Canonical Duality-Triality:

Complete Sets of Analytical Solutions to a Class

of Challenging Problems in Nonconvex Analysis

and Complex Systems

David Yang Gao

Supported by US Air Force AFOSR grants:

US $740k + $445k (2009 - 2015)

1. Unified Modeling and Challenging Problems in

Nonlinear Analysis/Elasticity

2. Canonical Duality Principle and General Analytic Solution

3. Triality and Global Extremal Solutions

Canonical Duality-Triality Theory

A methodological theory comprises mainly

1. Canonical dual transformation

2. Complementary-Dual Principle

3. Triality Theory

Unified Modeling

Unified Solution

Identify both global and local extrema

Design powerful algorithms

Unified understanding complexities

Gao-Strang, 1989 MIT and 1991 Harvard

min = max

min = min

max = max

s

x

P Pd

W(v) : Objective function

W( Q v ) = W( v ) QT = Q -1, det Q = 1

v* = ∂W(v ) Constitutive duality

A(u) = f input f u

In ether case, A(u) = f can be written in

min P(u) = W(Du) – U(u )

2. A(u) is a non potential operator (even order)

1. A(u) is a potential operator (odd order) A(u) = ∂W(u)

min P(u) = W( u ) – ( u , f )

Least squares min P(u) = || A(u) – f || 2

U(u) = (u, f ) subjective function,

action-reaction duality u* = ∂U = f

Unified Modeling in Complex Systems out puts

v* v

D* D

∂W

u u* = f ∂U

( u , f )

( v ; v* )

D: Linea (differential) operator P(u)=0 Euler-Lagrange eqn: A(u) = D* ∂ W( Du ) = f

Exam: W(v) = ½ ( | v |2 - 1 )2

|Q v |2 = vTQTQ v = | v |2

u3 + au2 + b u = f u2 = f

u

W(u)

u

W

Unified Model in Mathematical Physics

- u = f

e s

D = grad D* = - div

(s : e ) s = H e

Au = - m u’’ = f

p v d

dt D = d

dt

D* = -

( p ; v )

p = m v

(u*, u) f = u* u

System:

Au = f input

f

out put

u

Dynamic system: (1) v = D u

Au = D* m Du = f (2) v* = m v

(3) u* = D* v*

Newton: m = const.

Einstein: m = m0 /(1- v2/c2)½

Nash: 2 person game theory

Static system:

Au =- divH grad u = f

p.d H Elliptic

i.d H Hyperbolic

det H = 0 Parabolic

H = I - u = f ( p ; D u) = p u’ dt = - p’ u dt = (D* p , v )

min P(u) = W(Du) - (u , u*)

Buckling Phenomenon 1. Euler-Bernoulli Elastic Beam Model (1750)

0 = uxxxx + uxx

Total Potential: P(u) = ½[uxx2 - ux

2 ]dx

- rutt Chaos

Total Action: [½ r ut2 - P(u) ] dt min?

If < c = min uxx2 dx / ux

2 dx convex P(u) pre-buckling

If > c Concave P(u) crush

u

P

f > 0 f < 0

f

von Karman’s nonlinear plate is linear in 1-D!

- aux2 uxx

2. Nonlinear Beam (Gao, 1996) P(u)= ½ [uxx

2 - ux2 + 1/6 a ux

4 ]dx

e = ux

- f udx

Linear

perturbation

Perfect beam + f

Nonconvexity leads to chaos

New Phenomena in Chaos (Gao, 2005)

t t

u

umax

umax

Tri-Chaos: life of beam vibration Meta-Chaos: sign for changing

Unified nonlinear problems in R∞ Min P(u) = W(Du)+ ½ (u, Qu) – (u, f ), W = ½ (½ |e|2 - a)2

Qu + D* ∂ W(Du) = f

3. Cahn-Hillard: Q = + curl curl , D=I

u + curl curl u + ( ½ |u|2 - a) u = f

5. Nonlinear Schrodinger: Q = ∂tt – i ∂x , D=I

utt - i ux + ( ½ |u|2 - a) u = f

1. Duffing: Q = - ∂tt , D = I

- u,tt + ( ½ u2 - a) u = f 2. Landau-Ginzburg: Q = - , D = I

- u + ( ½ |u|2 - a) u = f

e

W

6. Large Deformed Beam (Gao, 1996): Q = - ∂tt + ∂xxxx , D= ∂x

- utt + uxxxx - [( ½ ux2 - a ) ux]x = f u(t , x) = q(t) sin w x

Duffing eqn

Matlab ode 23 Matlab ode 15s

4. Nonlinear Gordon: Q = - ∂tt + , D=I - utt + u + ( ½ |u|2 - a) u = f

Classification of Nonlinearities In nonlinear analysis, a PDE A(u, ∂u) = f is called

1. Semi-Linear: A is nonlinear in u but linear in e = ∂ u

2. Quasi-Linear: A is linear in the highest order of ∂u

3. Fully Nonlinear: A is nonlinear in the highest order of ∂ u

This classification does not make sense in global analysis!

