Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University 14. Numerical Approaches for Solving Optimization Problems
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Hae-Jin ChoiSchool of Mechanical Engineering,
Chung-Ang University
14. Numerical Approaches for Solving
Optimization Problems
SCHOOL OF MECHANICAL ENG.
CHUNG-ANG UNIVERSITY-1-
SCHOOL OF MECHANICAL ENG.
CHUNG-ANG UNIVERSITY
If an optimization problem involves the objective/constraint functions that are
not stated as explicit functions or too complicated to manipulate, we cannot
solve it with the analytical approaches learned in the previous lectures.
Numerical approaches for optimization problems can be analogous to the
numerical techniques, such as Lunge-Kutta method and Simpson rule, for
mathematical solutions of differentiation and integration.
Numerical approaches are classified into several categories depending on the
types of optimization problems
MATLAB toolbox, called ‘optimization toolbox’ is a useful tool for practical use
of optimization techniques in various engineering optimization problems.
DOE and Optimization
Introduction to Numerical Approach
SCHOOL OF MECHANICAL ENG.
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CHUNG-ANG UNIVERSITYDOE and Optimization
MATLAB Functions for Solving Opt. Prob.
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MATLAB GUI Tool for Optimization
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Elimination methods
Unrestricted search
Exhaustive search
Dichotomous search
Fibonacci method
Golden section method
Interpolation method
Quadratic interpolation method
Cubic interpolation method
Direct root methods
DOE and Optimization
Techniques for One-variable Opt. Prob.
Golden section method
Quadratic interpolation method
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Direct search methods
Random search method
Grid search method
Univariate method
Pattern search methods
Decent methods
Steepest Descent (Cauchy) method
Newton’s method
Quasi-Newton method
DOE and Optimization
Techniques for Unconstrained Opt. Prob.
Grid search method
Steepest descent method
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Techniques for solving linear constrained
problems.
Linear Programming (Linear objective
function)
Simplex Method -> next week
Revised Simplex Method
Dual Simplex Method
Quadratic Programming (Quadratic
objective function)
Kuhn-Tucker necessary condition +Simplex
Method
DOE and Optimization
Techniques for Linear Constrained Prob.
Simplex method
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Direct methods
Random search methods
Sequential Linear Programming (SLP)
Sequential Quadratic Programming (SQP)
Most recent and popular method for constrained optimization problem
Convert original problem into stepwise Quadratic Programming
Linearization of the constraints
Solve QP for search direction and find new evaluation point
Iteratively process the above steps until converge to optimum point.
Indirect methods
Interior penalty function method
Exterior penalty function method
DOE and Optimization
Techniques for Nonlinear Constrained Prob.
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In a genetic algorithm, a population of which
encode candidate solutions to an optimization
problem, evolves toward better solutions.
Traditionally, design variables are represented in
binary as strings of 0s and 1s.
The evolution usually starts from a population of
randomly generated individuals
DOE and Optimization
Modern Methods – Genetic Algorithm (GA)
In each generation, the fitness of every individual in the population is evaluated, multiple
individuals are stochastically selected from the current population and modified (recombined
and possibly randomly mutated) to form a new population. The new population is then used in
the next iteration of the algorithm.
The algorithm terminates when either a maximum number of generations has been produced,
or a satisfactory fitness level has been reached for the population. If the algorithm has terminated
due to a maximum number of generations, a satisfactory solution may or may not have been
reached.
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The name and inspiration come from annealing in
metallurgy. The heat causes the atoms to become unstuck
from their initial positions (a local minimum of the
internal energy) and wander randomly through states of
higher energy; the slow cooling gives them more chances
of finding configurations with lower internal energy than
the initial one.
DOE and Optimization
Modern Methods – Simulated Annealing (SA)
By analogy with this physical process, each step of the SA algorithm attempts to replace the
current solution by a new random solution near the current solution. The new solution may then
be accepted with a probability of P = exp(-∆E/kT).
The choice between the previous and current solution is almost random when T is large, but
increasingly selects the better or "downhill" solution (for a minimization problem) as T goes to
zero. The allowance for "uphill" moves potentially saves the method from becoming stuck
at local optima.
The algorithm is terminated when new solutions are repeatedly rejected with the higher
temperature setting.
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Topology optimization example
A mathematical approach that optimizes material layout within a given
design space, for a given set of loads and boundary conditions such that the
resulting layout meets a prescribed set of performance targets.
http://www.youtube.com/user/OSDELab#p/u/5/bvf6Dm_HF4A
Autodesk Inventor Shape Generator
https://www.youtube.com/watch?v=GGOK-gqtCqM
https://www.youtube.com/watch?v=ZgtdIKo9gmw
Multi-material topology design and manufacturing
https://www.youtube.com/watch?v=EOxX892yg5g
https://www.youtube.com/watch?v=L3CkzQQFZXs&t=45s
DOE and Optimization
Example of Numerical Optimization
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Design is multi-objective by nature, so we will look at some multi-objective
formulations first.
“The typical role of a design engineer is to resolve conflicting objectives and arrive at a design that represents an acceptable or desired balance of all objectives.” (Mattson & Messac 2002)
Classical examples of conflicting objectives:
Truss Design: Weight versus Strength
Flywheel design: Kinetic Energy stored versus Weight
Finite Element Meshes: Aspect Ratio versus Distortion Parameter
Standard problem definition (Textbook’s notation):
Minimize f = [ f1(x), f2(x), … , fm(x) ],
where each fi is an objective function
Subject to x Ω (constraints on space of design variables)
DOE and Optimization
Multi-Objective Optimization -> Next week
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