on Complementarity Duality and Global Optimization in Science and Engineering February 28March 2 2007 Industrial and Systems Engineering Department A CategoryTheoretic Approach to Duality ID: 617618
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CDGO 2007: 2nd International Conference on Complementarity, Duality and Global Optimization in Science and Engineering February 28-March 2, 2007 Industrial and Systems Engineering Department Slide2
A Category-Theoretic Approach to Duality Sabah E. Karam, Information Specialist
Morgan State University
Planning & Information Technology
Baltimore, MD 21251
tel
:
443-885-4597
email: Sabah.Karam@ morgan.edu
CDGO
2007
Slide3
Historical notesCategories were first introduced by S.Eilenberg and S. MacLane during the years 1942-1945, in connection with algebraic topology, a branch of mathematics in which tools from abstract algebra are used to study topological spaces. Category theory has come to occupy a central position in pure mathematics and theoretical computer science.Categories are algebraic structures with many complementary natures, e.g., geometric
, logical, computational, combinatorial.
Category Theory is an alternative to classical
set theory
as a
foundation for mathematics
. The primitive, set-theoretic concept of "element" or "membership" is replaced by that of "
function." CDGO 2007 Slide4
Applications of CTMathematics and Computer ScienceQuantum Physics and Tensor CTGenomes and Computational BiologyInformation Systems (databases, OOT)Unified Modeling Language and Software EngineeringCompiler OptimizationLogic and PhilosophyNatural Transformation Models in Molecular Biology
Neural Network Analysis and Design
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Slide5
Reasons to use CTit is a unifying language for discussing different mathematical models and other logic-based structures,it reveals common structures in seemingly unrelated systems and a framework for comparing them,it reveals invertible structures, i.e. for every categorical construct there is a dual formed by reversing all the transformations,it consolidates the description of similar operations such as
'products'
found in set theory, group theory, linear algebra, and topology, and
it produces
graphical models
which are
intuitive
, formal, declarative, and subject to further analysis.CDGO 2007
Slide6
A category consists of 3 entities: objects, morphisms, and compositionsa class of objects (A, B, C, …)a class of morphisms between objects symbolized by ‘’ For each morphism one object, A, is the domain of f and another object, B, is the codomain, f: AB.
a binary operation called
composition
. For each pair of morphisms f: A
B and g: B
C, a composite morphism, g ○ f: AC is defined.
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Slide7
Morphisms have two propertiesAssociativity: If f : A → B, g : B → C and h : C → D then h
○ (g ○
f
) = (
h
○
g
) ○ f , and Identity : For every object A, there exists a morphism 1A : A → A called the identity morphism for A, such that for every morphism f : A → B, we have 1
A
○
f
=
f
=
f
○ 1
A
.
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Slide8
SOME MORE TERMINOLOGYEvery morphism has a source object, called the domain, and a target object, called the codomain. If f is a morphism with X as its source and Y as its target, we write f: X → Y. We write Hom(X,Y) for the set of morphisms from X to Y. In traditional set theory morphisms are nothing more than the set of functions from X to Y.Hom( ) is short for Homology.
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Slide9
What is a homology?A correspondence or structural parallel. In biology, two or more structures are said to be homologous if they are alike because of shared ancestry. This could be evolutionary ancestry, e.g. the wings of bats and the arms of humans, or developmental ancestry, e.g. the ovaries of female humans and the testicles of males.Scientists use physical structures to reconstruct evolutionary history.
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Slide10
What is a homology (cond’t)?In mathematics, especially algebraic topology and abstract algebra, homology is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). In anthropology and archaeology, homology refers to a type of analogy whereby two human beliefs, practices or arte-facts are separated by time but share similarities due to genetic or historical connections. CDGO
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Elementary ExampleAny partial ordering, sequencing, or arrangement of the elements of a set (a) Objects are the elements of the partial order; numbers, sets, points in a plane, integers, people in a genealogy relationship, (b) Morphisms: , , divisibility relationship. (c) Composition works because of transitivity.
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Slide12
Mathematical CategoriesSet = sets with linear transfomations Vect = vector spaces with linear transfomations Poset = partially ordered sets with monotone functionsGrp = groups with group homomorphisms Top = topological spaces with continuous functionsDiff = smooth manifolds with smooth mapsRing = rings with ring homomorphisms
Met
= metric spaces with contraction maps
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Slide13
Functors, Natural Transformations, and AdjointsSaunders MacLane, one of the founders of category theory, remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Also called natural equivalence or isomorphism of functors.The context of Mac Lane's remark was the axiomatic theory of homology. With the language of natural transformations he could easily express: (i) how homology groups are compatible with morphisms between objects and (ii) how two equivalent homology theories not only have the same homology groups but also the same morphisms between those groups. CDGO
2007
Slide14
Definition of FunctorLet C and D be categories.A functor F from C to D is a mapping that (a) associates with each object X ε C an object F(X) ε D, and (b) associates with each morphism f: XY a morphism
F
(
f):
F
(X)
F(Y) such that the following two properties hold: (i) F(1A) = 1 F(A) for every object, and (ii) F(g ○ f ) = F( g) ○ F (f )
That is to say, functors
preserve
identity morphisms and composition of morphisms.