Min P(u) = W(Du) – (u, f ) A(u, ∂u) = f

convex P(u) monotone A easy problem

Nonconvex P(u) non-monotone A NP-hard problem

Canonical Problem: Min P(u) = V(L(u))– (u, f )

Geometrically nonlinear: L(u) is nonlinear (far from equilibrium)

Physically nonlinear: ∂V(e ) is nonlinear (canonical duality)

Fully Nonlinear: both geometrically and physically nonlinear

u

∂ P = A - f

Convexity vs Nonconvexity

Convex Nonconvex

Static systems Buckling

Dynamical systems Chaos

Computer Science NP-Hardness

Challenges and New Methodology

(VP): min P(u ) = [W(u ) - u f ]d

Nonconvex W(e) non monotone e* = ∂W(e)

P(u)=0 (PDE): - div ∂ W(u ) = f

J. Ball: outstanding open problems … ,

Basic Reason: e = u is not a strain measure

Exam: W = ½ k (e2 - )2

P(u) = 0 [k ux( ux2 - )] x = 0

Three ux = { 0, ± √ } ∞ solutions

x

u

D

e* e

D*

u u*= f

L

e* e

Lt*

e = ux

W(e)

e

V

e* = ∂V

e

e

e*= ∂W

Canonical Transformation: objective W(e) ,

an objective measure e = L( e)

and a convex V(e ) W( e) = V (L( e ) ) and e * = ∂ V(e ) is one-to-one (canonical duality) !

Canonical problem: (P): min P(u) = [V(L( Du )) - u f ] d

= e2

= ½ k (e- )2

Analytical Solutions to “Fully Nonlinear” Problems

(P): min P(u) = k [1 + u’ 2 ] ½ d x - f u dx

s.t. u(0) = 0 = u(1)

Convex W(e) = k [ 1 + e 2 ] ½ (Pd): max P d(s ) = [k 2 - s 2] ½ d x

s.t. - s ’ = f , |s | < k s (x) = f(t) dt + c = g(x) + c

W*(s ) = [k 2 - s 2] ½

u’ = e = ∂ W*(s )

Einstein’s special relativity:

W(v) = - m0 (1- v2/c2)½

Analytic solution (Gao, 2000)

f(x) u x

P(u) = 0 A (u) = - k [ u’ ( 1 + u’ 2 ) - ½ ]’ = f

Canonical Duality-Triality: Nonlinear Differential Eqn Algebraic Eqn

(P): min P(u) = [W(ux) – u f ]dx, W(e) = ½( ½ e2 - 1)2

Diff. eqn: [( ½ ux2 - ) ux]x= f non-unique ux at each x

(Pd): max Pd(s ) = - [½ s - + ½s 2 + s ]dx

s.t. x = f(x)

∂Pd(s ) = 0 Algebraic eqn:

2s 2(s + ) = (x)

Thm (G. 1996): For each critical point si (i=1,2,3),

ui= - x (t) si

-1(t) dt is a critical solution of (P) and

P(u1 ) = min P(u) = maxs > 0 Pd(s ) = Pd(s 1 )

W

e =ux

Newton method:

[s(uxk) ux

k+1] x= f

(x) = f(x) dx

s3 ≤ - 2/3 ≤ s2 ≤ 0 ≤ s1 s

s1 s2 s3 -/3

e = ½ e 2 W(e) = V(e) = ½ (e – 1 ) 2

P(u2 ) = min P(x) = mins < 0 Pd(s ) = Pd(s2 )

P(u3 ) = max P(x) = maxs < 0 Pd(s) = Pd(s3 )

s = ∂ V(e ) = e -

u(t) si (t)

Complete solutions for

(P): min { [½(ut2 – 1)2– u f ]dt = g(u)dt }

Newton method

[s(utk) ut

k+1]t = f(t)

which only produces smooth

numerical “solutions” !

