The Optimization Problem Formulation

Optimization algorithms work to minimize (or maximize) an objective function subject to constraints on design variables and responses.

The optimization to be solved is expressed as follows:
Minimize: ψ 0 ( x , b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdK3aaSbaaSqaaiaaicdaaeqaaOGaaiik aiaadIhacaGGSaGaamOyaiaacMcaaaa@43A9@ (objective function)
Subject to: ψ i ( x , b ) 0 ,   i = 1 , ...   p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdK3aaSbaaSqaaiaadMgaaeqaaOGaaiik aiaadIhacaGGSaGaamOyaiaacMcacqGHLjYScaaIWaGaaiilaiaabc cacaWGPbGaeyypa0JaaGymaiaacYcacaGGUaGaaiOlaiaac6cacaWG Wbaaaa@4E1A@ (inequality constraints)
  ψ i ( x , b ) = 0 ,   i = p + 1 , ... ,   m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdK3aaSbaaSqaaiaadMgaaeqaaOGaaiik aiaadIhacaGGSaGaamOyaiaacMcacqGH9aqpcaaIWaGaaiilaiaabc cacaWGPbGaeyypa0JaamiCaiabgUcaRiaaigdacaGGSaGaaiOlaiaa c6cacaGGUaGaamyBaaaa@4F2E@

b L b b U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyamaaBaaaleaacaWGmbaabeaakiabgsMi JkaadkgacqGHKjYOcaWGIbWaaSbaaSqaceaatLVaamyvaaqabaaaaa@4679@

(equality constraints)

(design limits)

     

The functions ψ k ( x , b ) ,   k = 0 , ... , m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdK3aaSbaaSqaaiaadUgaaeqaaOGaaiik aiaadIhacaGGSaGaamOyaiaacMcacaGGSaGaaeiiaiaadUgacqGH9a qpcaaIWaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWGTbaaaa@4C4A@ are assumed to have the form:

ψ k ( x , b ) = ψ k 0 ( x , b ) + T 0 T f L k ( x , b ) d t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdK3aaSbaaSqaaiaadUgaaeqaaOGaaiik aiaadIhacaGGSaGaamOyaiaacMcacqGH9aqpcqaHipqEdaWgaaWcba Gaam4AaiaaicdaaeqaaOGaaiikaiaadIhacaGGSaGaamOyaiaacMca cqGHRaWkdaWdXbqaaiaadYeadaWgaaWcbaGaam4AaaqabaGcdaqada qaaiaadIhacaGGSaGaamOyaaGaayjkaiaawMcaaiaadsgacaWG0baa leaacaWGubWaaSbaaWqaaiaaicdaaeqaaaWcbaGaamivaiaadAgaa0 Gaey4kIipaaaa@5B50@

In this formulation:
  • b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ is an n-dimensional vector of real-valued design variables
  • b L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ and b U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ are the lower and upper bounds, respectively, on the design variables
  • x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamiEaaaa@3DFB@ is the set of states that the solver uses to represent the system
  • ψ k 0 ( x , b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdK3aaSbaaSqaaiaadUgacaaIWaaabeaa kiaacIcacaWGIbGaaiykaaaa@42EC@ is the value of the function from a previous simulation. For the first simulation, it is always zero.

The goal of the optimization effort is to minimize the objective function ψ 0 ( x , b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdK3aaSbaaSqaaiaaicdaaeqaaOGaaiik aiaadIhacaGGSaGaamOyaiaacMcaaaa@43A9@ while satisfying the constraints ψ k ( x , b ) ,   k = 1, ...,   m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdK3aaSbaaSqaaiaadMgaaeqaaOGaaiik aiaadIhacaGGSaGaamOyaiaacMcacaGGSaGaaeiiaiaadMgacqGH9a qpcaaIXaGaaiOlaiaac6cacaGGUaGaamyBaaaa@4AE7@ . Constraints are assumed to be inherently nonlinear. They can be either inequality or equality constraints.

b L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ and b U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ define the lower and upper bounds for the elements of b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ . The set of all allowable values of b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ is known as the design space for the problem. A design point or a sample point is a particular set of values within the design space.

A design point is said to be feasible if and only if it satisfies all the constraints. Correspondingly, a design point is said to be infeasible if it violates one or more of the constraints. Our aim, of course, is to find a feasible design. Sometimes, due to the presence of constraints, this may not be possible.

Many different methods are available for solving the above-mentioned optimization problem. All of these iterate on the design b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ in some manner to find a better solution. A generic algorithm that describes this process is:
  • An initial value for the design variables b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ is provided to the optimizer.
  • The response quantities ψ k ( x , b ) ,   k = 0 , ... , m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdK3aaSbaaSqaaiaadUgaaeqaaOGaaiik aiaadIhacaGGSaGaamOyaiaacMcacaGGSaGaaeiiaiaadUgacqGH9a qpcaaIWaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWGTbaaaa@4C4A@ are computed by running a simulation.
  • Some algorithm, sometimes a sensitivity-based method, is applied to generate a new b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ that will either reduce the objective function, reduce the amount of infeasibility, or both.
  • When a sensitivity-based method is used, the optimizer also needs to compute the sensitivity of the functions ψ k ( x , b ) ,   k = 0 , ... , m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdK3aaSbaaSqaaiaadUgaaeqaaOGaaiik aiaadIhacaGGSaGaamOyaiaacMcacaGGSaGaaeiiaiaadUgacqGH9a qpcaaIWaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWGTbaaaa@4C4A@ with respect to the design b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ . This means the optimizer requires the matrix of partial derivatives, [ ψ ( x , b ) b ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaWaamWaaeaadaWcaaqaaiabgkGi2kabeI8a5jaa cIcacaWG4bGaaiilaiaadkgacaGGPaaabaGaeyOaIyRaamOyaaaaai aawUfacaGLDbaaaaa@486E@ .
  • In an iterative fashion, new designs ( b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaamOyaaaa@3DE5@ ) are generated by the optimizer until a determination is made that the optimizer has found a minimum or the iteration limits have been exceeded.

In some instances you may want to maximize the value of a certain objective. Without any loss of generality, you can convert it to a minimization problem by simply negating the objective you calculate. Thus, if you want to maximize a function χ ( x , b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeq4XdmMaaiikaiaadIhacaGGSaGaamOyaiaa cMcaaaa@42A2@ , you can convert it to a minimization problem by defining the cost function as ψ ( x , b ) = χ ( x , b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdKNaaiikaiaadIhacaGGSaGaamOyaiaa cMcacqGH9aqpcqGHsislcqaHhpWycaGGOaGaamiEaiaacYcacaWGIb Gaaiykaaaa@4A50@ .