Design Optimization School of Engineering University of Bradford 1 Numerical optimization techniques Unconstrained multi-parameter optimization techniques.

Slides:

Advertisements

Similar presentations

Curved Trajectories towards Local Minimum of a Function Al Jimenez Mathematics Department California Polytechnic State University San Luis Obispo, CA

Advertisements

ECE 471/571 - Lecture 13 Gradient Descent 10/13/14.

Optimization with Constraints

Engineering Optimization

Unconstrained optimization Gradient based algorithms –Steepest descent –Conjugate gradients –Newton and quasi-Newton Population based algorithms –Nelder.

Optimization of thermal processes

Optimization methods Review

Chapter 3 UNCONSTRAINED OPTIMIZATION

2.7.6 Conjugate Gradient Method for a Sparse System Shi & Bo.

Geometry Optimization Pertemuan VI. Geometry Optimization Backgrounds Real molecules vibrate thermally about their equilibrium structures. Finding minimum.

1 L-BFGS and Delayed Dynamical Systems Approach for Unconstrained Optimization Xiaohui XIE Supervisor: Dr. Hon Wah TAM.

Numerical Optimization

Unconstrained Optimization Rong Jin. Recap  Gradient ascent/descent Simple algorithm, only requires the first order derivative Problem: difficulty in.

1cs542g-term Notes  Extra class this Friday 1-2pm  If you want to receive s about the course (and are auditing) send me .

Methods For Nonlinear Least-Square Problems

Engineering Optimization

1 L-BFGS and Delayed Dynamical Systems Approach for Unconstrained Optimization Xiaohui XIE Supervisor: Dr. Hon Wah TAM.

Optimization Methods One-Dimensional Unconstrained Optimization

Optimization Mechanics of the Simplex Method

Optimization Methods One-Dimensional Unconstrained Optimization

Nonlinear programming

Engineering Optimization

Advanced Topics in Optimization

An Introduction to Optimization Theory. Outline Introduction Unconstrained optimization problem Constrained optimization problem.

Optimization Methods One-Dimensional Unconstrained Optimization

9 1 Performance Optimization. 9 2 Basic Optimization Algorithm p k - Search Direction  k - Learning Rate or.

UNCONSTRAINED MULTIVARIABLE

MATH 685/ CSI 700/ OR 682 Lecture Notes Lecture 9. Optimization problems.

1 Hybrid methods for solving large-scale parameter estimation problems Carlos A. Quintero 1 Miguel Argáez 1 Hector Klie 2 Leticia Velázquez 1 Mary Wheeler.

ENCI 303 Lecture PS-19 Optimization 2

84 b Unidimensional Search Methods Most algorithms for unconstrained and constrained optimisation use an efficient unidimensional optimisation technique.

Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 101 Quasi-Newton Methods.

Application of Differential Applied Optimization Problems.

Nonlinear programming Unconstrained optimization techniques.

1 Unconstrained Optimization Objective: Find minimum of F(X) where X is a vector of design variables We may know lower and upper bounds for optimum No.

1 Optimization Multi-Dimensional Unconstrained Optimization Part II: Gradient Methods.

Computer Animation Rick Parent Computer Animation Algorithms and Techniques Optimization & Constraints Add mention of global techiques Add mention of calculus.

Engineering Optimization Chapter 3 : Functions of Several Variables (Part 1) Presented by: Rajesh Roy Networks Research Lab, University of California,

559 Fish 559; Lecture 5 Non-linear Minimization. 559 Introduction Non-linear minimization (or optimization) is the numerical technique that is used by.

Institute of Biophysics and Biomedical Engineering - Bulgarian Academy of Sciences OLYMPIA ROEVA 105 Acad. George Bonchev Str Sofia, Bulgaria

Quasi-Newton Methods of Optimization Lecture 2. General Algorithm n A Baseline Scenario Algorithm U (Model algorithm for n- dimensional unconstrained.

Lecture 13. Geometry Optimization References Computational chemistry: Introduction to the theory and applications of molecular and quantum mechanics, E.

1 Chapter 6 General Strategy for Gradient methods (1) Calculate a search direction (2) Select a step length in that direction to reduce f(x) Steepest Descent.

L24 Numerical Methods part 4

Exam 1 Oct 3, closed book Place ITE 119, Time:12:30-1:45pm

Non-Linear Programming © 2011 Daniel Kirschen and University of Washington 1.

Gradient Methods In Optimization

Survey of unconstrained optimization gradient based algorithms

Chapter 2-OPTIMIZATION G.Anuradha. Contents Derivative-based Optimization –Descent Methods –The Method of Steepest Descent –Classical Newton’s Method.

Data assimilation for weather forecasting G.W. Inverarity 06/05/15.

INTRO TO OPTIMIZATION MATH-415 Numerical Analysis 1.

Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 117 Penalty and Barrier Methods General classical constrained.

Computational Biology BS123A/MB223 UC-Irvine Ray Luo, MBB, BS.

FINITE DIFFERENCE In numerical analysis, two different approaches are commonly used: The finite difference and the finite element methods. In heat transfer.