Canonical dual:

s 2(s +) = ½

( t) = - f(t)dt s

( t)

s3(t) ≤ s2 (t) ≤ 0 ≤ s1(t)

s1

s2

s3

u1(t)

u2(t) u3(t)

u(t)

order

disorder

Nonlinear ODE: [s (ut) ut]t = f(t)

f(t) applied force

t

Analytic Sol. ui(t) = si- (t) dt

u(t)

g(u)

g(u)

(x) si(x) ui(x)

ALL Newton-type methods fail to converge

Complete Solutions to 3-D “Fully Nonlinear” PDE

Cauchy Strain tensor: C = F T F

St. Venant-Kichhoff: V(C) = ½ C :H: C = m tr (CC ) + ½ ( tr C )2

Pure Complementary Energy (G. 1999) (Pd): max Pd(T) = [½ tr ( T -1 ) - V*(T)] d , s.t. - div = f

V*(T) =

Thm: For a given s.t. - div = f ,

(Pd) has only one T 0 , at most nine Ti < 0, and 17 indefinite Ti

at each point x .

2nd Piola-Kichhoff stress: T = H -1 : C

(P): min P(u )= [ W(u) - u f ]d W

F Deformation gradient: F = u + I

Each Ti analytical solution

ui (x) = Ti -1 d x i = 1, … 24

Nonsmooth Elasto-Plastic Problem (Gao,2000)

(P): min P(u) = [V(e(ux)) – u f ]dx

Canonical strain: e = L(u) = ½ ux2

V(e) = { ½ k1 e

2 if e ec

½ k1 ec2 + ½ k2 (e – ec ) + s p (e – ec ) if e > ec

Nonsmooth strain energy:

s ∂V (e) discontinuous constitutive law

V s

V* V

e e

V*

s

e

s

V*

V

Fenchel Trans: E *(s ) = sup ( e ; s ) - E( e)

ec

s p

Understand Chaos

Why periodic three Chaos Reason: Nonconvexity (double-well)

f(x,t)

Euler Iteration: F(s (uk)) uk+1 = f

x

P

umax

v = ut

Nonlinear G. Beam: utt - uxxxx + [( - ½ ux ) ux] x

= f

u

Canonical Dual Control Against Chaos Non-convex Dynamical System: q,tt + n q,t = q( - ½ q2 ) + f (t)

(q) = en t [ ½ q,t2 - ½ ( ½ q2 - )2 + q f ] dt

d(p,s ) =en t [( p,t+f )2/(2s) - (p2/2 -s 2/2)] dt

d(p,s ) = 0 Dual Duffing System (DAE):

2s 2(s + ) = ( f - p,t- n q)

[( p,t +n q - f )s - ],t + p = 0

Chaotic &Bifurcation Criterion:

b(t) = ( f - p,t -n q)

If b (t) > bo = (/3 ) 3 one s(t) > 0

If b (t) < bo three s3 ≤s ≤ 0 ≤ s

b b = bo

s

Fig. Dual feedback control of a smart beam

Variation (R∞) Optimization (Rn) min P(x)= W(D x) – ( x , f )

x = {x k }

d inf P(u)= W(Du) – (u , f )

u = {ui (x) }d

Numerical

discretization

Network flow

Transportation

TSP, TTP Problems

W = ½ Huxx2 dx (E-B) W = ½ (Dx)T H(Dx)

= ½ xTAx (A= DT HD )

Structural mech.

f1

f2

f3

f

+ ½ a( |ux| – )2 dx (G, 1996)

ux

P

x

+ S ½akp(|xk – xp|2 – kp

2 )2

Database analysis

Sensor localization

u

Continuum mech.

f < 0

Complete Solutions for Post-Buckled Beam

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.0 0.2 0.4 0.6 0.8 1.0

Beam Axis/m

Beam

defle

ction

/m .

Global max

Local max

Local minne=20

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.0 0.2 0.4 0.6 0.8 1.0

Beam Axis/m

Bea

m d

efle

ctio

n/m

.

Global max

Local max

Local min

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.0 0.2 0.4 0.6 0.8 1.0

Beam Axis/mBe

am d

eflec

tion/

m .

Global max

Local max

Local min

ne=30 ne=40

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.0 0.2 0.4 0.6 0.8 1.0

Beam Axis/m

Beam

def

lectio

n/m

.

Global max

Local max

Local min

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.0 0.2 0.4 0.6 0.8 1.0

Beam Axis/m

Beam

def

lectio

n/m

.

