MER 160, Prof. Bruno1 Optimization The idea behind “optimization” is to find the “best”...
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Transcript of MER 160, Prof. Bruno1 Optimization The idea behind “optimization” is to find the “best”...
MER 160, Prof. Bruno 1
OptimizationOptimization
• The idea behind “optimization” is to find the “best” solution from a domain of “possible” solutions.
• Optimization methods provide mathematical tools that allow the search for this “best” solution to carried out in a rational and efficient way.
• Before these tools can be applied the design problem needs to be recast in an appropriate form.
MER 160, Prof. Bruno 2
Optimization in Design
Need Identified
Problem Definition
Concept Generation
Modeling/Simulation
Workable Design
Optimization/Optimal Design
Concept Selection
MER 160, Prof. Bruno 3
OptimizationOptimization
x1
x2
*
Set of all “workable”or “functional designs”(Allowed by physics,orange boarder)
Set of all “acceptable”designs. (allowed by constraints, yellow boarder)
Optimal DesignU(x1,x2) = Umax
H2 : X2> c2
H1 : X1> c1
MER 160, Prof. Bruno 4
LingoLingo
• Objective Function: represents the quantity (U) which is to be optimized (the “objective”) as a function of one or more independent variables (x1, x2, x3…)
– The best form to put the objective function in depends on the optimization technique to be employed.
U = U(x1, x2, x3…)→Uopt
MER 160, Prof. Bruno 5
LingoLingo
• Design Variables: The independent variables (x1, x2, x3…) that the objective function depends on.– It is generally best to minimize the number of
design variables … the more variables the tougher the optimization will be.
MER 160, Prof. Bruno 6
Lingo Cont.Lingo Cont.• Constraints: Relations which limit the possible
(physical limitations) or the permissible (external constraints) solutions to the objective function.
• Constraints come in two mathematical “flavors.”– Equality Constraints: Often come from fundamental
physics considerations (e.g. cons. of mass) – Inequality Constraints: Often from safety, cost, space,
material strength limits etc.
• Generally Equality Constraints are easier to deal with than inequality constraints.
• Generally it is desirable to reduce the number of constraints.
MER 160, Prof. Bruno 7
Mathematical FormulationMathematical Formulation• Objective Function of n independent design
variables:For U( x1, x2, x3…xn) Find Uopt
• Equality Constraints: Gi( x1, x2, x3…)=0 i=1,2,…,m
• Inequality Constraints:Hj(x1, x2, x3…) < or > Cj j=1,2,…l
If n>m → An Optimization problem resultsIf n=m → A unique solution exists…just solve all
equations simultaneouslyIf n<m → The problem is “overconstrained” no solution
which satisfies all of the constraints is possible
MER 160, Prof. Bruno 8
OptimizationOptimization
x1
x2
*
Set of all “workable”or “functional designs”(Allowed by physics,orange boarder)
Set of all “acceptable”designs. (allowed by constraints, yellow boarder)
Optimal DesignU(x1,x2) = Umax
H2 : X2> c2
H1 : X1> c1
MER 160, Prof. Bruno 9
Classification of Optimization TechniquesClassification of Optimization Techniques• Calculus based Techniques
– Lagrange Multipliers
• Search Methods– Elimination Methods
• Exhaustive• Fibonacci• golden section search
– “Hill Climbing” techniques• Lattice Search• Steepest ascent
• “Programming” methods– Linear Programming– Geometric Programming
MER 160, Prof. Bruno 10
Techniques Cont.Techniques Cont.
• Calculus Methods: Related in principle to the simple “minimization” and “maximization” techniques that you used in Calculus. – Require all equations to be differentiable
(a.k.a. Continuous, “well behaved” and explicit)
– This puts severe limitations on the usefulness of this technique
MER 160, Prof. Bruno 11
Techniques Cont.Techniques Cont.• Search Methods: Are exactly what they
sound like … you keep trying different solutions searching for the “best.”– These techniques are Very flexible, making
them applicable to a broad array of problems.– The “techniques” provide algorithms which
help to find the optimum in the minimum possible number of searches.
– Very useful when a design variable can only take on certain discrete values.
– These methods do result in a finite amount of uncertainty about the optimum.
MER 160, Prof. Bruno 12
Exhaustive SearchExhaustive Search
x1
x2
*
Optimal DesignU(x1,x2) = Umax
H2 : X2> c2
H1 : X1> c1
Note: None of the searchpoints exactly hits the optimum. The spacebetween search points isknown as the “interval of uncertainty”
The aptly named
MER 160, Prof. Bruno 14
““Programming” MethodsProgramming” Methods• These methods have nothing to do with “Programming” in
the sense that you usually think of it!• Linear Programming: Very powerful, but very limited!
– Applies only when the objective function and all constraints can be expressed as Linear Functions … Not often the case in thermal/fluid systems
• Dynamic Programming: Related to optimizing a “process”– Lets you find the “best” order to do steps in– Very useful in Project Management and Assembly optimization
• Geometric Programming: Similar to Linear Programming, but now all functions must be polynomials.– Reduces the restrictions on linear programming quite a bit– Very useful where empirical correlations for system behavior are
known– More difficult and computationally intensive than Linear Prog.
MER 160, Prof. Bruno 15
ExampleExample
• Set up an optimization problem formulation for a water Chilling system to minimize first cost.
MER 160, Prof. Bruno 16
Homework?Homework?• What exactly are you trying to optimize?
→ “What is your Objective Function?”• What is the Absolute maximum that one
would be willing to pay? → “Is there a cost inequality constraint that we can use to help limit our design domain?”
• What are your “design variables” ?• What is the nature of your functions?
(continuous / discrete) (linear/non-linear) etc.