Optimization Problem Types

Optimization problems types include unconstrained optimization, simple bound constraints, and nonlinearly constrained optimization.

Unconstrained Optimization
An unconstrained problem has no constraints. Thus there are no equality or inequality constraints that the solution b, has to satisfy. Furthermore there are no design limits either.
Simple Bound Constraints
A bound-constrained problem has only lower and upper bounds on the design parameters. There are no equality or inequality constraints that the solution b, has to satisfy. In the finite element world, these are also known as side constraints.
Nonlinearly Constrained Optimization
This is the most complex variation of the optimization problem. The solution has to satisfy some nonlinear constraints (inequality and/or equality) and there are bounds on the design variables that specify limits on the values they can assume.

It is important to know about these problem types because several optimization search methods are available in MotionSolve. Some of these methods work only for specific types of optimization problems.

The optimization problem formulation is:
minimize ψ 0 ( x , b ) (objective function)
subject to ψ i ( x , b ) ≥ 0 ,   i = 1 , ... p (inequality constraints)
ψ i ( x , b ) = 0 ,   i = p + 1 , ... m (equality constraints)
b L ≤ b ≤ b U (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 ( b ) + ∫ t 0 t f L k ( x , b , t ) d t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaeqiYdK3aaSbaaSqaaiaadUgaaeqaaOGaaiik aiaadIhacaGGSaGaamOyaiaacMcacqGH9aqpcqaHipqEdaWgaaWcba Gaam4AaiaaicdaaeqaaOGaaiikaiaadkgacaGGPaGaey4kaSYaa8qC aeaacaWGmbWaaSbaaSqaaiaadUgaaeqaaOWaaeWaaeaacaWG4bGaai ilaiaadkgacaGGSaGaamiDaaGaayjkaiaawMcaaiaadsgacaWG0baa leaacaWG0bGaaGimaaqaaiaadshacaWGMbaaniabgUIiYdaaaa@5B54@