Global max

Local max

Local min

ne=50 ne = 60

- utt + uxxxx - [( ½ ux2 - f ) ux]x = q

What is Chaos and its global optimal solution

Logistic equation in population growth (Ruan and Gao 2012-2014)

ut = r u (1 – k -1 u ) – C(t)

Nonlinear dynamic system u’ = f( t, u ), u (t0 ) = u0

Least squares nonconvex global optimization problem

(P): min || xk+1 - xk - h f (tk , xk ) ||

2 , x R n

u(t) x R n ,

xk+1 = xk + h f (tk , xk ) , k = 1, …, n-1

tn tn-1

tk t2

t1 t0

FDM for t [0, T]

What kind of nonlinear f Chaotic u(t) ?

u 2 – a = 0 a > 0, two real solutions, a < 0, no real solution

W

Li-Zhou-Gao: Nonlinear Dynamics, 2014

Navier-Stokes equation (Euler n = 0 ) :

ut + u .u + p = f + n u, div u = 0

Weak form + (FEM , FDM) Nonlinear quadratic algebraic system

Least squares nonconvex global optimization problem

(P): min || N(uk) - f (tk , xk ) || 2

W

Finite Element Simulations for Large Deformation Hyper-Elastic Digraph (Ionita-Gao, 2001)

Nonconvex, nonsmooth, neoconservative dynamical problem

Computational Biology A novel canonical dual computational approach for prion

AGAAAAGA amyloid fibril molecular modeling

by Zhang, Gao, Yearwood, J. Theoretical Biology 284 (2011) 149–157

min

ij = (x3i-2 – x3j -2) 2 + (x3i-1 – x3j -1 )

2 + (x3i – x3j ) 2

Lennard-Jones Potential

Applications to Discrete Systems Given ai , eik and dij , find xi Rd, such that

| xi - xj | 2 = 2 dij

2 i, j N

| xi – ak | ≤ eik i, k B

NP–complete problem even d =1 !

xi

xj dik

ak

3D Tensegrity graph Biotensegrity

Tensegrity in a cell

(P): min P(x) = S½aij (½ |Di xj|2– dij

2 )2

s.t. x Rd x n , Di xj = xi – xj

Canonical Dual Trans. (Cauchy-type strain)

e = {e ij } = { ½ |Di xj |2 } 0

W(Dx)= E(e (x))= S ½ aij ( ei j – dij2 )2 convex

Sensor localization

Example

(Ruan-Gao, 2014)

Perturbation for

Sensor Network

d = 2, n = 50, a = 4

Min P (u, r)

s.t. r (x) {0, 1}, x , u(x) U

Topology Optimization

Finite Element Method:

= e , u(x) = N(x) q , x e

Mixed Integer Programming

Min { P (q, r) | r {0, 1}m , q Rn }

Topology Optimization provides the most challenging (NP-Hard)

problems in global optimization and computer science!

Canonical transformation: e i = r i (r i – ) = 0

0 < r <

Problems that can be solved

Benchmark Problems:

1. Rosenbrock function

2. Lennard-Jones potential minimization

3. Three Hump Camel Back Problem

4. Goldstein-Price Problem

5. 2n order polynomials minimizations

6. Canonical functions … New math– Nonlinear space

Duality in Nonconvex Systems: Theory, Methods and Application Kluwer Academic Publishers (Springer) , 2000, 454pp

Part I Symmetry in Convex Systems

1. Mono-duality in static systems

2. Bi-duality in dynamical systems

Part II Symmetry Breaking:

Triality Theory in Nonconvex Systems

3. Tri-duality in nonconvex systems

4. Multi-duality and classifications of

general systems Part III Duality in Canonical Systems

5. Duality in geometrically linear systems

6. Duality in finite deformation systems

7. Applications, open problems and concluding remarks

duality in fluid mechanics ? Recent papers:

1. Advances in Canonical Duality-Triality Theory, Two Special Issues of

Math Mech. Solids

2. Canonical Duality-Triality: Unified Understanding Bifurcation, Chaos, and

NP-Hard Problems in Complex Systems

Mixed Integer Optim. Supply Chain Process

Nonconvex/nonsmooth Variational/V.I. Analysis

Graph, lattice, fuzzy max-plus algebra

FEM, FDM, FVM, SDP Meshless, Wavelet, SIP

Discrete optimization

Continuous Optimization

Unified Global Optimization

Combinatorial

Algebra

Numerical

Analysis

Combinatorial Optim. Integer Programming

Canonical

Duality-Triality

Theory

International Society of Global Optimization www.iSoGOp.org

Discussion and Open Problems

1. To model complex phenomena within a unified framework

3. To solve general global optimization/computational sciences problems

2. To understand and identify NP-hard problems

4. Nonlinear PDEs Nonlinear algebraic equations All solutions

5. To understand and solve nonlinear (chaotic) dynamic systems

6. To check, verify and predict model-theory-event

7. Unified understanding natural phinomena

Open Problems: (P) is NP-Hard

if (Pd) has no solution in Sa+

f

x u(x) ≥ 0 nonconvex V.I. + free boundary

f > fc s.t. u(x) = 0 for certain x

u

u

x

u

Thanks!