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CATEGORIES:Theoretical computer science seminar
SUMMARY:Solution of an old problem in vector optimisation:
Support function of the generalised Jacobian - Ab
bas Edalat (Imperial College)
DTSTART:20170428T100000Z
DTEND:20170428T110000Z
UID:TALK2410AT
URL:/talk/index/2410
DESCRIPTION:Many optimisation problems are non-smooth in the s
ense that the\nobjective function is not different
iable. This is invariably the case in\nmany areas
of application when the objective function contain
s simple\nconstructs such as maximum or minimum o
f several differentiable\nfunctions or contains\,
for example\, the absolute value function. The no
tion of the\nClarke sub-gradient of locally Lipsch
itz maps has been used successfully\nin the past d
ecades to solve non-smooth non-convex optimisation
problems\nby methods such as the sub-gradient des
cent or the more powerful bundle\nmethod.\n\nThe g
eneralised Jacobian of a locally Lipschitz vector
function\, as\nintroduced by Clarke in mid 1970's
[1]\, plays the same role in non-smooth\nvector (m
ulti-objective) optimisation that the (Clarke) sub
-gradient or\nsub-differential plays in the usual
non-smooth scalar optimisation. The\nsub-gradient
of a non-smooth objective function is a non-empty\
, compact\nand convex subset of the finite Euclide
an space of the same dimension as\nthe domain of t
he objective function. Similarly\, the generalised
\nJacobian is a non-empty\, compact and convex sub
set of the space of real\nmxn matrices where n is
the dimension of input and m is the dimension of\n
the output of the multi-objective function. Howeve
r\, until now\, there\nhas been a huge discrepancy
in the way they have been defined. The\nsub-gradi
ent of a scalar function at a point was originally
defined\nconstructively by its support function w
hich provides a simple\nexpression for its boundar
y. In contrast\, the generalised Jacobian has\nbee
n defined non-constructively by using a theorem of
Rademacher in\nanalysis which states that every L
ipschitz map between finite\ndimensional Euclidean
spaces is differentiable for almost all its\nargu
ments.\n\nThe non-constructive nature of this defi
nition has created a major\nobstacle in the applic
ation of the generalised Jacobian in vector\noptim
isation and several other fields of computation su
ch as non-smooth\nNewton method or solution of par
ametric ODE's with non-smooth right hand\nside. Th
ere have been two attempts to tackle the problem o
f finding the\nsupport function of the generalised
Jacobian. Hiriart-Urruty [2]\nderived the support
function of an approximation to the generalised\n
Jacobian\, namely its so-called plenary hull. Imb
ert [3] used a\nGreen-Stokes formula to obtain an
intractable analytic expression for\nthe support f
unction of the generalized Jacobian that involves
a limsup\noperation over surface integrals of the
divergence of the derived\nfunction on a shrinking
sequence of hypercubes.\n\nIn this talk\, I will
derive a simple expression to compute the support\
nfunction of the generalised Jacobian after 40 yea
rs. It determines the\nboundary of the non-empty c
ompact convex set that the generalised\nJacobian r
epresents in the space of mxn matrices. The deriva
tion is\nelementary\, does not invoke Rademacher's
theorem and is in the same\nspirit as that of the
sub-gradient of a scalar objective function.\nMor
eover\, I will show that the same technique can be
used to compute the\ngeneralised Jacobian of loca
lly Lipschitz multi-objective functions\ndefined o
n an infinite dimensional Banach space\, which all
ows\noptimisation over function spaces.\n\n[1] F.
H. Clarke\, Optimization and Nonsmooth Analysis\,
Society for Industrial and Applied Mathematics (SI
AM)\, 1990.\n\n[2] J.-B. Hiriart-Urruty\, Characte
rizations of the plenary hull of the generalized J
acobian matrix\, Mathematical Programming Study\,
17 (198\n2) 1-12\n\n[3] C. Imbert\, Support functi
ons of the Clarke generalized Jacobian and of its
plenary hull\, Nonlinear Analysis: Theory\, Method
s & Applicat\nions archive\nVolume 49 Issue 8\, Ju
ly 2002\nPages 1111-1125 \n\n
LOCATION:Computer Science Sloman Lounge
CONTACT:Paul Taylor
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