On the Proximal Minimization Algorithm with D ...

A. BERMAN,

212 THEOREM 4.

If

M. GOLDBERG,

AND D. HERSHKOWI’IZ

n > 2 und T : H,, -P H,, is a rank-2 preserver, then T is of the form

(*) or (**). REFERENCES 1

P. Botta,

Linear

maps preserving

Multilinear Algebra 20:197-201 2

J. W.

Helton

matrices, 3

and L.

Rodman,

and S. Pierce,

C. R. Johnson

Linear

and S. Pierce,

less than or equal preserving

linear

maps on Hermitian

M. H. Lim, Linear

Linear and

Linear

transformation

maps of Hermitian

matrices:

The stabilizer

of

matrices:

The stabilizer

of

(1985).

Bull. 28:401-404

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an inertia class, II, Linear and Multilinear Algebra 19:21-31 5

to one,

(1985).

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an inertia class, Canad. Math. 4

Signature

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C. R. Johnson

rank

(1987).

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matrices,

(1986). Linear

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M. H. Lim, Linear

mappings

on second

symmetric

product

spaces that preserve

rank less than or equal to one, Linear and M&linear 7

R. Loewy, Linear transformations R. Loewy, Linear R. Loewy,

maps that preserve

Linear

maps which preserve

Appl. 134:165-179 10

S. Pierce

inertia,

a balanced

by YAIR CENSOR’

Linear

preservers

class,

Linear Algebra

of the class of Hermitian

matrices

(1988).

and STAVROS A. ZENIOS3

Introduction proximal minimization

algorithm deals with the optimization minimize

convex

problem

F(x)

to

(1.1)

XEX,

F : W” -t LFlis a given proper convex function and X c W” is a nonempty subset

of the

* Department Haifa 31999,

n-dimensional

of Mathematics

Euclidean

and Computer

space

W”. The

approach

Science,

University

of Haifa,

closed

is based

of Decision

PA 19104-6366.

Sciences,

The

Wharton

School,

University

on

Mt. Carmel,

Israel.

3 Department Philadelphia,

Appl.

with D-Functions

subject where

inertia

SZAM J. Matrix Anal. AppZ. 9:461-472

On the Proximal Minimization Algorithm

1.

class, SZAM J. Matrix Anal.

(1990).

and L. Rodman,

with balanced

The

an inertia

(1990).

11:107-1123 9

which preserve

(1989).

Appl. 121:151-161 8

Algebra 26:187-193 (1990). or decrease rank, Linear Algebra

of Pennsylvania,

CONFERENCE converting

213

REPORT

(1.1) into a sequence

functions

obtained

of optimization

by adding quadratic

The origins of the algorithm this algorithm

the dual problem problem

classes

of a strictly these

In [7] we generalize

obtained

for which convergence

were introduced

by Bregman

and Iusem

from our scheme

linear

programming

optimization e.g.,

(F

problems,

and

sequence

of entropy

problems

the structure

of the algorithm proximal

choice

row-action

the quadratic

and properties

can be preserved. by Censor

minimization

of a D-function.

are all linear)

for which

work is contained

computations.

of

These

and Lent

algorithm

A different

is

choice

algorithm with an entropy additive term. In the case of

x EX

several

[S, 6, 131. Such an approach

relevant

for parallel

on some specialized

algorithm by replacing

original

by one special

and can be

For several important

[3] and studied further

[B]. The

leads to a proximal minimization

is differentiable

problems.

the proximal minimization

some such D-functions

point

tool. This is so because

ascent.

can be decomposed

D : R” x W” + R and specifying

term with a function

[4] and De Pierro

problem

like dual coordinate

transportation

[15], and Rockafellar

in the family of proximal

computational

[19] in this issue, which reports

for a class of nonlinear

D-functions

interest

convex optimization

procedures

dual algorithms

See also the synopsis algorithms

theoretical

is also an important

solved by simple iterative

with strictly convex objective

go back to Minty [14], Moreau

[16, 171. In addition to considerable algorithms,

problems

terms to F(x).

the latter

leads

good special-purpose

of replacing

a linear

was heuristically

in [l, 2, 9-11,

to purely

algorithms

programming

suggested

entropy

exist;

problem

by Eriksson

see,

with a

[12]. Further

181.

The Proximal Minimization Algorithm with D-Functions

2. Let

S be a nonempty

open convex

closure

of S, and

A is the domain

twice

continuously

differentiable

gradient

and its Hessian

continuous

and strictly

these assumptions From f(x)

at every

f

such

that

s E A, where

5 is the

: A E R” -+ R. Assume that _f( x) is

XE S, and denote

matrix at x, respectively.

by Vf(x)

Furthermore,

and V’f(x)

assume

where

(

D,-( x, y), Df : .? x S c Rz” + & by

(2.1)

. , * ) denotes the usual inner product in R”. Df-functions Df-projections

optimization

algorithms

The following

is

to as an auxiliary function.

the D-function

Df(r,Y)=f(x)-f(Y)-(Vf(Y),x-Y).

defining

its

that f(x)

convex on .?. The set S is called the zone of f, and f obeying

is referred

construct

set in R”

of a function

onto

convex

sets

and play a key role

are instrumental in the

in

primal-dual

in [3, 4, 81.

additional

properties

tions, their zones, and the Df-functions

need to be postulated constructed

for the auxiliary

from them. Denote,

func-

for any cx E LQ,

by

L>(% Y) = {dpf(X, the partial

level sets of

Y) Q

Df( x, y).

a),

L$(x,a) =

{y~SjZ+(x,

y)