y,-y) + <u,y> 4 0 ==> F*0+(-A*y,y) - <u,y>

4 0], one has p(u) 8 R and the "sup" defining p(u) is actually "max."

THEOREM V. Let H be closed proper convex-concave [9] on X x Y, put

F(x,u) : sup{H(x,y) - <u,y>ly e Y), and assume this F together with A satisfy

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!77A--.

the hypotheses of Theorem IV. Then

- < min sup{H(x,y) - <Ax,y> - <u,y>}xeD(A) y

= sup, inf{H(x,y) - <x,A*y> - <u,y>1 =: p(u)yeD(A ) x

< min sup <(x,u),(v,-y)> = sup <u,-y> = p0+ (u)xeD(A) (y,v)eA yeA0

where A := {(y,v)ly e D(A*), G(y,v + A y) > - -}, A0 := {yl(y,0) e A} for G(y,v)

= inf{H(x,y) - <x,v>lx 8 X1. For any E e [0,-), -y e a p(u) <=->

y e Arg C-max inf{H(x,y) - <x,A y> - <u,;>)yeD(A ) x

y e D(A*) and ax e D(A) such that<==> ,

() sup{H(x,y) - <Ax,y> - <u,y>} 4 e + inf{H(x,y) - <x,A y> - <u,y>}

y x

y e D(A*) and ax e D(A) with (**) (A*y,-y) e a F(x,u + Ax).If also (iii) A is w.l.c. then, for each u such that0

n {yJH(x,)0 +(y) - <u + Ax,y> ) 01 is a subspace,xeC ( D(A)

where C := {xIH(x,.) is somewhere finite} and H(x,.)(y) := usc H(x,.)(y), one

has p(u) e R and the "sup" defining p(u) is actually "max."

Relation (*) (resp. (**)) for E = 0 can be regarded as the abstract

Hamiltonian (resp. Euler-Lagrange) system for the setting of Theorems V, IV (cf.

Main outlines of proof. (I) follows from (IV) by replacing A by -A and

putting F(x,u) := h(x) + *{o}(u), where generally *S := 0 on S and

off S. (V) is deduced from (IV) plus saddle function theory. (IV) splits into two

cases: that of A continuous is proved by refining for the map (x,u) + u the

proof of (III) sketched below; that of A densely defined is deduced by applying

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*, L

the first to F1 (x,u) := F(x,u + Ax) + D(A)(x). Here, one proves as preliminary

facts formulas for F10 +, F a F1. (II) involves two steps: the case m - 2 is

proved by refining for the map (xlx 2 ) + x1 + x2 the proof of (III); then the

continuous case of (IV) permits replacing f2 by f2 0 ... fm"

Sketch for (III). Observe (Bk)* 4 Ah always. Let (Bk)*(a) < -. Define a

l.s.c. proper convex F by F(w,wl,w 2 ) := g 1 (w - wl) + g2 (w - w2 ) - <wi>, for

W = W 1 = W2 = V x y, Z = Z = Z2 = X x U, = (0,), gl = *G(B)' g2 = k + *y. One

shows a := inf F(w,0,0) and 8 := inf F*(0,zl,z 2 ) satisfy a = -8, with infw z 1 z2

in 8 attained, if for p(wlw 2 ) inf F(w,wl,w 2 ) one has (a) a > -, (b)w

(0,0) and dom p cannot be properly separated, and (c) 3 l.s.c. proper convex

P > p such that {(zZ 2 )IP (Zl Z2) Y 1}, Y e R, are w.l.c. Now a = -(Bk) (a)

==> (a), and (ii) <==> (b). For (c), take p(wl,w 2 ) := F(w + w1 ,wl1 w2 ) for

fixed w e dom g1. Then P(wlw 2 ) = g2 (w + wl - w2 ) - <wlZ> + c, c :=g 1 (w) -

< ,i>, yields P (z1 ,z 2 ) = g(z) - <;,z> - c if (1,0) + (zl,z 2 ) = (z,-z) for

some z e Z, and p =- otherwise. Then (i) ==> (c). Since F*(Z,Zl,z 2 ) =

h(-x 2 ) if z = (x,u) and zi - (xi,ui) satisfy z + z1 + = z, (xl,-ul) e G(A),

0, and F = otherwise, S = (Ah)(a). Thus, (1) holds. The 3

assertions are proved with the aid of (1). For (2), observe epi Ah lies between

(A x I)epi h and its closure, where I is the identity on R; similarly for

epi A(h0+). Using (i), (ii) and T, G(T) := G(A x I) n (epi h x U x R), one

proves R(T) = (A x I)epi h closed. Then, using (i), (ii) and S, G(S)