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Slide15
Example of a functorGiven a set S = {a, b, c, 1, 2, 3, @, # ,$)Objects: List(S) = {a, b2, c$, 3#a3, ...}, L = [s1, s2, s3, s4, …] Morphisms: f: S S’ (e.g. a sort routine)
Identity
: we also need to define an associative binary concatenation operator,
call it *, and an identity operator, call it [ ], such that [ ] * L = L = l * [ ].
Functor
: F(f):
List
(S) List(S’) List(f)( L) = [ f(s1), f(s2), f(s3), f(s4), …]Equivalent to the java class
mapList.
It can be used to create a dictionary by reading a collection of words and definitions.
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Slide16
Object-Oriented (OO) TechnologyObjects are the principle building blocks of object-oriented programs. Each object is a programming unit consisting of data (instance variables) and functionality (instance methods). CDGO 2007
Customer_Order
CustomerID
customerName
dateShipped
dateReceived
datePayed
checkInventory( )
contactCustomer( )
Ship( )
refund( )
calculateSale( )Slide17
Definition of Natural TransformationLet X and Z be two categories and let F and G be two functors F: X Z, and G: X Z. Let f: A Bη is a NT from F to G, written η:F G, if the diagram commutes.CDGO
2007
η
B
F(f)
F(A)
F(B)
F(B)
η
A
G(f)
F(A)Slide18
Examples of Natural Transformations (NT)NT’s are structure preserving mappings from one functor to another functor. Two decks of playing cards, all analog wrist watches, and all tie shoes with the same number of holes are isomorphic. Consider f(x + y) = f(x) + f(y). Then f(x) = 4x is one such preserving map, since f(x + y) = 4(x + y) = 4x + 4y = f(x
) + f(y).
Consider f
(
a
+
b
) = f(a) * f(b), Then f(x) = ex satisfies this condition since 5 + 7 = 12 translates into e5 * e7 = e12. In group theory, every group is naturally isomorphic to its opposite group in which the preserving map, F(a*b) = b*a, inverts the binary operation. The dual of dual is a “natural transformation” in category theory
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Slide19
Analogous electric and mechanical systemsElectrical and mechanical system have differential equations of the same form and can be considered isomorphic. Electrical: e = iR e = voltage, i = current, R = resistance L = inductance, C = capacitance, Q = charge LQ'' + RQ' + Q/C = 0 Mechanical: f = vB v = velocity, f = force, B = friction, M = mass spring-mass differential eq. mx'' + bx' + kx = 0
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Slide20
Functional programming languageObjects: Int, Real, Bool, Char, RefMorphisms: isZero: Int Bool (test for zero) not: Bool Bool (negation) succInt: Int Int (successor) toReal: Int Real (conversion)Constants: zero (Int) , true/false (Bool)
Composition
: false = not
○
true
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Slide21
Contravariant functors & Dual SpacesThere are many constructions in mathematics which would be functors but for the fact that they "turn morphisms around" and reverse the direction of composition F(g ○ f ) = F( f ) ○ F (g ).Dual vector spaces, maps which assign to every vector space its dual space reflect, in an abstract way, the relationship between row vectors and column
vectors. The dual or transpose is a
contravariant functor
from the category of all vector spaces over a fixed field to itself.
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Primal/Dual & Minimax TheoremA dual mathematical program has the property that its objective is always a bound on the original mathematical program, called the primalMinimax theorem proven by von Neumann in 1928, it is a cornerstone of duality and of game theoryLet X and Y be mixed strategies for players A and B. Let A be the payoff matrix. Then max min XT
AY = min max XTAY
X
Y
Y XCDGO 2007 Slide23
Commutative diagrams and LPmin{max F(x,y): y in Y}: x in X} = max{min F(x,y): x in X}: y in Y} F: X*Y R and X and Y are non-empty, convex, compact sets.CDGO
2007
min
min
max
max
LOSS
GAIN
Optimal solution
Original problemSlide24
Primal/dual commutative diagram CDGO 2007
transpose
matrix A
new obj. function
The original
objective function
Optimal solution
max c
T
x
subject to Ax
≤ b
min b
T
y
subject to A
T
x
≥ bSlide25
Commutative diagrams as ProofsCommutative diagrams play the role in category theory that equations play in algebra.Commutative diagrams can be used to assert the validity of program transformations. Diagram chasing is a method of mathematical proof especially in homological algebra and computer science.CDGO 2007
succ
Real
toReal
Int
Int
Real
Real
succ
Int
toRealSlide26
Relationship between M-theory and Type IIA supergravity/string theoryIn the strong coupling limit type IIA string theory approaches an 11 dimensional Lorentz invariant theory.Commutative diagrams are used to assert the validity of relationships between theories. CDGO 2007
S1 compactification
low energy
limit
M theory
Type IIA string theory
D=11 supergravity
Type IIA supergravity
S1 compactification
low energy limitSlide27
Classification of DualsDorn's dual primal/dual are convex quadratic programs Fenchel's Conjugate Dual Generalized penalty-function/surrogate Dual Geometric dual Inference dual Lagrangian Dual
LP Dual. This is the cornerstone of duality. In canonical form: Primal: Min
{cx: x >= 0, Ax >= b}.