Optimal Control.

Non-linear Minimization

Classification Analytical methods classical methods

Computational Optimization

Iterative Non-Linear Optimization Methods

CS5321 Numerical Optimization

Optimization Part II G.Anuradha.

Introduction to Scientific Computing II

Introduction to Scientific Computing II

Numerical Linear Algebra

Performance Optimization

Section 3: Second Order Methods

Steepest Descent Optimization

First-Order Methods.

Presentation transcript:

Design Optimization School of Engineering University of Bradford 1 Numerical optimization techniques Unconstrained multi-parameter optimization techniques Direct search (no information on derivatives used): Hooke-Jeeves’ pattern search Nelder-Mead’s sequential simplex method Powell's conjugate directions method various evolutionary techniques OPTIMIZATION TECHNIQUES

Design Optimization School of Engineering University of Bradford 2 Numerical optimization techniques Unconstrained multi-parameter optimization techniques Gradient-based methods (information on derivatives is used): Steepest Descent Fletcher-Reeves' Conjugate Gradient method Second order methods (information on the second derivatives is used): Newton's Method Quasi-Newton Method (constructs an approximation of the matrix of second derivatives OPTIMIZATION TECHNIQUES

Design Optimization School of Engineering University of Bradford 3 Numerical optimization techniques. Example 1. OPTIMIZATION TECHNIQUES Unconstrained multi- parameter optimization techniques. Difficulties in typical problems. Rosenbrock’s “banana” function F=100(x 2 -x 1 2 ) 2 +(x 1 -1) 2

Design Optimization School of Engineering University of Bradford 4 Numerical optimization techniques. Example 1. Nelder-Mead’s sequential simplex method OPTIMIZATION TECHNIQUES

Design Optimization School of Engineering University of Bradford 5 Numerical optimization techniques. Example 1. Steepest Descent OPTIMIZATION TECHNIQUES

Design Optimization School of Engineering University of Bradford 6 OPTIMIZATION TECHNIQUES Effect of scaling

Design Optimization School of Engineering University of Bradford 7 Numerical optimization techniques. Example 1. Powell's conjugate directions method OPTIMIZATION TECHNIQUES

Design Optimization School of Engineering University of Bradford 8 Numerical optimization techniques. Example 1. Fletcher-Reeves' Conjugate Gradient method OPTIMIZATION TECHNIQUES

Design Optimization School of Engineering University of Bradford 9 Numerical optimization techniques. Example 1. Newton's Method OPTIMIZATION TECHNIQUES

Design Optimization School of Engineering University of Bradford 10 Numerical optimization techniques. Example 1. Quasi-Newton Method OPTIMIZATION TECHNIQUES

Design Optimization School of Engineering University of Bradford 11 Numerical optimization techniques OPTIMIZATION TECHNIQUES Unconstrained multi- parameter optimization techniques. Example 2.

Design Optimization School of Engineering University of Bradford 12 Numerical optimization techniques OPTIMIZATION TECHNIQUES Example 2. Newton, trajectories of search from three starting points

Design Optimization School of Engineering University of Bradford 13 Numerical optimization techniques OPTIMIZATION TECHNIQUES Example 2. Newton, trajectory of search from the first starting point. First iteration.

Design Optimization School of Engineering University of Bradford 14 Numerical optimization techniques OPTIMIZATION TECHNIQUES Example 2. Newton, trajectory of search from the first starting point. Second iteration, optimization stops at the saddle point

Design Optimization School of Engineering University of Bradford 15 Numerical optimization techniques OPTIMIZATION TECHNIQUES Example 2. Newton, trajectory of search from the third starting point. First iteration.

Design Optimization School of Engineering University of Bradford 16 Numerical optimization techniques OPTIMIZATION TECHNIQUES Example 2. Newton, trajectory of search from the third starting point. Second iteration.

Design Optimization School of Engineering University of Bradford 17 Numerical optimization techniques Penalty function approaches Exterior penalty function OPTIMIZATION TECHNIQUES

Design Optimization School of Engineering University of Bradford 18 Numerical optimization techniques OPTIMIZATION TECHNIQUES Exterior penalty function approach. First iteration

Design Optimization School of Engineering University of Bradford 19 Numerical optimization techniques OPTIMIZATION TECHNIQUES Exterior penalty function approach. Second iteration

Design Optimization School of Engineering University of Bradford 20 Numerical optimization techniques OPTIMIZATION TECHNIQUES Exterior penalty function approach. Third iteration

Design Optimization School of Engineering University of Bradford 21 Numerical optimization techniques Penalty function approaches Interior penalty function OPTIMIZATION TECHNIQUES

Design Optimization School of Engineering University of Bradford 22 Numerical optimization techniques OPTIMIZATION TECHNIQUES Interior penalty function approach. First iteration

Design Optimization School of Engineering University of Bradford 23 Numerical optimization techniques OPTIMIZATION TECHNIQUES Interior penalty function approach. Second iteration

Design Optimization School of Engineering University of Bradford 24 Numerical optimization techniques OPTIMIZATION TECHNIQUES Interior penalty function approach. Third iteration