G(I x A x I) r (cl H x R x U x R), where H := (1,x,C)l > 0, (x,E) e epi h),

one proves R(S) = {(G,u,P)lo > 0, (u,v) e (A x I)(o epi h) U

{(0,u')1(u,tj) e (A x i)0 +epi h) is closed. Since

J :[{C(1,u,IJ)IO > 0, (u,ti) e (A x I)epi h) satisfies J C R(S)C cl J =

j U {(0,u,P)I(u,P) e 0+(A x I)epi h}, R(S) = cl J follows, whence

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(A x 1)0+epi h = 0+(A x I)epi h. The nontrivial half of this plus (ii) imply

(A x I)epi h contains no "vertical" line, completing (2).

Details and related results will appear elsewhere.

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REFERENCES

[1) J.-P. Aubin, Mathematical methods of game and economic theory, North-Holland,

1979.

(2] V. Barbu and T. Precupanu, Convexity and optimization in Banach spaces,

Sijthoff and Noordhoff, 1978.

[3] I. Ekeland and R. Temam, Analyse convexe et problemes variationnels, Dunod,

1974.

[4) A. D. loffe and V. M. Tikhomirov, Teoriia ekstremal'nykh zadach, Nauka, 1974.

[5] J. L. Joly and P.-J. Laurent, Stability and duality in convex minimization

problems, Rev. Francaise Informat. Recherche Op6rationnelle R-2, 1971, 3-42.

[6] P.-J. Laurent, Approximation et optimisation, Hermann, 1972.

(7] J. J. Moreau, Fonctionnelles convexes, lecture notes, Seminaire Equations aux

derivees partielles, College de France, 1966-67.

[81 R. T. Rockafellar, Convex analysis, Princeton Univ. Press, 1970.

(9] R. T. Rockafellar, Saddle-points and convex analysis, Differential Games and

Related Topics (H. W. Kuhn and G. P. SzegB, eds.), North-Holland, 1971,

109-127.

(10] R. T. Rockafellar, Conjugate duality and optimization, SIAM, 1974.

IX: scr

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SECURITY CLASSIFICATION OF THIS PAGE ("en Data Entered

REPORT DOCUMENTATION PAGE BRE COMPLETIOR1. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER

2708 Yy ,

4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED

Summary Report - no specificQUASI STABLE PARAMETRIC OPTIMIZATION WITHOUT reporting periodCOMPACT LEVEL SETS 6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(&) 1. CONTRACT OR GRANT NUMBER(*)

L. McLinden DAAGZ9-80-C-0041MCS-8102684

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA & WORK UNIT NUMBERSMathematics Research Center, University of Work Unit Number 5 - Optimiza610 Walnut Street Wisconsin tion and Large Scale System!Madison. Wisconsin 53706 i

11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

June 1984See Item 18 below. 13. NUMBER OF PAGES

714. MONITORING AGENCY NAME & ADDRESS(If different from Controflln Office) IS. SECURITY CL -f thile report)

UNCL TFIED-So. DECL ASSIF "ION/DOWNGRADING

SCMEDULf

IS. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, It different from Report)

IS. SUPPLEMENTARY NOTESU. S. Army Research Office National Science FoundationP. 0. Box 12211 Washington, D. C. 20550Research Triangle ParkNorth Carolina 27709

19. KEY WORDS (Continue on reveree side if neceeeary nd Identify by block number)convex optimization weak local compactness perturbational dualitynoncoercive functionals recession theory abstract Hamiltonian systemunbounded operators dual operations abstract Euler-Lagrangeconvex processes c-subdifferentials systemc-solutions saddlepoint problems

20. ABSTRACT (Continue on reverse olde If nececeery nd Identlfy by block number)The well-known perturbational duality theory for convex optimization is

refined to handle directly, in locally convex Hausdorff spaces, problemsinvolving noncoercive convex functionals together with unbounded densely definedlinear operators or, more generally, convex processes. The theory presentedincludes conjugacy, recession, and E-subdifferential formulas for the twofundamental pairs of dual operations and also includes systematic treatment ofE-solutions.

DD IJOAN73 1473 EDITION OF I NOV65 IS OSOLETE UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE (When Date Entered)

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