Dual
:
Max
{yb: y >= 0, yA <= c}. CDGO 2007 Slide28
Duals (cond’t)Self Dual when a dual is equivalent to its primal - LP problems Semi-infinite Dual Superadditive Dual Surrogate Dual Symmetric Dual Wolfe's Dual
Hooker: A relaxation dual in which there is a finite algorithm
for solving the relaxation is an inference dual. An
inference dual
in which the
proofs are parameterized
is a relaxation dual. CDGO 2007 Slide29
Natural Transformation Models in Molecular BiologyMolecular models in terms of categories, functors and natural transformations are introduced for: (a)unimolecular chemical transformations, (b) multi-molecular chemical, and (c) biochemical transformations
Several applications of such natural transformations are then presented to analyze
(a)
protein biosynthesis
,
(b)
embryogenesis
and (c) nuclear transplant experimentsCDGO 2007 Slide30
Complex graph matching problems as combinatorial optimization Phletora of confusing terms: computational complexity; neural networks; linear programming; weighted graph matching; quadratic optimization; simplex-based algorithm; Hungarian method; eigendecomposition; pattern recognition; symmetric polynomial transform; genetic algorithms; probabilistic relaxation; clustering techniques.CDGO 2007 Slide31
Duality, polarity, complementarityElectronics: two devices or two circuits having mathematical descriptions that are identical except that voltages in one formula correspond to currents in the other formula. Chemistry: Conjugate Acid/Base pairs Electromagnetic theory: electric fields are dual to magnetic fields.Meterology: precipitation/evaporationMathematics: projective geometry, category theory, Morgans laws (logic), set theory, operations (+/-, x/÷, ∫/Dx
).
Biology
:
dualism is the theory that blood cells have two origins, from the lymphatic system and from the bone marrow.
Physics
:
particle/wave nature of light, electrical-mechanical duality of the differential equations.CDGO
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Duality, polarity, complementarity Genetics: DNA base-pairing (A-T, C-G)Philosophy: yin-yang basis of Chinese medicineEndocrinology: metabolic processes that assemble/ disassemble (anabolic/catabolic) molecules in the body = a hormonal processTheology: Koranic verses describe created pairs Molecular biology: code-duality (analog/digital)
Language:
structure from which meaning is derived
Psychology
:
referent and probe in judgment
General theory of relativity
: 4 elementary forcesQuantum field theory: fermion-boson dualityCDGO 2007 Slide33
Establishing a curriculum based on a Duality PrincipleDr. Glenda Prime, Coordinator, Doctoral Programs in Mathematics and Science Education Glenda.Prime@morgan.edu The School of Education and Urban Studies, through the Doctoral Programs in Mathematics Education and Science Education seeks to enhance the quality of science and mathematics education by preparing a cadre of highly qualified mathematics and science educators, supervisors and curriculum specialists.
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ReferencesS. Eilenberg and S. MacLane, "Natural Isomorphisms in Group Theory," Proceedings of the National Academy of Sciences, 28, (1942), pp. 537-543 S. Mac Lane, “Categories for the Working Mathematician” 2nd edition, Springer (2000) J.N. Hooker, Duality in Optimization and Constraint Satisfaction, Carnegie Mellon Univ., Pittsburgh, PA (2006) http://wpweb2.tepper.cmu.edu/jnh/duals.pdf M. Barr and C. Wells, “Category Theory for Computing Science” 3rd edition, CRM (1999)Proceedings SIAM & Society for Mathematical Biology Meeting N/A(3), pages pp. 230-232, Colorado, 1983.
http://glossary.computing.society.informs.org/second.php?page=duals.html
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End of PresentationWe would like to thank Organizers: Panos M. Pardalos & Altannar Chinchuluun, Univ. of FloridaAdvisory Committee: David Y. Gao & Hanif D. Sherali, Virginia Tech Univ.And to everyone who attended this session CDGO
2007