Unconstrained and Constrained Optimization Algorithms By Soman K.P
Optimization Optimization is the methodology for obtaining the best alternative from all possible...
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Optimization • Optimization is the methodology for obtaining the best alternative from all possible ones. • Unconstrained optimization has only an objective function. – What is the route and travel speed that minimize the distance, or travel time, or number of turns from UF to your home? • Design variables are road segments and speeds. • In constrained optimization we add constraints. – No more than 3 stop lights on the way (inequality). – Drive exactly at speed limit (equality constraint).
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Standard formulation The standard formulation of an optimization problem is: Minimize: What do we do for maximization? Inequalities have less than zero. What do we do when we have the opposite sense?
Transcript of Optimization Optimization is the methodology for obtaining the best alternative from all possible...
PowerPoint PresentationOptimization
Optimization is the methodology for obtaining the best alternative from all possible ones.
Unconstrained optimization has only an objective function.
What is the route and travel speed that minimize the distance, or travel time, or number of turns from UF to your home?
Design variables are road segments and speeds.
In constrained optimization we add constraints.
No more than 3 stop lights on the way (inequality).
Drive exactly at speed limit (equality constraint).
Standard formulation
Minimize: What do we do for maximization?
Inequalities have less than zero. What do we do when we have the opposite sense?
Example 1.4.1 (but differently)
Choose the fiber fraction for Kevlar epoxy, so as to minimize it, and satisfy the following constraints
Objective function is equal to the single design variable.
To formulate constraints we need the moduli of fiber and matrix
Formulation
Optimization problem
Need analysis to calculate constraints in terms of design variable (often comes from a computer code rather than an equation).
Analysis
It pays to invest in a bit of algebra
This reveals that the solution will not depend on the individual moduli but only on their ratio.
For =0 get
Solution near =0.2
Vf=linspace(0,0.4,101);
E2oE1=1./E1oE2;
Feasible domain when both are negative
Vf=linspace(0.1,0.3,101);
E2oE1=1./E1oE2;
legend('g1','g2','Location','North')
Solution is 0.1675. Textbook says 0.18 (one student found that the denominator should be 1-r not 1+r, giving 0.1771
Stacking sequence optimization
Most of this course is concerned with optimization of the angles of the fiber in a laminate composed of unidirectional plies.
The objective function is either the thickness (total number of plies) or the load that can be carried by a laminate of given thickness.
(
)
Optimization is the methodology for obtaining the best alternative from all possible ones.
Unconstrained optimization has only an objective function.
What is the route and travel speed that minimize the distance, or travel time, or number of turns from UF to your home?
Design variables are road segments and speeds.
In constrained optimization we add constraints.
No more than 3 stop lights on the way (inequality).
Drive exactly at speed limit (equality constraint).
Standard formulation
Minimize: What do we do for maximization?
Inequalities have less than zero. What do we do when we have the opposite sense?
Example 1.4.1 (but differently)
Choose the fiber fraction for Kevlar epoxy, so as to minimize it, and satisfy the following constraints
Objective function is equal to the single design variable.
To formulate constraints we need the moduli of fiber and matrix
Formulation
Optimization problem
Need analysis to calculate constraints in terms of design variable (often comes from a computer code rather than an equation).
Analysis
It pays to invest in a bit of algebra
This reveals that the solution will not depend on the individual moduli but only on their ratio.
For =0 get
Solution near =0.2
Vf=linspace(0,0.4,101);
E2oE1=1./E1oE2;
Feasible domain when both are negative
Vf=linspace(0.1,0.3,101);
E2oE1=1./E1oE2;
legend('g1','g2','Location','North')
Solution is 0.1675. Textbook says 0.18 (one student found that the denominator should be 1-r not 1+r, giving 0.1771
Stacking sequence optimization
Most of this course is concerned with optimization of the angles of the fiber in a laminate composed of unidirectional plies.
The objective function is either the thickness (total number of plies) or the load that can be carried by a laminate of given thickness.
(
)
