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Bi-Duality, Triality and Extended Lagrangian in Global Optimization – Discovered by David Gao, 1997
(see text below)

 

 

   

Global Optimization is concerned with the computation and characterization of global minima (or maxima) of nonlinear, nonconvex and even nonsmooth functions. Global optimization problems are widespread in the mathematical modeling of real world systems for a very broad range of applications. Such applications include structural engineering, civil and aerospace mechanical design, finite element methods, network, economics of scale, fixed charges, finance, allocation and location problems, operations research, statistics, transportation problems, chip design and database problems, chemical engineering design and control, molecular biology and a number of other combinatorial optimization problems such as integer programming and related graph theory. With the development and implementation of practical global optimization algorithms, more and more scientists in diverse disciplines have been using global optimization techniques to solve problems. Global optimization is now playing a significant role in many sciences.

 Conferences Organized Recently & Keynote speaker for:

 

People in the field:

Panos Pardalos

 

Associated groups:



Selected Recent Publications:
    Monograph: 

     Encyclopedia Articles:

  1. Gao, David Y., Duality-Mathematics,   Wiley Encyclopedia of Electrecal  and Electronical Engineering 6, 1999, 68-77 ps file
  2. Gao, David Y., Mono-duality in Convex Systems, ENCYCLOPEDIA OF  OPTIMIZATION, Kluwer Academic Publishers
  3. Gao, David Y., Bi-duality in convex dynamically systems and D.C. programming, ENCYCLOPEDIA OF  OPTIMIZATION, Kluwer Academic Publishers
  4. Gao, David Y., Triality in Global Optimization,  ENCYCLOPEDIA OF  OPTIMIZATION Kluwer Academic Publishers

     
 
Bi-Duality and Triality in Global Optimization and Variational Analysis

Consider the following nonconvex problem:

Min P(x) = .5 ( .5 x2 - a )2 - c x  

For a given positive parameter a and a constant (input) c, the graph of P(x)  is a double-well function with two minimizers and one local maximizer. All these three extremum values are critical points of P(x), i.e. the roots of the following equilibrium equation:

P'(x) =  x ( .5 x2 - a ) - c = 0.                                                                        


By the canonical dual transformation method developed by D. Gao, the so-called extended Lagrangian associated with this nonconvex problem is

L(x,y) =   (.5 x2  - a )  y - .5 y 2 - c x

 

Its graph is a surface in 2 dimensional space shown in the picture. From the picture we can see that if the dual variable y > 0, then L(x,y)  is a saddle-point function, i.e.

L(x,y) is convex in x and concave in y. However if y < 0, then   L(x,y) is a so-called super-Lagrangian (see Gao's book on Duality Principles in Nonconvex Systems). Thus, a very interesting triality theory was proposed in 1997 (see the original paper by David Gao: Dual extremum principles in finite deformation theory with applications to post-buckling analysis of extended nonlinear beam theory , Applied Mechanics Reviews, ASME,50, no. 11, Part 2, November 1997, S64-S71. ps file )


                        

Graph of P(x) (red), Pd (y) (blue) and Lagrangian


Triality Theorem (Gao, 1997):  For a given parameter a > 0 and constant c such that (xc , yc ) is a critical point of L(x,y).  If  yc > 0, then  (xc , yc ) is a local saddle point of  L(x,y) and

minx maxy L(x,y) =  L( xc , yc ) = maxy minx L(x,y)

If  yc < 0, then  (xc , yc ) is a local super-critical point of  L(x,y) and in this case, we have either

minx maxy L(x,y) =  L( xc , yc ) = miny maxx L(x,y)

or
 

maxx maxy L(x,y) =  L( xc , yc ) = maxy maxx L(x,y)


From the extended Lagrangian L(x,y) the so-called canonical dual function can be obtained by solving the following stationary problem: for a given non zero y

Pd(y)= stax L(x,y) = - .5 c2/y  - .5 y2  - a y


Then the canonical dual problem associated with the original primal problem is
to find stationary points yc of Pd(y) such that
 

Pd(yc ) = Sta { Pd(y)=  - .5 c2/y - .5 y2  - a y | for all non zero y } .

The criticality condition of  Pd(y)  leads to the following dual equilibrium equation:

 

2 y2 (  y  +  a  ) = c2.

This cubic algebraic dual equilibrium  equation has at most three real roots satisfying:

y3 <=  y2 < 0 < y1 .

Tri-duality Theorem (Gao, 1999):  Suppose that (xc , yc ) is a critical point of L(x,y).  Then we have

P(xc ) = L( xc , yc ) = Pd( yc ).

If  yc > 0, then  xc is a global minimizer of  P( xc )  and yc  is a global maximizer of  Pd( yc ), i.e.

P(xc ) = minx P( x ) =   maxy  Pd( y ) = Pd( yc ).

If  yc < 0, then   xc  and  yc  are either    local minimizers or local maximizers of P(x) and Pd( y ),  respectively, i.e.   we have either

P(xc ) = minx P( x ) =   miny  Pd( y ) = Pd( yc )

or
 

P(xc ) = maxx P( x ) =   maxy  Pd( y ) = Pd( yc ).

 

 

 

This triality theory has important applications in nonconvex variational analysis, PDEs, chaotic dynamics,  global optimization and numerical methods.
Detailed information can be found in Gao's recent papers and books at his web sites: http://www2.latech.edu/~dgao/ and http://www.math.vt.edu/people/gao/

 

 

Research Projects:


1. Eigenvalue Problem on Optimal Surface with Fixed Boundary
New Phenomena: Singularity on the surface when domain is getting smaller
Discovered by Gao and Yau at Harvard in 1991.
 

 

There are many very interested phenomena and associated mathematical problems, numerical methods needed to be investigate for this very important project, which might bring some new insights into many fields!

 

2. Optimal Shape  Design for  Beam  Model  Subjected  to  Given  Total Materials, distributed load and Boundary  Conditions.
       There is also a singularity on the optimal shape of the beam discovered by Gao in 1994 (see the following shape evolution)

 

 

3. Feedback Control Against Chaotic Vibration and Smart Beam
Mathematically speaking, the total potential of a chaotic system is usually nonconvex or even nonsmooth. Very small perturbations of the system's initial conditions and parameters may lead the system to different operating points with significantly different performance characteristics. The
numerical results vary with the methods used. This is the one of main reasons why the traditional perturbation analysis and the direct approaches cannot successfully be applied to chaotic systems.
Duality is a fundamental concept that underlies all most all natural phenomena. Based on the triality theory discovered recently, some chaotic bifurcation criteria have been proposed which can be used to control chaotic vibration of nonconvex structures like large deformed beam and plates.
See recent paper: Canonical Dual Control for Nonconvex Distributed-Parameter Systems: Theory and Method, in Control of Nonlinear Distributed Parameter Systems, Goong Chen, Irena Lasiecka and Jianxin Zhou (eds). Marcel Dekker, 2001, 85-112.